Daily Papers

In 2011, einsum was introduced to NumPy as a practical and convenient notation for tensor expressions in machine learning, quantum circuit simulation, and other fields. It has since been implemented in additional Python frameworks such as PyTorch and TensorFlow, as well as in other programming languages such as Julia. Despite its practical success, the einsum notation still lacks a solid theoretical basis, and is not unified across the different frameworks, limiting opportunities for formal reasoning and systematic optimization. In this work, we discuss the terminology of tensor expressions and provide a formal definition of the einsum language. Based on this definition, we formalize and prove important equivalence rules for tensor expressions and highlight their relevance in practical applications.

Boosting the runtime performance of deep neural networks (DNNs) is critical due to their wide adoption in real-world tasks. Existing approaches to optimizing the tensor algebra expression of a DNN only consider expressions representable by a fixed set of predefined operators, missing possible optimization opportunities between general expressions. We propose OLLIE, the first derivation-based tensor program optimizer. OLLIE optimizes tensor programs by leveraging transformations between general tensor algebra expressions, enabling a significantly larger expression search space that includes those supported by prior work as special cases. OLLIE uses a hybrid derivation-based optimizer that effectively combines explorative and guided derivations to quickly discover highly optimized expressions. Evaluation on seven DNNs shows that OLLIE can outperform existing optimizers by up to 2.73times (1.46times on average) on an A100 GPU and up to 2.68times (1.51times) on a V100 GPU, respectively.

Going beyond stochastic gradient descent (SGD), what new phenomena emerge in wide neural networks trained by adaptive optimizers like Adam? Here we show: The same dichotomy between feature learning and kernel behaviors (as in SGD) holds for general optimizers as well, including Adam -- albeit with a nonlinear notion of "kernel." We derive the corresponding "neural tangent" and "maximal update" limits for any architecture. Two foundational advances underlie the above results: 1) A new Tensor Program language, NEXORT, that can express how adaptive optimizers process gradients into updates. 2) The introduction of bra-ket notation to drastically simplify expressions and calculations in Tensor Programs. This work summarizes and generalizes all previous results in the Tensor Programs series of papers.

Tensor networks are efficient representations of high-dimensional tensors which have been very successful for physics and mathematics applications. We demonstrate how algorithms for optimizing such networks can be adapted to supervised learning tasks by using matrix product states (tensor trains) to parameterize models for classifying images. For the MNIST data set we obtain less than 1% test set classification error. We discuss how the tensor network form imparts additional structure to the learned model and suggest a possible generative interpretation.

Progress in AI is hindered by the lack of a programming language with all the requisite features. Libraries like PyTorch and TensorFlow provide automatic differentiation and efficient GPU implementation, but are additions to Python, which was never intended for AI. Their lack of support for automated reasoning and knowledge acquisition has led to a long and costly series of hacky attempts to tack them on. On the other hand, AI languages like LISP an Prolog lack scalability and support for learning. This paper proposes tensor logic, a language that solves these problems by unifying neural and symbolic AI at a fundamental level. The sole construct in tensor logic is the tensor equation, based on the observation that logical rules and Einstein summation are essentially the same operation, and all else can be reduced to them. I show how to elegantly implement key forms of neural, symbolic and statistical AI in tensor logic, including transformers, formal reasoning, kernel machines and graphical models. Most importantly, tensor logic makes new directions possible, such as sound reasoning in embedding space. This combines the scalability and learnability of neural networks with the reliability and transparency of symbolic reasoning, and is potentially a basis for the wider adoption of AI.

In this paper, we delve into the foundational principles of tensor categories, harnessing the universal property of the tensor product to pioneer novel methodologies in deep network architectures. Our primary contribution is the introduction of the Tensor Attention and Tensor Interaction Mechanism, a groundbreaking approach that leverages the tensor category to enhance the computational efficiency and the expressiveness of deep networks, and can even be generalized into the quantum realm.

Multi-channel imaging data is a prevalent data format in scientific fields such as astronomy and biology. The structured information and the high dimensionality of these 3-D tensor data makes the analysis an intriguing but challenging topic for statisticians and practitioners. The low-rank scalar-on-tensor regression model, in particular, has received widespread attention and has been re-formulated as a tensor Gaussian Process (Tensor-GP) model with multi-linear kernel in Yu et al. (2018). In this paper, we extend the Tensor-GP model by integrating a dimensionality reduction technique, called tensor contraction, with a Tensor-GP for a scalar-on-tensor regression task with multi-channel imaging data. This is motivated by the solar flare forecasting problem with high dimensional multi-channel imaging data. We first estimate a latent, reduced-size tensor for each data tensor and then apply a multi-linear Tensor-GP on the latent tensor data for prediction. We introduce an anisotropic total-variation regularization when conducting the tensor contraction to obtain a sparse and smooth latent tensor. We then propose an alternating proximal gradient descent algorithm for estimation. We validate our approach via extensive simulation studies and applying it to the solar flare forecasting problem.

The development of efficient machine learning models for molecular systems representation is becoming crucial in scientific research. We introduce TensorNet, an innovative O(3)-equivariant message-passing neural network architecture that leverages Cartesian tensor representations. By using Cartesian tensor atomic embeddings, feature mixing is simplified through matrix product operations. Furthermore, the cost-effective decomposition of these tensors into rotation group irreducible representations allows for the separate processing of scalars, vectors, and tensors when necessary. Compared to higher-rank spherical tensor models, TensorNet demonstrates state-of-the-art performance with significantly fewer parameters. For small molecule potential energies, this can be achieved even with a single interaction layer. As a result of all these properties, the model's computational cost is substantially decreased. Moreover, the accurate prediction of vector and tensor molecular quantities on top of potential energies and forces is possible. In summary, TensorNet's framework opens up a new space for the design of state-of-the-art equivariant models.

In the classical transformer attention scheme, we are given three n times d size matrices Q, K, V (the query, key, and value tokens), and the goal is to compute a new n times d size matrix D^{-1} exp(QK^top) V where D = diag( exp(QK^top) {bf 1}_n ). In this work, we study a generalization of attention which captures triple-wise correlations. This generalization is able to solve problems about detecting triple-wise connections that were shown to be impossible for transformers. The potential downside of this generalization is that it appears as though computations are even more difficult, since the straightforward algorithm requires cubic time in n. However, we show that in the bounded-entry setting (which arises in practice, and which is well-studied in both theory and practice), there is actually a near-linear time algorithm. More precisely, we show that bounded entries are both necessary and sufficient for quickly performing generalized computations: bullet On the positive side, if all entries of the input matrices are bounded above by o(sqrt[3]{log n}) then we show how to approximate the ``tensor-type'' attention matrix in n^{1+o(1)} time. bullet On the negative side, we show that if the entries of the input matrices may be as large as Omega(sqrt[3]{log n}), then there is no algorithm that runs faster than n^{3-o(1)} (assuming the Strong Exponential Time Hypothesis from fine-grained complexity theory). We also show that our construction, algorithms, and lower bounds naturally generalize to higher-order tensors and correlations. Interestingly, the higher the order of the tensors, the lower the bound on the entries needs to be for an efficient algorithm. Our results thus yield a natural tradeoff between the boundedness of the entries, and order of the tensor one may use for more expressive, efficient attention computation.

Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring the formation of very high-dimensional matrices and tensors, and therefore being computationally infeasible. In this paper I show how a linear simplification of recursive neural tensor network models can be mapped directly onto the categorical approach, giving a way of computing the required matrices and tensors. This mapping suggests a number of lines of research for both categorical compositional vector space models of meaning and for recursive neural network models of compositionality.

Multimodal research is an emerging field of artificial intelligence, and one of the main research problems in this field is multimodal fusion. The fusion of multimodal data is the process of integrating multiple unimodal representations into one compact multimodal representation. Previous research in this field has exploited the expressiveness of tensors for multimodal representation. However, these methods often suffer from exponential increase in dimensions and in computational complexity introduced by transformation of input into tensor. In this paper, we propose the Low-rank Multimodal Fusion method, which performs multimodal fusion using low-rank tensors to improve efficiency. We evaluate our model on three different tasks: multimodal sentiment analysis, speaker trait analysis, and emotion recognition. Our model achieves competitive results on all these tasks while drastically reducing computational complexity. Additional experiments also show that our model can perform robustly for a wide range of low-rank settings, and is indeed much more efficient in both training and inference compared to other methods that utilize tensor representations.

Tensorial neural networks (TNNs) combine the successes of multilinear algebra with those of deep learning to enable extremely efficient reduced-order models of high-dimensional problems. Here, I describe a deep neural network architecture that fuses multiple TNNs into a larger network, intended to solve a broader class of problems than a single TNN. I evaluate this architecture, referred to as a "stacked tensorial neural network" (STNN), on a parametric PDE with three independent variables and three parameters. The three parameters correspond to one PDE coefficient and two quantities describing the domain geometry. The STNN provides an accurate reduced-order description of the solution manifold over a wide range of parameters. There is also evidence of meaningful generalization to parameter values outside its training data. Finally, while the STNN architecture is relatively simple and problem agnostic, it can be regularized to incorporate problem-specific features like symmetries and physical modeling assumptions.

As its width tends to infinity, a deep neural network's behavior under gradient descent can become simplified and predictable (e.g. given by the Neural Tangent Kernel (NTK)), if it is parametrized appropriately (e.g. the NTK parametrization). However, we show that the standard and NTK parametrizations of a neural network do not admit infinite-width limits that can learn features, which is crucial for pretraining and transfer learning such as with BERT. We propose simple modifications to the standard parametrization to allow for feature learning in the limit. Using the *Tensor Programs* technique, we derive explicit formulas for such limits. On Word2Vec and few-shot learning on Omniglot via MAML, two canonical tasks that rely crucially on feature learning, we compute these limits exactly. We find that they outperform both NTK baselines and finite-width networks, with the latter approaching the infinite-width feature learning performance as width increases. More generally, we classify a natural space of neural network parametrizations that generalizes standard, NTK, and Mean Field parametrizations. We show 1) any parametrization in this space either admits feature learning or has an infinite-width training dynamics given by kernel gradient descent, but not both; 2) any such infinite-width limit can be computed using the Tensor Programs technique. Code for our experiments can be found at github.com/edwardjhu/TP4.

Coecke, Sadrzadeh, and Clark (arXiv:1003.4394v1 [cs.CL]) developed a compositional model of meaning for distributional semantics, in which each word in a sentence has a meaning vector and the distributional meaning of the sentence is a function of the tensor products of the word vectors. Abstractly speaking, this function is the morphism corresponding to the grammatical structure of the sentence in the category of finite dimensional vector spaces. In this paper, we provide a concrete method for implementing this linear meaning map, by constructing a corpus-based vector space for the type of sentence. Our construction method is based on structured vector spaces whereby meaning vectors of all sentences, regardless of their grammatical structure, live in the same vector space. Our proposed sentence space is the tensor product of two noun spaces, in which the basis vectors are pairs of words each augmented with a grammatical role. This enables us to compare meanings of sentences by simply taking the inner product of their vectors.

Tensor Networks (TN) offer a powerful framework to efficiently represent very high-dimensional objects. TN have recently shown their potential for machine learning applications and offer a unifying view of common tensor decomposition models such as Tucker, tensor train (TT) and tensor ring (TR). However, identifying the best tensor network structure from data for a given task is challenging. In this work, we leverage the TN formalism to develop a generic and efficient adaptive algorithm to jointly learn the structure and the parameters of a TN from data. Our method is based on a simple greedy approach starting from a rank one tensor and successively identifying the most promising tensor network edges for small rank increments. Our algorithm can adaptively identify TN structures with small number of parameters that effectively optimize any differentiable objective function. Experiments on tensor decomposition, tensor completion and model compression tasks demonstrate the effectiveness of the proposed algorithm. In particular, our method outperforms the state-of-the-art evolutionary topology search [Li and Sun, 2020] for tensor decomposition of images (while being orders of magnitude faster) and finds efficient tensor network structures to compress neural networks outperforming popular TT based approaches [Novikov et al., 2015].

In this work, we demonstrate that a simple two-layer neural network with standard activation functions can learn an arbitrary word operation in any finite group, provided sufficient width is available and exhibits grokking while doing so. To explain the mechanism by which this is achieved, we reframe the problem as that of learning a particular 3-tensor, which we show is typically of low rank. A key insight is that low-rank implementations of this tensor can be obtained by decomposing it along triplets of basic self-conjugate representations of the group and leveraging the fusion structure to rule out many components. Focusing on a phenomenologically similar but more tractable surrogate model, we show that the network is able to find such low-rank implementations (or approximations thereof), thereby using limited width to approximate the word-tensor in a generalizable way. In the case of the simple multiplication word, we further elucidate the form of these low-rank implementations, showing that the network effectively implements efficient matrix multiplication in the sense of Strassen. Our work also sheds light on the mechanism by which a network reaches such a solution under gradient descent.

E(3)-equivariant neural networks have demonstrated success across a wide range of 3D modelling tasks. A fundamental operation in these networks is the tensor product, which interacts two geometric features in an equivariant manner to create new features. Due to the high computational complexity of the tensor product, significant effort has been invested to optimize the runtime of this operation. For example, Luo et al. (2024) recently proposed the Gaunt tensor product (GTP) which promises a significant speedup. In this work, we provide a careful, systematic analysis of a number of tensor product operations. In particular, we emphasize that different tensor products are not performing the same operation. The reported speedups typically come at the cost of expressivity. We introduce measures of expressivity and interactability to characterize these differences. In addition, we realized the original implementation of GTP can be greatly simplified by directly using a spherical grid at no cost in asymptotic runtime. This spherical grid approach is faster on our benchmarks and in actual training of the MACE interatomic potential by 30%. Finally, we provide the first systematic microbenchmarks of the various tensor product operations. We find that the theoretical runtime guarantees can differ wildly from empirical performance, demonstrating the need for careful application-specific benchmarking. Code is available at https://github.com/atomicarchitects/PriceofFreedom.

We provide a rigorous analysis of implicit regularization in an overparametrized tensor factorization problem beyond the lazy training regime. For matrix factorization problems, this phenomenon has been studied in a number of works. A particular challenge has been to design universal initialization strategies which provably lead to implicit regularization in gradient-descent methods. At the same time, it has been argued by Cohen et. al. 2016 that more general classes of neural networks can be captured by considering tensor factorizations. However, in the tensor case, implicit regularization has only been rigorously established for gradient flow or in the lazy training regime. In this paper, we prove the first tensor result of its kind for gradient descent rather than gradient flow. We focus on the tubal tensor product and the associated notion of low tubal rank, encouraged by the relevance of this model for image data. We establish that gradient descent in an overparametrized tensor factorization model with a small random initialization exhibits an implicit bias towards solutions of low tubal rank. Our theoretical findings are illustrated in an extensive set of numerical simulations show-casing the dynamics predicted by our theory as well as the crucial role of using a small random initialization.

We study the estimation of a planted signal hidden in a recently introduced nested matrix-tensor model, which is an extension of the classical spiked rank-one tensor model, motivated by multi-view clustering. Prior work has theoretically examined the performance of a tensor-based approach, which relies on finding a best rank-one approximation, a problem known to be computationally hard. A tractable alternative approach consists in computing instead the best rank-one (matrix) approximation of an unfolding of the observed tensor data, but its performance was hitherto unknown. We quantify here the performance gap between these two approaches, in particular by deriving the precise algorithmic threshold of the unfolding approach and demonstrating that it exhibits a BBP-type transition behavior. This work is therefore in line with recent contributions which deepen our understanding of why tensor-based methods surpass matrix-based methods in handling structured tensor data.

We introduce Einstein Fields, a neural representation that is designed to compress computationally intensive four-dimensional numerical relativity simulations into compact implicit neural network weights. By modeling the metric, which is the core tensor field of general relativity, Einstein Fields enable the derivation of physical quantities via automatic differentiation. However, unlike conventional neural fields (e.g., signed distance, occupancy, or radiance fields), Einstein Fields are Neural Tensor Fields with the key difference that when encoding the spacetime geometry of general relativity into neural field representations, dynamics emerge naturally as a byproduct. Einstein Fields show remarkable potential, including continuum modeling of 4D spacetime, mesh-agnosticity, storage efficiency, derivative accuracy, and ease of use. We address these challenges across several canonical test beds of general relativity and release an open source JAX-based library, paving the way for more scalable and expressive approaches to numerical relativity. Code is made available at https://github.com/AndreiB137/EinFields

Tucker decomposition is a powerful tensor model to handle multi-aspect data. It demonstrates the low-rank property by decomposing the grid-structured data as interactions between a core tensor and a set of object representations (factors). A fundamental assumption of such decomposition is that there are finite objects in each aspect or mode, corresponding to discrete indexes of data entries. However, real-world data is often not naturally posed in this setting. For example, geographic data is represented as continuous indexes of latitude and longitude coordinates, and cannot fit tensor models directly. To generalize Tucker decomposition to such scenarios, we propose Functional Bayesian Tucker Decomposition (FunBaT). We treat the continuous-indexed data as the interaction between the Tucker core and a group of latent functions. We use Gaussian processes (GP) as functional priors to model the latent functions. Then, we convert each GP into a state-space prior by constructing an equivalent stochastic differential equation (SDE) to reduce computational cost. An efficient inference algorithm is developed for scalable posterior approximation based on advanced message-passing techniques. The advantage of our method is shown in both synthetic data and several real-world applications. We release the code of FunBaT at https://github.com/xuangu-fang/Functional-Bayesian-Tucker-Decomposition.

Tuning tensor program generation involves searching for various possible program transformation combinations for a given program on target hardware to optimize the tensor program execution. It is already a complex process because of the massive search space and exponential combinations of transformations make auto-tuning tensor program generation more challenging, especially when we have a heterogeneous target. In this research, we attempt to address these problems by learning the joint neural network and hardware features and transferring them to the new target hardware. We extensively study the existing state-of-the-art dataset, TenSet, perform comparative analysis on the test split strategies and propose methodologies to prune the dataset. We adopt an attention-inspired approach for tuning the tensor programs enabling them to embed neural network and hardware-specific features. Our approach could prune the dataset up to 45\% of the baseline without compromising the Pairwise Comparison Accuracy (PCA). Further, the proposed methodology can achieve on-par or improved mean inference time with 25%-40% of the baseline tuning time across different networks and target hardware.

Graph Neural Networks (GNN) are inherently limited in their expressive power. Recent seminal works (Xu et al., 2019; Morris et al., 2019b) introduced the Weisfeiler-Lehman (WL) hierarchy as a measure of expressive power. Although this hierarchy has propelled significant advances in GNN analysis and architecture developments, it suffers from several significant limitations. These include a complex definition that lacks direct guidance for model improvement and a WL hierarchy that is too coarse to study current GNNs. This paper introduces an alternative expressive power hierarchy based on the ability of GNNs to calculate equivariant polynomials of a certain degree. As a first step, we provide a full characterization of all equivariant graph polynomials by introducing a concrete basis, significantly generalizing previous results. Each basis element corresponds to a specific multi-graph, and its computation over some graph data input corresponds to a tensor contraction problem. Second, we propose algorithmic tools for evaluating the expressiveness of GNNs using tensor contraction sequences, and calculate the expressive power of popular GNNs. Finally, we enhance the expressivity of common GNN architectures by adding polynomial features or additional operations / aggregations inspired by our theory. These enhanced GNNs demonstrate state-of-the-art results in experiments across multiple graph learning benchmarks.

Recently, triple decomposition has attracted increasing attention for decomposing third-order tensors into three factor tensors. However, this approach is limited to third-order tensors and enforces uniformity in the lower dimensions across all factor tensors, which restricts its flexibility and applicability. To address these issues, we propose the Multiple decomposition, a novel framework that generalizes triple decomposition to arbitrary order tensors and allows the short dimensions of the factor tensors to differ. We establish its connections with other classical tensor decompositions. Furthermore, implicit neural representation (INR) is employed to continuously represent the factor tensors in Multiple decomposition, enabling the method to generalize to non-grid data. We refer to this INR-based Multiple decomposition as Implicit Multiple Tensor Decomposition (IMTD). Then, the Proximal Alternating Least Squares (PALS) algorithm is utilized to solve the IMTD-based tensor reconstruction models. Since the objective function in IMTD-based models often lacks the Kurdyka-Lojasiewicz (KL) property, we establish a KL-free convergence analysis for the algorithm. Finally, extensive numerical experiments further validate the effectiveness of the proposed method.

We review some recent applications of machine learning to algebraic geometry and physics. Since problems in algebraic geometry can typically be reformulated as mappings between tensors, this makes them particularly amenable to supervised learning. Additionally, unsupervised methods can provide insight into the structure of such geometrical data. At the heart of this programme is the question of how geometry can be machine learned, and indeed how AI helps one to do mathematics. This is a chapter contribution to the book Machine learning and Algebraic Geometry, edited by A. Kasprzyk et al.

Design of neural networks that incorporate symmetry is crucial for geometric deep learning. Central to this effort is the development of invariant and equivariant operations. This works presents a systematic method for constructing valid invariant and equivariant operations. It can handle inputs and outputs in the form of Cartesian tensors with different rank, as well as spherical tensors with different types. In addition, our method features a graphical representation utilizing the symmetric tensor network, which simplifies both the proofs and constructions related to invariant and equivariant functions. We also apply this approach to design the equivariant interaction message for the geometry graph neural network, and equivariant machine learning model to learn the constitutive law of materials.

Understanding sentences that contain mathematical expressions in text form poses significant challenges. To address this, the importance of converting these expressions into formula images has been highlighted. For instance, the expression ``x equals minus b plus or minus the square root of b squared minus four a c, all over two a'' is more readily comprehensible when displayed as an image x = -b pm sqrt{b^2 - 4ac}{2a}. To develop a text-to-image conversion system, we can break down the process into text-to-LaTeX and LaTeX-to-image conversions, with the latter being managed with by existing various LaTeX engines. However, the former approach has been notably hindered by the severe scarcity of text-to-LaTeX paired data, presenting a significant challenge in the field.In this context, we introduce MathBridge, the first extensive dataset for translating mathematical spoken English into LaTeX, which aims to establish a robust baseline for future research in text-to-LaTeX translation. MathBridge comprises approximately 23 million LaTeX formulas paired with corresponding spoken English expressions. Through comprehensive evaluations, including fine-tuning and testing with data, we discovered that MathBridge significantly enhances pre-trained language models' capabilities for text-to-LaTeX translation. Specifically, for the T5-large model, the sacreBLEU score increased from 4.77 to 46.8, demonstrating substantial enhancement. Our findings indicate the necessity for a new metric specifically for text-to-LaTeX conversion evaluation.

Attention operators have been applied on both 1-D data like texts and higher-order data such as images and videos. Use of attention operators on high-order data requires flattening of the spatial or spatial-temporal dimensions into a vector, which is assumed to follow a multivariate normal distribution. This not only incurs excessive requirements on computational resources, but also fails to preserve structures in data. In this work, we propose to avoid flattening by assuming the data follow matrix-variate normal distributions. Based on this new view, we develop Kronecker attention operators (KAOs) that operate on high-order tensor data directly. More importantly, the proposed KAOs lead to dramatic reductions in computational resources. Experimental results show that our methods reduce the amount of required computational resources by a factor of hundreds, with larger factors for higher-dimensional and higher-order data. Results also show that networks with KAOs outperform models without attention, while achieving competitive performance as those with original attention operators.

In multi-task learning, the conventional approach involves training a model on multiple tasks simultaneously. However, the training signals from different tasks can interfere with one another, potentially leading to negative transfer. To mitigate this, we investigate if modular language models can facilitate positive transfer and systematic generalization. Specifically, we propose a novel modular language model (TensorPoly), that balances parameter efficiency with nuanced routing methods. For modules, we reparameterize Low-Rank Adaptation (LoRA) by employing an entangled tensor through the use of tensor product operations and name the resulting approach TLoRA. For routing function, we tailor two innovative routing functions according to the granularity: TensorPoly-I which directs to each rank within the entangled tensor while TensorPoly-II offers a finer-grained routing approach targeting each order of the entangled tensor. The experimental results from the multi-task T0-benchmark demonstrate that: 1) all modular LMs surpass the corresponding dense approaches, highlighting the potential of modular language models to mitigate negative inference in multi-task learning and deliver superior outcomes. 2) TensorPoly-I achieves higher parameter efficiency in adaptation and outperforms other modular LMs, which shows the potential of our approach in multi-task transfer learning.

Tensor computations, with matrix multiplication being the primary operation, serve as the fundamental basis for data analysis, physics, machine learning, and deep learning. As the scale and complexity of data continue to grow rapidly, the demand for tensor computations has also increased significantly. To meet this demand, several research institutions have started developing dedicated hardware for tensor computations. To further improve the computational performance of tensor process units, we have reexamined the issue of computation reuse that was previously overlooked in existing architectures. As a result, we propose a novel EN-T architecture that can reduce chip area and power consumption. Furthermore, our method is compatible with existing tensor processing units. We evaluated our method on prevalent microarchitectures, the results demonstrate an average improvement in area efficiency of 8.7\%, 12.2\%, and 11.0\% for tensor computing units at computational scales of 256 GOPS, 1 TOPS, and 4 TOPS, respectively. Similarly, there were energy efficiency enhancements of 13.0\%, 17.5\%, and 15.5\%.

Tensor Cores have been an important unit to accelerate Fused Matrix Multiplication Accumulation (MMA) in all NVIDIA GPUs since Volta Architecture. To program Tensor Cores, users have to use either legacy wmma APIs or current mma APIs. Legacy wmma APIs are more easy-to-use but can only exploit limited features and power of Tensor Cores. Specifically, wmma APIs support fewer operand shapes and can not leverage the new sparse matrix multiplication feature of the newest Ampere Tensor Cores. However, the performance of current programming interface has not been well explored. Furthermore, the computation numeric behaviors of low-precision floating points (TF32, BF16, and FP16) supported by the newest Ampere Tensor Cores are also mysterious. In this paper, we explore the throughput and latency of current programming APIs. We also intuitively study the numeric behaviors of Tensor Cores MMA and profile the intermediate operations including multiplication, addition of inner product, and accumulation. All codes used in this work can be found in https://github.com/sunlex0717/DissectingTensorCores.

TensorFlow is an interface for expressing machine learning algorithms, and an implementation for executing such algorithms. A computation expressed using TensorFlow can be executed with little or no change on a wide variety of heterogeneous systems, ranging from mobile devices such as phones and tablets up to large-scale distributed systems of hundreds of machines and thousands of computational devices such as GPU cards. The system is flexible and can be used to express a wide variety of algorithms, including training and inference algorithms for deep neural network models, and it has been used for conducting research and for deploying machine learning systems into production across more than a dozen areas of computer science and other fields, including speech recognition, computer vision, robotics, information retrieval, natural language processing, geographic information extraction, and computational drug discovery. This paper describes the TensorFlow interface and an implementation of that interface that we have built at Google. The TensorFlow API and a reference implementation were released as an open-source package under the Apache 2.0 license in November, 2015 and are available at www.tensorflow.org.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

We show how the Schur-Weyl duality that exists between the partition algebra and the symmetric group results in a stronger theoretical foundation for characterising all of the possible permutation equivariant neural networks whose layers are some tensor power of the permutation representation M_n of the symmetric group S_n. In doing so, we unify two separate bodies of literature, and we correct some of the major results that are now widely quoted by the machine learning community. In particular, we find a basis of matrices for the learnable, linear, permutation equivariant layer functions between such tensor power spaces in the standard basis of M_n by using an elegant graphical representation of a basis of set partitions for the partition algebra and its related vector spaces. Also, we show how we can calculate the number of weights that must appear in these layer functions by looking at certain paths through the McKay quiver for M_n. Finally, we describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

Tensor network (TN) is a powerful framework in machine learning, but selecting a good TN model, known as TN structure search (TN-SS), is a challenging and computationally intensive task. The recent approach TNLS~li2022permutation showed promising results for this task, however, its computational efficiency is still unaffordable, requiring too many evaluations of the objective function. We propose TnALE, a new algorithm that updates each structure-related variable alternately by local enumeration, greatly reducing the number of evaluations compared to TNLS. We theoretically investigate the descent steps for TNLS and TnALE, proving that both algorithms can achieve linear convergence up to a constant if a sufficient reduction of the objective is reached in each neighborhood. We also compare the evaluation efficiency of TNLS and TnALE, revealing that Omega(2^N) evaluations are typically required in TNLS for reaching the objective reduction in the neighborhood, while ideally O(N^2R) evaluations are sufficient in TnALE, where N denotes the tensor order and R reflects the ``low-rankness'' of the neighborhood. Experimental results verify that TnALE can find practically good TN-ranks and permutations with vastly fewer evaluations than the state-of-the-art algorithms.

Multiword expressions (MWEs) refer to idiomatic sequences of multiple words. MWE identification, i.e., detecting MWEs in text, can play a key role in downstream tasks such as machine translation. Existing datasets for MWE identification are inconsistently annotated, limited to a single type of MWE, or limited in size. To enable reliable and comprehensive evaluation, we created CoAM: Corpus of All-Type Multiword Expressions, a dataset of 1.3K sentences constructed through a multi-step process to enhance data quality consisting of human annotation, human review, and automated consistency checking. MWEs in CoAM are tagged with MWE types, such as Noun and Verb, to enable fine-grained error analysis. Annotations for CoAM were collected using a new interface created with our interface generator, which allows easy and flexible annotation of MWEs in any form, including discontinuous ones. Through experiments using CoAM, we find that a fine-tuned large language model outperforms the current state-of-the-art approach for MWE identification. Furthermore, analysis using our MWE type tagged data reveals that Verb MWEs are easier than Noun MWEs to identify across approaches.

Temporal and numerical expression understanding is of great importance in many downstream Natural Language Processing (NLP) and Information Retrieval (IR) tasks. However, much previous work covers only a few sub-types and focuses only on entity extraction, which severely limits the usability of identified mentions. In order for such entities to be useful in downstream scenarios, coverage and granularity of sub-types are important; and, even more so, providing resolution into concrete values that can be manipulated. Furthermore, most previous work addresses only a handful of languages. Here we describe a multi-lingual evaluation dataset - NTX - covering diverse temporal and numerical expressions across 14 languages and covering extraction, normalization, and resolution. Along with the dataset we provide a robust rule-based system as a strong baseline for comparisons against other models to be evaluated in this dataset. Data and code are available at https://aka.ms/NTX.

We completely classify the atomic summands in a graph product (M,varphi) = *_{v in G} (M_v,varphi_v) of von Neumann algebras with faithful normal states. Each type I factor summand (N,psi) is a tensor product of type I factor summands (N_v,psi_v) in the individual algebras. The existence of such a summand and its weight in the direct sum can be determined from the (N_v,psi_v)'s using explicit polynomials associated to the graph.

We propose Strivec, a novel neural representation that models a 3D scene as a radiance field with sparsely distributed and compactly factorized local tensor feature grids. Our approach leverages tensor decomposition, following the recent work TensoRF, to model the tensor grids. In contrast to TensoRF which uses a global tensor and focuses on their vector-matrix decomposition, we propose to utilize a cloud of local tensors and apply the classic CANDECOMP/PARAFAC (CP) decomposition to factorize each tensor into triple vectors that express local feature distributions along spatial axes and compactly encode a local neural field. We also apply multi-scale tensor grids to discover the geometry and appearance commonalities and exploit spatial coherence with the tri-vector factorization at multiple local scales. The final radiance field properties are regressed by aggregating neural features from multiple local tensors across all scales. Our tri-vector tensors are sparsely distributed around the actual scene surface, discovered by a fast coarse reconstruction, leveraging the sparsity of a 3D scene. We demonstrate that our model can achieve better rendering quality while using significantly fewer parameters than previous methods, including TensoRF and Instant-NGP.

This paper presents the UniMER dataset to provide the first study on Mathematical Expression Recognition (MER) towards complex real-world scenarios. The UniMER dataset consists of a large-scale training set UniMER-1M offering an unprecedented scale and diversity with one million training instances and a meticulously designed test set UniMER-Test that reflects a diverse range of formula distributions prevalent in real-world scenarios. Therefore, the UniMER dataset enables the training of a robust and high-accuracy MER model and comprehensive evaluation of model performance. Moreover, we introduce the Universal Mathematical Expression Recognition Network (UniMERNet), an innovative framework designed to enhance MER in practical scenarios. UniMERNet incorporates a Length-Aware Module to process formulas of varied lengths efficiently, thereby enabling the model to handle complex mathematical expressions with greater accuracy. In addition, UniMERNet employs our UniMER-1M data and image augmentation techniques to improve the model's robustness under different noise conditions. Our extensive experiments demonstrate that UniMERNet outperforms existing MER models, setting a new benchmark in various scenarios and ensuring superior recognition quality in real-world applications. The dataset and model are available at https://github.com/opendatalab/UniMERNet.

We analyse perception and memory, using mathematical models for knowledge graphs and tensors, to gain insights into the corresponding functionalities of the human mind. Our discussion is based on the concept of propositional sentences consisting of subject-predicate-object (SPO) triples for expressing elementary facts. SPO sentences are the basis for most natural languages but might also be important for explicit perception and declarative memories, as well as intra-brain communication and the ability to argue and reason. A set of SPO sentences can be described as a knowledge graph, which can be transformed into an adjacency tensor. We introduce tensor models, where concepts have dual representations as indices and associated embeddings, two constructs we believe are essential for the understanding of implicit and explicit perception and memory in the brain. We argue that a biological realization of perception and memory imposes constraints on information processing. In particular, we propose that explicit perception and declarative memories require a semantic decoder, which, in a simple realization, is based on four layers: First, a sensory memory layer, as a buffer for sensory input, second, an index layer representing concepts, third, a memoryless representation layer for the broadcasting of information ---the "blackboard", or the "canvas" of the brain--- and fourth, a working memory layer as a processing center and data buffer. We discuss the operations of the four layers and relate them to the global workspace theory. In a Bayesian brain interpretation, semantic memory defines the prior for observable triple statements. We propose that ---in evolution and during development--- semantic memory, episodic memory, and natural language evolved as emergent properties in agents' process to gain a deeper understanding of sensory information.

Transformers are widely used to extract semantic meanings from input tokens, yet they usually operate as black-box models. In this paper, we present a simple yet informative decomposition of hidden states (or embeddings) of trained transformers into interpretable components. For any layer, embedding vectors of input sequence samples are represented by a tensor h in R^{C times T times d}. Given embedding vector h_{c,t} in R^d at sequence position t le T in a sequence (or context) c le C, extracting the mean effects yields the decomposition \[ h_{c,t} = \mu + pos_t + ctx_c + resid_{c,t} \] where mu is the global mean vector, pos_t and ctx_c are the mean vectors across contexts and across positions respectively, and resid_{c,t} is the residual vector. For popular transformer architectures and diverse text datasets, empirically we find pervasive mathematical structure: (1) (pos_t)_{t} forms a low-dimensional, continuous, and often spiral shape across layers, (2) (ctx_c)_c shows clear cluster structure that falls into context topics, and (3) (pos_t)_{t} and (ctx_c)_c are mutually nearly orthogonal. We argue that smoothness is pervasive and beneficial to transformers trained on languages, and our decomposition leads to improved model interpretability.

We examine in detail the accuracy, efficiency and implementation issues that arise when a simplified code structure is employed to evaluate the partition function of the two-dimensional square Ising model on periodic lattices though repeated tensor contractions.

For any graph G having n vertices and its automorphism group Aut(G), we provide a full characterisation of all of the possible Aut(G)-equivariant neural networks whose layers are some tensor power of R^{n}. In particular, we find a spanning set of matrices for the learnable, linear, Aut(G)-equivariant layer functions between such tensor power spaces in the standard basis of R^{n}.

Fully Homomorphic Encryption (FHE) is an encryption scheme that allows for computation to be performed directly on encrypted data, effectively closing the loop on secure and outsourced computing. Data is encrypted not only during rest and transit, but also during processing. However, FHE provides a limited instruction set: SIMD addition, SIMD multiplication, and cyclic rotation of 1-D vectors. This restriction makes performing multi-dimensional tensor operations challenging. Practitioners must pack these tensors into 1-D vectors and map tensor operations onto this one-dimensional layout rather than their traditional nested structure. And while prior systems have made significant strides in automating this process, they often hide critical packing decisions behind layers of abstraction, making debugging, optimizing, and building on top of these systems difficult. In this work, we approach multi-dimensional tensor operations in FHE through Einstein summation (einsum) notation. Einsum notation explicitly encodes dimensional structure and operations in its syntax, naturally exposing how tensors should be packed and transformed. We decompose einsum expressions into a fixed set of FHE-friendly operations. We implement our design and present EinHops, a minimalist system that factors einsum expressions into a fixed sequence of FHE operations. EinHops enables developers to perform encrypted tensor operations using FHE while maintaining full visibility into the underlying packing strategy. We evaluate EinHops on a range of tensor operations from a simple transpose to complex multi-dimensional contractions. We show that the explicit nature of einsum notation allows us to build an FHE tensor system that is simple, general, and interpretable. We open-source EinHops at the following repository: https://github.com/baahl-nyu/einhops.

Systematic under-counting effects are observed in data collected across many disciplines, e.g., epidemiology and ecology. Under-counted tensor completion (UC-TC) is well-motivated for many data analytics tasks, e.g., inferring the case numbers of infectious diseases at unobserved locations from under-counted case numbers in neighboring regions. However, existing methods for similar problems often lack supports in theory, making it hard to understand the underlying principles and conditions beyond empirical successes. In this work, a low-rank Poisson tensor model with an expressive unknown nonlinear side information extractor is proposed for under-counted multi-aspect data. A joint low-rank tensor completion and neural network learning algorithm is designed to recover the model. Moreover, the UC-TC formulation is supported by theoretical analysis showing that the fully counted entries of the tensor and each entry's under-counting probability can be provably recovered from partial observations -- under reasonable conditions. To our best knowledge, the result is the first to offer theoretical supports for under-counted multi-aspect data completion. Simulations and real-data experiments corroborate the theoretical claims.

Processing of multi-word expressions (MWEs) is a known problem for any natural language processing task. Even neural machine translation (NMT) struggles to overcome it. This paper presents results of experiments on investigating NMT attention allocation to the MWEs and improving automated translation of sentences that contain MWEs in English->Latvian and English->Czech NMT systems. Two improvement strategies were explored -(1) bilingual pairs of automatically extracted MWE candidates were added to the parallel corpus used to train the NMT system, and (2) full sentences containing the automatically extracted MWE candidates were added to the parallel corpus. Both approaches allowed to increase automated evaluation results. The best result - 0.99 BLEU point increase - has been reached with the first approach, while with the second approach minimal improvements achieved. We also provide open-source software and tools used for MWE extraction and alignment inspection.

Most state of the art deep neural networks are overparameterized and exhibit a high computational cost. A straightforward approach to this problem is to replace convolutional kernels with its low-rank tensor approximations, whereas the Canonical Polyadic tensor Decomposition is one of the most suited models. However, fitting the convolutional tensors by numerical optimization algorithms often encounters diverging components, i.e., extremely large rank-one tensors but canceling each other. Such degeneracy often causes the non-interpretable result and numerical instability for the neural network fine-tuning. This paper is the first study on degeneracy in the tensor decomposition of convolutional kernels. We present a novel method, which can stabilize the low-rank approximation of convolutional kernels and ensure efficient compression while preserving the high-quality performance of the neural networks. We evaluate our approach on popular CNN architectures for image classification and show that our method results in much lower accuracy degradation and provides consistent performance.

How can we predict missing values in multi-dimensional data (or tensors) more accurately? The task of tensor completion is crucial in many applications such as personalized recommendation, image and video restoration, and link prediction in social networks. Many tensor factorization and neural network-based tensor completion algorithms have been developed to predict missing entries in partially observed tensors. However, they can produce inaccurate estimations as real-world tensors are very sparse, and these methods tend to overfit on the small amount of data. Here, we overcome these shortcomings by presenting a data augmentation technique for tensors. In this paper, we propose DAIN, a general data augmentation framework that enhances the prediction accuracy of neural tensor completion methods. Specifically, DAIN first trains a neural model and finds tensor cell importances with influence functions. After that, DAIN aggregates the cell importance to calculate the importance of each entity (i.e., an index of a dimension). Finally, DAIN augments the tensor by weighted sampling of entity importances and a value predictor. Extensive experimental results show that DAIN outperforms all data augmentation baselines in terms of enhancing imputation accuracy of neural tensor completion on four diverse real-world tensors. Ablation studies of DAIN substantiate the effectiveness of each component of DAIN. Furthermore, we show that DAIN scales near linearly to large datasets.

In this paper, we introduce a mesh-free two-level hybrid Tucker tensor format for approximation of multivariate functions, which combines the product Chebyshev interpolation with the ALS-based Tucker decomposition of the tensor of Chebyshev coefficients. It allows to avoid the expenses of the rank-structured approximation of function-related tensors defined on large spacial grids, while benefiting from the Tucker decomposition of the rather small core tensor of Chebyshev coefficients. This leads to nearly optimal Tucker rank parameters which are close to the results for well established Tucker-ALS algorithm applied to the large grid-based tensors. These rank parameters inherited from the Tucker-ALS decomposition of the coefficient tensor can be much less than the polynomial degrees of the initial Chebyshev interpolant via function independent basis set. Furthermore, the tensor product Chebyshev polynomials discretized on a tensor grid leads to a low-rank two-level orthogonal algebraic Tucker tensor that approximates the initial function with controllable accuracy. It is shown that our techniques could be gainfully applied to the long-range part of the electrostatic potential of multi-particle systems approximated in the range-separated tensor format. Error and complexity estimates of the proposed methods are presented. We demonstrate the efficiency of the suggested method numerically on examples of the long-range components of multi-particle interaction potentials generated by 3D Newton kernel for large bio-molecule systems and lattice-type compounds.

Deep learning optimizers are often motivated through a mix of convex and approximate second-order theory. We select three such methods -- Adam, Shampoo and Prodigy -- and argue that each method can instead be understood as a squarely first-order method without convexity assumptions. In fact, after switching off exponential moving averages, each method is equivalent to steepest descent under a particular norm. By generalizing this observation, we chart a new design space for training algorithms. Different operator norms should be assigned to different tensors based on the role that the tensor plays within the network. For example, while linear and embedding layers may have the same weight space of R^{mtimes n}, these layers play different roles and should be assigned different norms. We hope that this idea of carefully metrizing the neural architecture might lead to more stable, scalable and indeed faster training.

Learning features from data is one of the defining characteristics of deep learning, but our theoretical understanding of the role features play in deep learning is still rudimentary. To address this gap, we introduce a new tool, the interaction tensor, for empirically analyzing the interaction between data and model through features. With the interaction tensor, we make several key observations about how features are distributed in data and how models with different random seeds learn different features. Based on these observations, we propose a conceptual framework for feature learning. Under this framework, the expected accuracy for a single hypothesis and agreement for a pair of hypotheses can both be derived in closed-form. We demonstrate that the proposed framework can explain empirically observed phenomena, including the recently discovered Generalization Disagreement Equality (GDE) that allows for estimating the generalization error with only unlabeled data. Further, our theory also provides explicit construction of natural data distributions that break the GDE. Thus, we believe this work provides valuable new insight into our understanding of feature learning.

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of R^{n} for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of R^{n} when the group is O(n) or SO(n), and in the symplectic basis of R^{n} when the group is Sp(n).

Optimizing neural networks with loss that contain high-dimensional and high-order differential operators is expensive to evaluate with back-propagation due to O(d^{k}) scaling of the derivative tensor size and the O(2^{k-1}L) scaling in the computation graph, where d is the dimension of the domain, L is the number of ops in the forward computation graph, and k is the derivative order. In previous works, the polynomial scaling in d was addressed by amortizing the computation over the optimization process via randomization. Separately, the exponential scaling in k for univariate functions (d=1) was addressed with high-order auto-differentiation (AD). In this work, we show how to efficiently perform arbitrary contraction of the derivative tensor of arbitrary order for multivariate functions, by properly constructing the input tangents to univariate high-order AD, which can be used to efficiently randomize any differential operator. When applied to Physics-Informed Neural Networks (PINNs), our method provides >1000times speed-up and >30times memory reduction over randomization with first-order AD, and we can now solve 1-million-dimensional PDEs in 8 minutes on a single NVIDIA A100 GPU. This work opens the possibility of using high-order differential operators in large-scale problems.

In this paper we show how tensor networks help in developing explainability of machine learning algorithms. Specifically, we develop an unsupervised clustering algorithm based on Matrix Product States (MPS) and apply it in the context of a real use-case of adversary-generated threat intelligence. Our investigation proves that MPS rival traditional deep learning models such as autoencoders and GANs in terms of performance, while providing much richer model interpretability. Our approach naturally facilitates the extraction of feature-wise probabilities, Von Neumann Entropy, and mutual information, offering a compelling narrative for classification of anomalies and fostering an unprecedented level of transparency and interpretability, something fundamental to understand the rationale behind artificial intelligence decisions.

Graph neural networks that model 3D data, such as point clouds or atoms, are typically desired to be SO(3) equivariant, i.e., equivariant to 3D rotations. Unfortunately equivariant convolutions, which are a fundamental operation for equivariant networks, increase significantly in computational complexity as higher-order tensors are used. In this paper, we address this issue by reducing the SO(3) convolutions or tensor products to mathematically equivalent convolutions in SO(2) . This is accomplished by aligning the node embeddings' primary axis with the edge vectors, which sparsifies the tensor product and reduces the computational complexity from O(L^6) to O(L^3), where L is the degree of the representation. We demonstrate the potential implications of this improvement by proposing the Equivariant Spherical Channel Network (eSCN), a graph neural network utilizing our novel approach to equivariant convolutions, which achieves state-of-the-art results on the large-scale OC-20 and OC-22 datasets.

Linear structural equation models are multivariate statistical models encoded by mixed graphs. In particular, the set of covariance matrices for distributions belonging to a linear structural equation model for a fixed mixed graph G=(V, D,B) is parameterized by a rational function with parameters for each vertex and edge in G. This rational parametrization naturally allows for the study of these models from an algebraic and combinatorial point of view. Indeed, this point of view has led to a collection of results in the literature, mainly focusing on questions related to identifiability and determining relationships between covariances (i.e., finding polynomials in the Gaussian vanishing ideal). So far, a large proportion of these results has focused on the case when D, the directed part of the mixed graph G, is acyclic. This is due to the fact that in the acyclic case, the parametrization becomes polynomial and there is a description of the entries of the covariance matrices in terms of a finite sum. We move beyond the acyclic case and give a closed form expression for the entries of the covariance matrices in terms of the one-connections in a graph obtained from D through some small operations. This closed form expression then allows us to show that if G is simple, then the parametrization map is generically finite-to-one. Finally, having a closed form expression for the covariance matrices allows for the development of an algorithm for systematically exploring possible polynomials in the Gaussian vanishing ideal.

CNNs achieve remarkable performance by leveraging deep, over-parametrized architectures, trained on large datasets. However, they have limited generalization ability to data outside the training domain, and a lack of robustness to noise and adversarial attacks. By building better inductive biases, we can improve robustness and also obtain smaller networks that are more memory and computationally efficient. While standard CNNs use matrix computations, we study tensor layers that involve higher-order computations and provide better inductive bias. Specifically, we impose low-rank tensor structures on the weights of tensor regression layers to obtain compact networks, and propose tensor dropout, a randomization in the tensor rank for robustness. We show that our approach outperforms other methods for large-scale image classification on ImageNet and CIFAR-100. We establish a new state-of-the-art accuracy for phenotypic trait prediction on the largest dataset of brain MRI, the UK Biobank brain MRI dataset, where multi-linear structure is paramount. In all cases, we demonstrate superior performance and significantly improved robustness, both to noisy inputs and to adversarial attacks. We rigorously validate the theoretical validity of our approach by establishing the link between our randomized decomposition and non-linear dropout.

Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models. We achieve results that outperform commercial Computer Algebra Systems such as Matlab or Mathematica.

Computing the numerical solution to high-dimensional tensor differential equations can lead to prohibitive computational costs and memory requirements. To reduce the memory and computational footprint, dynamical low-rank approximation (DLRA) has proven to be a promising approach. DLRA represents the solution as a low-rank tensor factorization and evolves the resulting low-rank factors in time. A central challenge in DLRA is to find time integration schemes that are robust to the arising small singular values. A robust parallel basis update & Galerkin integrator, which simultaneously evolves all low-rank factors, has recently been derived for matrix differential equations. This work extends the parallel low-rank matrix integrator to Tucker tensors and general tree tensor networks, yielding an algorithm in which all bases and connecting tensors are evolved in parallel over a time step. We formulate the algorithm, provide a robust error bound, and demonstrate the efficiency of the new integrators for problems in quantum many-body physics, uncertainty quantification, and radiative transfer.

We introduce a large-scale dataset of math word problems and an interpretable neural math problem solver that learns to map problems to operation programs. Due to annotation challenges, current datasets in this domain have been either relatively small in scale or did not offer precise operational annotations over diverse problem types. We introduce a new representation language to model precise operation programs corresponding to each math problem that aim to improve both the performance and the interpretability of the learned models. Using this representation language, our new dataset, MathQA, significantly enhances the AQuA dataset with fully-specified operational programs. We additionally introduce a neural sequence-to-program model enhanced with automatic problem categorization. Our experiments show improvements over competitive baselines in our MathQA as well as the AQuA dataset. The results are still significantly lower than human performance indicating that the dataset poses new challenges for future research. Our dataset is available at: https://math-qa.github.io/math-QA/

Scaling language models to handle longer input sequences typically necessitates large key-value (KV) caches, resulting in substantial memory overhead during inference. In this paper, we propose Tensor Product Attention (TPA), a novel attention mechanism that uses tensor decompositions to represent queries, keys, and values compactly, significantly shrinking KV cache size at inference time. By factorizing these representations into contextual low-rank components (contextual factorization) and seamlessly integrating with RoPE, TPA achieves improved model quality alongside memory efficiency. Based on TPA, we introduce the Tensor ProducT ATTenTion Transformer (T6), a new model architecture for sequence modeling. Through extensive empirical evaluation of language modeling tasks, we demonstrate that T6 exceeds the performance of standard Transformer baselines including MHA, MQA, GQA, and MLA across various metrics, including perplexity and a range of renowned evaluation benchmarks. Notably, TPAs memory efficiency enables the processing of significantly longer sequences under fixed resource constraints, addressing a critical scalability challenge in modern language models. The code is available at https://github.com/tensorgi/T6.

The ability of neural networks to represent more features than neurons makes interpreting them challenging. This phenomenon, known as superposition, has spurred efforts to find architectures that are more interpretable than standard multilayer perceptrons (MLPs) with elementwise activation functions. In this note, I examine bilinear layers, which are a type of MLP layer that are mathematically much easier to analyze while simultaneously performing better than standard MLPs. Although they are nonlinear functions of their input, I demonstrate that bilinear layers can be expressed using only linear operations and third order tensors. We can integrate this expression for bilinear layers into a mathematical framework for transformer circuits, which was previously limited to attention-only transformers. These results suggest that bilinear layers are easier to analyze mathematically than current architectures and thus may lend themselves to deeper safety insights by allowing us to talk more formally about circuits in neural networks. Additionally, bilinear layers may offer an alternative path for mechanistic interpretability through understanding the mechanisms of feature construction instead of enumerating a (potentially exponentially) large number of features in large models.

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

Web applications are increasingly becoming the primary platform for AI service delivery, making in-browser deep learning (DL) inference more prominent. However, current in-browser inference systems fail to effectively utilize advanced web programming techniques and customize kernels for various client devices, leading to suboptimal performance. To address the issues, this paper presents the first in-browser inference system, nn-JIT.web, which enables just-in-time (JIT) auto-generation of optimized kernels for both CPUs and GPUs during inference. The system achieves this by using two novel web programming techniques that can significantly reduce kernel generation time, compared to other tensor compilers such as TVM, while maintaining or even improving performance. The first technique, Tensor-Web Compiling Co-Design, lowers compiling costs by unifying tensor and web compiling and eliminating redundant and ineffective compiling passes. The second technique, Web-Specific Lite Kernel Optimization Space Design, reduces kernel tuning costs by focusing on web programming requirements and efficient hardware resource utilization, limiting the optimization space to only dozens. nn-JIT.web is evaluated for modern transformer models on a range of client devices, including the mainstream CPUs and GPUs from ARM, Intel, AMD and Nvidia. Results show that nn-JIT.web can achieve up to 8.2x faster within 30 seconds compared to the baselines across various models.

Lattices are architected metamaterials whose properties strongly depend on their geometrical design. The analogy between lattices and graphs enables the use of graph neural networks (GNNs) as a faster surrogate model compared to traditional methods such as finite element modelling. In this work, we generate a big dataset of structure-property relationships for strut-based lattices. The dataset is made available to the community which can fuel the development of methods anchored in physical principles for the fitting of fourth-order tensors. In addition, we present a higher-order GNN model trained on this dataset. The key features of the model are (i) SE(3) equivariance, and (ii) consistency with the thermodynamic law of conservation of energy. We compare the model to non-equivariant models based on a number of error metrics and demonstrate its benefits in terms of predictive performance and reduced training requirements. Finally, we demonstrate an example application of the model to an architected material design task. The methods which we developed are applicable to fourth-order tensors beyond elasticity such as piezo-optical tensor etc.

The deployment and training of neural networks on edge computing devices pose many challenges. The low memory nature of edge devices is often one of the biggest limiting factors encountered in the deployment of large neural network models. Tensor rematerialization or recompute is a way to address high memory requirements for neural network training and inference. In this paper we consider the problem of execution time minimization of compute graphs subject to a memory budget. In particular, we develop a new constraint programming formulation called Moccasin with only O(n) integer variables, where n is the number of nodes in the compute graph. This is a significant improvement over the works in the recent literature that propose formulations with O(n^2) Boolean variables. We present numerical studies that show that our approach is up to an order of magnitude faster than recent work especially for large-scale graphs.

We provide a construction for categorical representation learning and introduce the foundations of "categorifier". The central theme in representation learning is the idea of everything to vector. Every object in a dataset S can be represented as a vector in R^n by an encoding map E: Obj(S)toR^n. More importantly, every morphism can be represented as a matrix E: Hom(S)toR^{n}_{n}. The encoding map E is generally modeled by a deep neural network. The goal of representation learning is to design appropriate tasks on the dataset to train the encoding map (assuming that an encoding is optimal if it universally optimizes the performance on various tasks). However, the latter is still a set-theoretic approach. The goal of the current article is to promote the representation learning to a new level via a category-theoretic approach. As a proof of concept, we provide an example of a text translator equipped with our technology, showing that our categorical learning model outperforms the current deep learning models by 17 times. The content of the current article is part of the recent US patent proposal (patent application number: 63110906).

Convolution is a broadly useful operation with applications including signal processing, machine learning, probability, optics, polynomial multiplication, and efficient parsing. Usually, however, this operation is understood and implemented in more specialized forms, hiding commonalities and limiting usefulness. This paper formulates convolution in the common algebraic framework of semirings and semimodules and populates that framework with various representation types. One of those types is the grand abstract template and itself generalizes to the free semimodule monad. Other representations serve varied uses and performance trade-offs, with implementations calculated from simple and regular specifications. Of particular interest is Brzozowski's method for regular expression matching. Uncovering the method's essence frees it from syntactic manipulations, while generalizing from boolean to weighted membership (such as multisets and probability distributions) and from sets to n-ary relations. The classic trie data structure then provides an elegant and efficient alternative to syntax. Pleasantly, polynomial arithmetic requires no additional implementation effort, works correctly with a variety of representations, and handles multivariate polynomials and power series with ease. Image convolution also falls out as a special case.

The computing cost and memory demand of deep learning pipelines have grown fast in recent years and thus a variety of pruning techniques have been developed to reduce model parameters. The majority of these techniques focus on reducing inference costs by pruning the network after a pass of full training. A smaller number of methods address the reduction of training costs, mostly based on compressing the network via low-rank layer factorizations. Despite their efficiency for linear layers, these methods fail to effectively handle convolutional filters. In this work, we propose a low-parametric training method that factorizes the convolutions into tensor Tucker format and adaptively prunes the Tucker ranks of the convolutional kernel during training. Leveraging fundamental results from geometric integration theory of differential equations on tensor manifolds, we obtain a robust training algorithm that provably approximates the full baseline performance and guarantees loss descent. A variety of experiments against the full model and alternative low-rank baselines are implemented, showing that the proposed method drastically reduces the training costs, while achieving high performance, comparable to or better than the full baseline, and consistently outperforms competing low-rank approaches.

We propose a machine learning method to model molecular tensorial quantities, namely the magnetic anisotropy tensor, based on the Gaussian-moment neural-network approach. We demonstrate that the proposed methodology can achieve an accuracy of 0.3--0.4 cm^{-1} and has excellent generalization capability for out-of-sample configurations. Moreover, in combination with machine-learned interatomic potential energies based on Gaussian moments, our approach can be applied to study the dynamic behavior of magnetic anisotropy tensors and provide a unique insight into spin-phonon relaxation.

As deep learning models grow, sparsity is becoming an increasingly critical component of deep neural networks, enabling improved performance and reduced storage. However, existing frameworks offer poor support for sparsity. Specialized sparsity engines focus exclusively on sparse inference, while general frameworks primarily focus on sparse tensors in classical formats and neglect the broader sparsification pipeline necessary for using sparse models, especially during training. Further, existing frameworks are not easily extensible: adding a new sparse tensor format or operator is challenging and time-consuming. To address this, we propose STen, a sparsity programming model and interface for PyTorch, which incorporates sparsity layouts, operators, and sparsifiers, in an efficient, customizable, and extensible framework that supports virtually all sparsification methods. We demonstrate this by developing a high-performance grouped n:m sparsity layout for CPU inference at moderate sparsity. STen brings high performance and ease of use to the ML community, making sparsity easily accessible.

Input constraints are useful for many software development tasks. For example, input constraints of a function enable the generation of valid inputs, i.e., inputs that follow these constraints, to test the function deeper. API functions of deep learning (DL) libraries have DL specific input constraints, which are described informally in the free form API documentation. Existing constraint extraction techniques are ineffective for extracting DL specific input constraints. To fill this gap, we design and implement a new technique, DocTer, to analyze API documentation to extract DL specific input constraints for DL API functions. DocTer features a novel algorithm that automatically constructs rules to extract API parameter constraints from syntactic patterns in the form of dependency parse trees of API descriptions. These rules are then applied to a large volume of API documents in popular DL libraries to extract their input parameter constraints. To demonstrate the effectiveness of the extracted constraints, DocTer uses the constraints to enable the automatic generation of valid and invalid inputs to test DL API functions. Our evaluation on three popular DL libraries (TensorFlow, PyTorch, and MXNet) shows that the precision of DocTer in extracting input constraints is 85.4%. DocTer detects 94 bugs from 174 API functions, including one previously unknown security vulnerability that is now documented in the CVE database, while a baseline technique without input constraints detects only 59 bugs. Most (63) of the 94 bugs are previously unknown, 54 of which have been fixed or confirmed by developers after we report them. In addition, DocTer detects 43 inconsistencies in documents, 39 of which are fixed or confirmed.

Quantum many-body physics simulation has important impacts on understanding fundamental science and has applications to quantum materials design and quantum technology. However, due to the exponentially growing size of the Hilbert space with respect to the particle number, a direct simulation is intractable. While representing quantum states with tensor networks and neural networks are the two state-of-the-art methods for approximate simulations, each has its own limitations in terms of expressivity and inductive bias. To address these challenges, we develop a novel architecture, Autoregressive Neural TensorNet (ANTN), which bridges tensor networks and autoregressive neural networks. We show that Autoregressive Neural TensorNet parameterizes normalized wavefunctions, allows for exact sampling, generalizes the expressivity of tensor networks and autoregressive neural networks, and inherits a variety of symmetries from autoregressive neural networks. We demonstrate our approach on quantum state learning as well as finding the ground state of the challenging 2D J_1-J_2 Heisenberg model with different systems sizes and coupling parameters, outperforming both tensor networks and autoregressive neural networks. Our work opens up new opportunities for scientific simulations of quantum many-body physics and quantum technology.

Convolutional neural networks are now seeing widespread use in a variety of fields, including image classification, facial and object recognition, medical imaging analysis, and many more. In addition, there are applications such as physics-informed simulators in which accurate forecasts in real time with a minimal lag are required. The present neural network designs include millions of parameters, which makes it difficult to install such complex models on devices that have limited memory. Compression techniques might be able to resolve these issues by decreasing the size of CNN models that are created by reducing the number of parameters that contribute to the complexity of the models. We propose a compressed tensor format of convolutional layer, a priori, before the training of the neural network. 3-way kernels or 2-way kernels in convolutional layers are replaced by one-way fiters. The overfitting phenomena will be reduced also. The time needed to make predictions or time required for training using the original Convolutional Neural Networks model would be cut significantly if there were fewer parameters to deal with. In this paper we present a method of a priori compressing convolutional neural networks for finite element (FE) predictions of physical data. Afterwards we validate our a priori compressed models on physical data from a FE model solving a 2D wave equation. We show that the proposed convolutinal compression technique achieves equivalent performance as classical convolutional layers with fewer trainable parameters and lower memory footprint.

We show that protein sequences can be thought of as sentences in natural language processing and can be parsed using the existing Quantum Natural Language framework into parameterized quantum circuits of reasonable qubits, which can be trained to solve various protein-related machine-learning problems. We classify proteins based on their subcellular locations, a pivotal task in bioinformatics that is key to understanding biological processes and disease mechanisms. Leveraging the quantum-enhanced processing capabilities, we demonstrate that Quantum Tensor Networks (QTN) can effectively handle the complexity and diversity of protein sequences. We present a detailed methodology that adapts QTN architectures to the nuanced requirements of protein data, supported by comprehensive experimental results. We demonstrate two distinct QTNs, inspired by classical recurrent neural networks (RNN) and convolutional neural networks (CNN), to solve the binary classification task mentioned above. Our top-performing quantum model has achieved a 94% accuracy rate, which is comparable to the performance of a classical model that uses the ESM2 protein language model embeddings. It's noteworthy that the ESM2 model is extremely large, containing 8 million parameters in its smallest configuration, whereas our best quantum model requires only around 800 parameters. We demonstrate that these hybrid models exhibit promising performance, showcasing their potential to compete with classical models of similar complexity.

We provide a full characterisation of all of the possible alternating group (A_n) equivariant neural networks whose layers are some tensor power of R^{n}. In particular, we find a basis of matrices for the learnable, linear, A_n-equivariant layer functions between such tensor power spaces in the standard basis of R^{n}. We also describe how our approach generalises to the construction of neural networks that are equivariant to local symmetries.

We define the bicategory of Graph Convolutional Neural Networks GCNN_n for an arbitrary graph with n nodes. We show it can be factored through the already existing categorical constructions for deep learning called Para and Lens with the base category set to the CoKleisli category of the product comonad. We prove that there exists an injective-on-objects, faithful 2-functor GCNN_n to Para(CoKl(R^{n times n} times -)). We show that this construction allows us to treat the adjacency matrix of a GCNN as a global parameter instead of a a local, layer-wise one. This gives us a high-level categorical characterisation of a particular kind of inductive bias GCNNs possess. Lastly, we hypothesize about possible generalisations of GCNNs to general message-passing graph neural networks, connections to equivariant learning, and the (lack of) functoriality of activation functions.

Mathematical formulas are a fundamental and widely used component in various scientific fields, serving as a universal language for expressing complex concepts and relationships. While state-of-the-art transformer models excel in processing and understanding natural language, they encounter challenges with mathematical notation, which involves a complex structure and diverse representations. This study focuses on the development of specialized training datasets to enhance the encoding of mathematical content. We introduce Math Mutator (MAMUT), a framework capable of generating equivalent and falsified versions of a given mathematical formula in LaTeX notation, effectively capturing the mathematical variety in notation of the same concept. Based on MAMUT, we have generated four large mathematical datasets containing diverse notation, which can be used to train language models with enhanced mathematical embeddings.

Large language models have led to state-of-the-art accuracies across a range of tasks. However, training these models efficiently is challenging for two reasons: a) GPU memory capacity is limited, making it impossible to fit large models on even a multi-GPU server, and b) the number of compute operations required to train these models can result in unrealistically long training times. Consequently, new methods of model parallelism such as tensor and pipeline parallelism have been proposed. Unfortunately, naive usage of these methods leads to fundamental scaling issues at thousands of GPUs, e.g., due to expensive cross-node communication or devices spending significant time waiting on other devices to make progress. In this paper, we show how different types of parallelism methods (tensor, pipeline, and data parallelism) can be composed to scale to thousands of GPUs and models with trillions of parameters. We survey techniques for pipeline parallelism and propose a novel interleaved pipeline parallelism schedule that can improve throughput by 10+% with memory footprint comparable to existing approaches. We quantitatively study the trade-offs between tensor, pipeline, and data parallelism, and provide intuition as to how to configure distributed training of a large model. Our approach allows us to perform training iterations on a model with 1 trillion parameters at 502 petaFLOP/s on 3072 GPUs with achieved per-GPU throughput of 52% of theoretical peak. Our code is open sourced at https://github.com/nvidia/megatron-lm.

Using fewer bits to represent model parameters and related tensors during pre-training has become a required technique for improving GPU efficiency without sacrificing accuracy. Microscaling (MX) formats introduced in NVIDIA Blackwell generation of GPUs represent a major advancement of this technique, making it practical to combine narrow floating-point data types with finer granularity per-block scaling factors. In turn, this enables both quantization of more tensors than previous approaches and more efficient execution of operations on those tensors. Effective use of MX-formats requires careful choices of various parameters. In this paper we review these choices and show how MXFP8-E4M3 datatype and a specific number conversion algorithm result in training sessions that match those carried out in BF16. We present results using models with up to 8B parameters, trained on high-quality datasets of up to 15T tokens.

Memory complexity and data scarcity have so far prohibited learning solution operators of partial differential equations (PDEs) at high resolutions. We address these limitations by introducing a new data efficient and highly parallelizable operator learning approach with reduced memory requirement and better generalization, called multi-grid tensorized neural operator (MG-TFNO). MG-TFNO scales to large resolutions by leveraging local and global structures of full-scale, real-world phenomena, through a decomposition of both the input domain and the operator's parameter space. Our contributions are threefold: i) we enable parallelization over input samples with a novel multi-grid-based domain decomposition, ii) we represent the parameters of the model in a high-order latent subspace of the Fourier domain, through a global tensor factorization, resulting in an extreme reduction in the number of parameters and improved generalization, and iii) we propose architectural improvements to the backbone FNO. Our approach can be used in any operator learning setting. We demonstrate superior performance on the turbulent Navier-Stokes equations where we achieve less than half the error with over 150x compression. The tensorization combined with the domain decomposition, yields over 150x reduction in the number of parameters and 7x reduction in the domain size without losses in accuracy, while slightly enabling parallelism.

We present a novel application of category theory for deep learning. We show how category theory can be used to understand and work with the linear layer functions of group equivariant neural networks whose layers are some tensor power space of R^{n} for the groups S_n, O(n), Sp(n), and SO(n). By using category theoretic constructions, we build a richer structure that is not seen in the original formulation of these neural networks, leading to new insights. In particular, we outline the development of an algorithm for quickly computing the result of a vector that is passed through an equivariant, linear layer for each group in question. The success of our approach suggests that category theory could be beneficial for other areas of deep learning.

Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of geometric deep learning. Often problems in the physical sciences deal with relatively small sets of points in two- or three-dimensional space wherein translation, rotation, and permutation equivariance are important or even vital for models to be useful in practice. In this work, we present rotation- and permutation-equivariant architectures for deep learning on these small point clouds, composed of a set of products of terms from the geometric algebra and reductions over those products using an attention mechanism. The geometric algebra provides valuable mathematical structure by which to combine vector, scalar, and other types of geometric inputs in a systematic way to account for rotation invariance or covariance, while attention yields a powerful way to impose permutation equivariance. We demonstrate the usefulness of these architectures by training models to solve sample problems relevant to physics, chemistry, and biology.

As the parameter size of large language models (LLMs) continues to expand, the need for a large memory footprint and high communication bandwidth have become significant bottlenecks for the training and inference of LLMs. To mitigate these bottlenecks, various tensor compression techniques have been proposed to reduce the data size, thereby alleviating memory requirements and communication pressure. Our research found that video codecs, despite being originally designed for compressing videos, show excellent efficiency when compressing various types of tensors. We demonstrate that video codecs can be versatile and general-purpose tensor codecs while achieving the state-of-the-art compression efficiency in various tasks. We further make use of the hardware video encoding and decoding module available on GPUs to create a framework capable of both inference and training with video codecs repurposed as tensor codecs. This greatly reduces the requirement for memory capacity and communication bandwidth, enabling training and inference of large models on consumer-grade GPUs.

Neural networks are deployed widely in natural language processing tasks on the industrial scale, and perhaps the most often they are used as compounds of automatic machine translation systems. In this work, we present a simple approach to fool state-of-the-art machine translation tools in the task of translation from Russian to English and vice versa. Using a novel black-box gradient-free tensor-based optimizer, we show that many online translation tools, such as Google, DeepL, and Yandex, may both produce wrong or offensive translations for nonsensical adversarial input queries and refuse to translate seemingly benign input phrases. This vulnerability may interfere with understanding a new language and simply worsen the user's experience while using machine translation systems, and, hence, additional improvements of these tools are required to establish better translation.

A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Currently, high-dimensional data is ubiquitous in data science, which necessitates the development of techniques to decompose and interpret such multidimensional (aka tensor) datasets. Finding a low dimensional representation of the data, that is, its inherent structure, is one of the approaches that can serve to understand the dynamics of low dimensional latent features hidden in the data. Nonnegative RESCAL is one such technique, particularly well suited to analyze self-relational data, such as dynamic networks found in international trade flows. Nonnegative RESCAL computes a low dimensional tensor representation by finding the latent space containing multiple modalities. Estimating the dimensionality of this latent space is crucial for extracting meaningful latent features. Here, to determine the dimensionality of the latent space with nonnegative RESCAL, we propose a latent dimension determination method which is based on clustering of the solutions of multiple realizations of nonnegative RESCAL decompositions. We demonstrate the performance of our model selection method on synthetic data and then we apply our method to decompose a network of international trade flows data from International Monetary Fund and validate the resulting features against empirical facts from economic literature.

Knowledge Graphs (KGs) are symbolically structured storages of facts. The KG embedding contains concise data used in NLP tasks requiring implicit information about the real world. Furthermore, the size of KGs that may be useful in actual NLP assignments is enormous, and creating embedding over it has memory cost issues. We represent KG as a 3rd-order binary tensor and move beyond the standard CP decomposition by using a data-specific generalized version of it. The generalization of the standard CP-ALS algorithm allows obtaining optimization gradients without a backpropagation mechanism. It reduces the memory needed in training while providing computational benefits. We propose a MEKER, a memory-efficient KG embedding model, which yields SOTA-comparable performance on link prediction tasks and KG-based Question Answering.

Given a matrix Ain R^{ntimes d} and a vector bin R^n, we consider the regression problem with ell_infty guarantees: finding a vector x'in R^d such that |x'-x^*|_infty leq epsilon{d}cdot |Ax^*-b|_2cdot |A^dagger| where x^*=argmin_{xin R^d}|Ax-b|_2. One popular approach for solving such ell_2 regression problem is via sketching: picking a structured random matrix Sin R^{mtimes n} with mll n and SA can be quickly computed, solve the ``sketched'' regression problem argmin_{xin R^d} |SAx-Sb|_2. In this paper, we show that in order to obtain such ell_infty guarantee for ell_2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m=epsilon^{-2}dlog^3(n/delta) such that solving the sketched regression problem gives the ell_infty guarantee, with probability at least 1-delta. Moreover, the matrix SA can be computed in time O(ndlog n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in [Price, Song and Woodruff, ICALP'17], in which a super-linear in d rows, m=Omega(epsilon^{-2}d^{1+gamma}) for gamma=Theta(frac{loglog n{log d}}) is required. We also develop a novel analytical framework for ell_infty guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in [Song and Yu, ICML'21]. Our analysis is arguably much simpler and more general than [Price, Song and Woodruff, ICALP'17], and it extends to dense sketches for tensor product of vectors.

We develop a general approach to distill symbolic representations of a learned deep model by introducing strong inductive biases. We focus on Graph Neural Networks (GNNs). The technique works as follows: we first encourage sparse latent representations when we train a GNN in a supervised setting, then we apply symbolic regression to components of the learned model to extract explicit physical relations. We find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example-a detailed dark matter simulation-and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.

TensorFlow is a machine learning system that operates at large scale and in heterogeneous environments. TensorFlow uses dataflow graphs to represent computation, shared state, and the operations that mutate that state. It maps the nodes of a dataflow graph across many machines in a cluster, and within a machine across multiple computational devices, including multicore CPUs, general-purpose GPUs, and custom designed ASICs known as Tensor Processing Units (TPUs). This architecture gives flexibility to the application developer: whereas in previous "parameter server" designs the management of shared state is built into the system, TensorFlow enables developers to experiment with novel optimizations and training algorithms. TensorFlow supports a variety of applications, with particularly strong support for training and inference on deep neural networks. Several Google services use TensorFlow in production, we have released it as an open-source project, and it has become widely used for machine learning research. In this paper, we describe the TensorFlow dataflow model in contrast to existing systems, and demonstrate the compelling performance that TensorFlow achieves for several real-world applications.

The NeuroEvolution of Augmenting Topologies (NEAT) algorithm has received considerable recognition in the field of neuroevolution. Its effectiveness is derived from initiating with simple networks and incrementally evolving both their topologies and weights. Although its capability across various challenges is evident, the algorithm's computational efficiency remains an impediment, limiting its scalability potential. In response, this paper introduces a tensorization method for the NEAT algorithm, enabling the transformation of its diverse network topologies and associated operations into uniformly shaped tensors for computation. This advancement facilitates the execution of the NEAT algorithm in a parallelized manner across the entire population. Furthermore, we develop TensorNEAT, a library that implements the tensorized NEAT algorithm and its variants, such as CPPN and HyperNEAT. Building upon JAX, TensorNEAT promotes efficient parallel computations via automated function vectorization and hardware acceleration. Moreover, the TensorNEAT library supports various benchmark environments including Gym, Brax, and gymnax. Through evaluations across a spectrum of robotics control environments in Brax, TensorNEAT achieves up to 500x speedups compared to the existing implementations such as NEAT-Python. Source codes are available at: https://github.com/EMI-Group/tensorneat.

We propose SMMF (Square-Matricized Momentum Factorization), a memory-efficient optimizer that reduces the memory requirement of the widely used adaptive learning rate optimizers, such as Adam, by up to 96%. SMMF enables flexible and efficient factorization of an arbitrary rank (shape) of the first and second momentum tensors during optimization, based on the proposed square-matricization and one-time single matrix factorization. From this, it becomes effectively applicable to any rank (shape) of momentum tensors, i.e., bias, matrix, and any rank-d tensors, prevalent in various deep model architectures, such as CNNs (high rank) and Transformers (low rank), in contrast to existing memory-efficient optimizers that applies only to a particular (rank-2) momentum tensor, e.g., linear layers. We conduct a regret bound analysis of SMMF, which shows that it converges similarly to non-memory-efficient adaptive learning rate optimizers, such as AdamNC, providing a theoretical basis for its competitive optimization capability. In our experiment, SMMF takes up to 96% less memory compared to state-of-the-art memory efficient optimizers, e.g., Adafactor, CAME, and SM3, while achieving comparable model performance on various CNN and Transformer tasks.

A mechanistic understanding of how MLPs do computation in deep neural networks remains elusive. Current interpretability work can extract features from hidden activations over an input dataset but generally cannot explain how MLP weights construct features. One challenge is that element-wise nonlinearities introduce higher-order interactions and make it difficult to trace computations through the MLP layer. In this paper, we analyze bilinear MLPs, a type of Gated Linear Unit (GLU) without any element-wise nonlinearity that nevertheless achieves competitive performance. Bilinear MLPs can be fully expressed in terms of linear operations using a third-order tensor, allowing flexible analysis of the weights. Analyzing the spectra of bilinear MLP weights using eigendecomposition reveals interpretable low-rank structure across toy tasks, image classification, and language modeling. We use this understanding to craft adversarial examples, uncover overfitting, and identify small language model circuits directly from the weights alone. Our results demonstrate that bilinear layers serve as an interpretable drop-in replacement for current activation functions and that weight-based interpretability is viable for understanding deep-learning models.

We introduce Kosmos-2, a Multimodal Large Language Model (MLLM), enabling new capabilities of perceiving object descriptions (e.g., bounding boxes) and grounding text to the visual world. Specifically, we represent refer expressions as links in Markdown, i.e., ``[text span](bounding boxes)'', where object descriptions are sequences of location tokens. Together with multimodal corpora, we construct large-scale data of grounded image-text pairs (called GrIT) to train the model. In addition to the existing capabilities of MLLMs (e.g., perceiving general modalities, following instructions, and performing in-context learning), Kosmos-2 integrates the grounding capability into downstream applications. We evaluate Kosmos-2 on a wide range of tasks, including (i) multimodal grounding, such as referring expression comprehension, and phrase grounding, (ii) multimodal referring, such as referring expression generation, (iii) perception-language tasks, and (iv) language understanding and generation. This work lays out the foundation for the development of Embodiment AI and sheds light on the big convergence of language, multimodal perception, action, and world modeling, which is a key step toward artificial general intelligence. Data, demo, and pretrained models are available at https://aka.ms/kosmos-2.

In this paper we generalize and extend an idea of low-rank adaptation (LoRA) of large language models (LLMs) based on Transformer architecture. Widely used LoRA-like methods of fine-tuning LLMs are based on matrix factorization of gradient update. We introduce LoTR, a novel approach for parameter-efficient fine-tuning of LLMs which represents a gradient update to parameters in a form of tensor decomposition. Low-rank adapter for each layer is constructed as a product of three matrices, and tensor structure arises from sharing left and right multipliers of this product among layers. Simultaneous compression of a sequence of layers with low-rank tensor representation allows LoTR to archive even better parameter efficiency then LoRA especially for deep models. Moreover, the core tensor does not depend on original weight dimension and can be made arbitrary small, which allows for extremely cheap and fast downstream fine-tuning.

High-quality, large-scale corpora are the cornerstone of building foundation models. In this work, we introduce MathPile, a diverse and high-quality math-centric corpus comprising about 9.5 billion tokens. Throughout its creation, we adhered to the principle of ``less is more'', firmly believing in the supremacy of data quality over quantity, even in the pre-training phase. Our meticulous data collection and processing efforts included a complex suite of preprocessing, prefiltering, language identification, cleaning, filtering, and deduplication, ensuring the high quality of our corpus. Furthermore, we performed data contamination detection on downstream benchmark test sets to eliminate duplicates. We hope our MathPile can help to enhance the mathematical reasoning abilities of language models. We plan to open-source different versions of \mathpile with the scripts used for processing, to facilitate future developments in this field.

The art of mathematical reasoning stands as a fundamental pillar of intellectual progress and is a central catalyst in cultivating human ingenuity. Researchers have recently published a plethora of works centered around the task of solving Math Word Problems (MWP) - a crucial stride towards general AI. These existing models are susceptible to dependency on shallow heuristics and spurious correlations to derive the solution expressions. In order to ameliorate this issue, in this paper, we propose a framework for MWP solvers based on the generation of linguistic variants of the problem text. The approach involves solving each of the variant problems and electing the predicted expression with the majority of the votes. We use DeBERTa (Decoding-enhanced BERT with disentangled attention) as the encoder to leverage its rich textual representations and enhanced mask decoder to construct the solution expressions. Furthermore, we introduce a challenging dataset, Psmall{ARAMAWPS}, consisting of paraphrased, adversarial, and inverse variants of selectively sampled MWPs from the benchmark Msmall{AWPS} dataset. We extensively experiment on this dataset along with other benchmark datasets using some baseline MWP solver models. We show that training on linguistic variants of problem statements and voting on candidate predictions improve the mathematical reasoning and robustness of the model. We make our code and data publicly available.

We introduce GraphNet, a dataset of 2.7K real-world deep learning computational graphs with rich metadata, spanning six major task categories across multiple deep learning frameworks. To evaluate tensor compiler performance on these samples, we propose the benchmark metric Speedup Score S(t), which jointly considers runtime speedup and execution correctness under tunable tolerance levels, offering a reliable measure of general optimization capability. Furthermore, we extend S(t) to the Error-aware Speedup Score ES(t), which incorporates error information and helps compiler developers identify key performance bottlenecks. In this report, we benchmark the default tensor compilers, CINN for PaddlePaddle and TorchInductor for PyTorch, on computer vision (CV) and natural language processing (NLP) samples to demonstrate the practicality of GraphNet. The full construction pipeline with graph extraction and compiler evaluation tools is available at https://github.com/PaddlePaddle/GraphNet .

We propose, implement, and compare with competitors a new architecture of equivariant neural networks based on geometric (Clifford) algebras: Generalized Lipschitz Group Equivariant Neural Networks (GLGENN). These networks are equivariant to all pseudo-orthogonal transformations, including rotations and reflections, of a vector space with any non-degenerate or degenerate symmetric bilinear form. We propose a weight-sharing parametrization technique that takes into account the fundamental structures and operations of geometric algebras. Due to this technique, GLGENN architecture is parameter-light and has less tendency to overfitting than baseline equivariant models. GLGENN outperforms or matches competitors on several benchmarking equivariant tasks, including estimation of an equivariant function and a convex hull experiment, while using significantly fewer optimizable parameters.

Vector space embedding models like word2vec, GloVe, fastText, and ELMo are extremely popular representations in natural language processing (NLP) applications. We present Magnitude, a fast, lightweight tool for utilizing and processing embeddings. Magnitude is an open source Python package with a compact vector storage file format that allows for efficient manipulation of huge numbers of embeddings. Magnitude performs common operations up to 60 to 6,000 times faster than Gensim. Magnitude introduces several novel features for improved robustness like out-of-vocabulary lookups.

Problems involving geometric data arise in a variety of fields, including computer vision, robotics, chemistry, and physics. Such data can take numerous forms, such as points, direction vectors, planes, or transformations, but to date there is no single architecture that can be applied to such a wide variety of geometric types while respecting their symmetries. In this paper we introduce the Geometric Algebra Transformer (GATr), a general-purpose architecture for geometric data. GATr represents inputs, outputs, and hidden states in the projective geometric algebra, which offers an efficient 16-dimensional vector space representation of common geometric objects as well as operators acting on them. GATr is equivariant with respect to E(3), the symmetry group of 3D Euclidean space. As a transformer, GATr is scalable, expressive, and versatile. In experiments with n-body modeling and robotic planning, GATr shows strong improvements over non-geometric baselines.

The approval success rate of drug candidates is very low with the majority of failure due to safety and efficacy. Increasingly available high dimensional information on targets, drug molecules and indications provides an opportunity for ML methods to integrate multiple data modalities and better predict clinically promising drug targets. Notably, drug targets with human genetics evidence are shown to have better odds to succeed. However, a recent tensor factorization-based approach found that additional information on targets and indications might not necessarily improve the predictive accuracy. Here we revisit this approach by integrating different types of human genetics evidence collated from publicly available sources to support each target-indication pair. We use Bayesian tensor factorization to show that models incorporating all available human genetics evidence (rare disease, gene burden, common disease) modestly improves the clinical outcome prediction over models using single line of genetics evidence. We provide additional insight into the relative predictive power of different types of human genetics evidence for predicting the success of clinical outcomes.

Theano is a Python library that allows to define, optimize, and evaluate mathematical expressions involving multi-dimensional arrays efficiently. Since its introduction, it has been one of the most used CPU and GPU mathematical compilers - especially in the machine learning community - and has shown steady performance improvements. Theano is being actively and continuously developed since 2008, multiple frameworks have been built on top of it and it has been used to produce many state-of-the-art machine learning models. The present article is structured as follows. Section I provides an overview of the Theano software and its community. Section II presents the principal features of Theano and how to use them, and compares them with other similar projects. Section III focuses on recently-introduced functionalities and improvements. Section IV compares the performance of Theano against Torch7 and TensorFlow on several machine learning models. Section V discusses current limitations of Theano and potential ways of improving it.

Handwritten mathematical expression recognition (HMER) is a challenging task that has many potential applications. Recent methods for HMER have achieved outstanding performance with an encoder-decoder architecture. However, these methods adhere to the paradigm that the prediction is made "from one character to another", which inevitably yields prediction errors due to the complicated structures of mathematical expressions or crabbed handwritings. In this paper, we propose a simple and efficient method for HMER, which is the first to incorporate syntax information into an encoder-decoder network. Specifically, we present a set of grammar rules for converting the LaTeX markup sequence of each expression into a parsing tree; then, we model the markup sequence prediction as a tree traverse process with a deep neural network. In this way, the proposed method can effectively describe the syntax context of expressions, alleviating the structure prediction errors of HMER. Experiments on three benchmark datasets demonstrate that our method achieves better recognition performance than prior arts. To further validate the effectiveness of our method, we create a large-scale dataset consisting of 100k handwritten mathematical expression images acquired from ten thousand writers. The source code, new dataset, and pre-trained models of this work will be publicly available.

Vectors are universal mathematical objects that can represent text, images, speech, or a mix of these data modalities. That happens regardless of whether data is represented by hand-crafted features or learnt embeddings. Collect a large enough quantity of such vectors and the question of retrieval becomes urgently relevant: Finding vectors that are more similar to a query vector. This monograph is concerned with the question above and covers fundamental concepts along with advanced data structures and algorithms for vector retrieval. In doing so, it recaps this fascinating topic and lowers barriers of entry into this rich area of research.

Large language models (LLMs) have demonstrated state-of-the-art performance across various tasks. However, the latency of inference and the large GPU memory consumption of LLMs restrict their deployment performance. Recently, there have been some efficient attempts to quantize LLMs, yet inference with large batch size or long sequence still has the issue of being compute-bound. Fine-grained quantization methods have showcased their proficiency in achieving low-bit quantization for LLMs, while requiring FP16 data type for linear layer computations, which is time-consuming when dealing with large batch size or long sequence. In this paper, we introduce a method called FlattenQuant, which significantly reduces the maximum value of the tensor by flattening the large channels in the tensor, to achieve low bit per-tensor quantization with minimal accuracy loss. Our experiments show that FlattenQuant can directly use 4 bits to achieve 48.29% of the linear layer calculation in LLMs, with the remaining layers using 8 bits. The 4-bit matrix multiplication introduced in the FlattenQuant method can effectively address the compute-bound caused by large matrix calculation. Our work achieves up to 2times speedup and 2.3times memory reduction for LLMs with negligible loss in accuracy.

Multimodal sentiment analysis is an increasingly popular research area, which extends the conventional language-based definition of sentiment analysis to a multimodal setup where other relevant modalities accompany language. In this paper, we pose the problem of multimodal sentiment analysis as modeling intra-modality and inter-modality dynamics. We introduce a novel model, termed Tensor Fusion Network, which learns both such dynamics end-to-end. The proposed approach is tailored for the volatile nature of spoken language in online videos as well as accompanying gestures and voice. In the experiments, our model outperforms state-of-the-art approaches for both multimodal and unimodal sentiment analysis.

In "Backprop as functor", the authors show that the fundamental elements of deep learning -- gradient descent and backpropagation -- can be conceptualized as a strong monoidal functor Para(Euc)toLearn from the category of parameterized Euclidean spaces to that of learners, a category developed explicitly to capture parameter update and backpropagation. It was soon realized that there is an isomorphism LearncongPara(Slens), where Slens is the symmetric monoidal category of simple lenses as used in functional programming. In this note, we observe that Slens is a full subcategory of Poly, the category of polynomial functors in one variable, via the functor Amapsto Ay^A. Using the fact that (Poly,otimes) is monoidal closed, we show that a map Ato B in Para(Slens) has a natural interpretation in terms of dynamical systems (more precisely, generalized Moore machines) whose interface is the internal-hom type [Ay^A,By^B]. Finally, we review the fact that the category p-Coalg of dynamical systems on any p in Poly forms a topos, and consider the logical propositions that can be stated in its internal language. We give gradient descent as an example, and we conclude by discussing some directions for future work.

Deep neural networks have achieved great success in many data processing applications. However, the high computational complexity and storage cost makes deep learning hard to be used on resource-constrained devices, and it is not environmental-friendly with much power cost. In this paper, we focus on low-rank optimization for efficient deep learning techniques. In the space domain, deep neural networks are compressed by low rank approximation of the network parameters, which directly reduces the storage requirement with a smaller number of network parameters. In the time domain, the network parameters can be trained in a few subspaces, which enables efficient training for fast convergence. The model compression in the spatial domain is summarized into three categories as pre-train, pre-set, and compression-aware methods, respectively. With a series of integrable techniques discussed, such as sparse pruning, quantization, and entropy coding, we can ensemble them in an integration framework with lower computational complexity and storage. Besides of summary of recent technical advances, we have two findings for motivating future works: one is that the effective rank outperforms other sparse measures for network compression. The other is a spatial and temporal balance for tensorized neural networks.

The advent of deep machine learning platforms such as Tensorflow and Pytorch, developed in expressive high-level languages such as Python, have allowed more expressive representations of deep neural network architectures. We argue that such a powerful formalism is missing in information retrieval (IR), and propose a framework called PyTerrier that allows advanced retrieval pipelines to be expressed, and evaluated, in a declarative manner close to their conceptual design. Like the aforementioned frameworks that compile deep learning experiments into primitive GPU operations, our framework targets IR platforms as backends in order to execute and evaluate retrieval pipelines. Further, we can automatically optimise the retrieval pipelines to increase their efficiency to suite a particular IR platform backend. Our experiments, conducted on TREC Robust and ClueWeb09 test collections, demonstrate the efficiency benefits of these optimisations for retrieval pipelines involving both the Anserini and Terrier IR platforms.

Natural language image-caption datasets, widely used for training Large Multimodal Models, mainly focus on natural scenarios and overlook the intricate details of mathematical figures that are critical for problem-solving, hindering the advancement of current LMMs in multimodal mathematical reasoning. To this end, we propose leveraging code as supervision for cross-modal alignment, since code inherently encodes all information needed to generate corresponding figures, establishing a precise connection between the two modalities. Specifically, we co-develop our image-to-code model and dataset with model-in-the-loop approach, resulting in an image-to-code model, FigCodifier and ImgCode-8.6M dataset, the largest image-code dataset to date. Furthermore, we utilize FigCodifier to synthesize novel mathematical figures and then construct MM-MathInstruct-3M, a high-quality multimodal math instruction fine-tuning dataset. Finally, we present MathCoder-VL, trained with ImgCode-8.6M for cross-modal alignment and subsequently fine-tuned on MM-MathInstruct-3M for multimodal math problem solving. Our model achieves a new open-source SOTA across all six metrics. Notably, it surpasses GPT-4o and Claude 3.5 Sonnet in the geometry problem-solving subset of MathVista, achieving improvements of 8.9% and 9.2%. The dataset and models will be released at https://github.com/mathllm/MathCoder.

Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free 2^nd-order optimizers for deep learning in low precision settings. Code: https://github.com/yorkerlin/StructuredNGD-DL

We propose a method that can generate an unambiguous description (known as a referring expression) of a specific object or region in an image, and which can also comprehend or interpret such an expression to infer which object is being described. We show that our method outperforms previous methods that generate descriptions of objects without taking into account other potentially ambiguous objects in the scene. Our model is inspired by recent successes of deep learning methods for image captioning, but while image captioning is difficult to evaluate, our task allows for easy objective evaluation. We also present a new large-scale dataset for referring expressions, based on MS-COCO. We have released the dataset and a toolbox for visualization and evaluation, see https://github.com/mjhucla/Google_Refexp_toolbox

Anchors (Ribeiro et al., 2018) is a post-hoc, rule-based interpretability method. For text data, it proposes to explain a decision by highlighting a small set of words (an anchor) such that the model to explain has similar outputs when they are present in a document. In this paper, we present the first theoretical analysis of Anchors, considering that the search for the best anchor is exhaustive. After formalizing the algorithm for text classification, we present explicit results on different classes of models when the vectorization step is TF-IDF, and words are replaced by a fixed out-of-dictionary token when removed. Our inquiry covers models such as elementary if-then rules and linear classifiers. We then leverage this analysis to gain insights on the behavior of Anchors for any differentiable classifiers. For neural networks, we empirically show that the words corresponding to the highest partial derivatives of the model with respect to the input, reweighted by the inverse document frequencies, are selected by Anchors.

We present a new class of fast polylog-linear algorithms based on the theory of structured matrices (in particular low displacement rank) for integrating tensor fields defined on weighted trees. Several applications of the resulting fast tree-field integrators (FTFIs) are presented, including (a) approximation of graph metrics with tree metrics, (b) graph classification, (c) modeling on meshes, and finally (d) Topological Transformers (TTs) (Choromanski et al., 2022) for images. For Topological Transformers, we propose new relative position encoding (RPE) masking mechanisms with as few as three extra learnable parameters per Transformer layer, leading to 1.0-1.5%+ accuracy gains. Importantly, most of FTFIs are exact methods, thus numerically equivalent to their brute-force counterparts. When applied to graphs with thousands of nodes, those exact algorithms provide 5.7-13x speedups. We also provide an extensive theoretical analysis of our methods.

A crucial component of many deep learning applications (such as FAQ and RAG) is dense retrieval, in which embedding models are used to convert raw text to numerical vectors and then get the most similar text by MIPS (Maximum Inner Product Search). Some text embedding benchmarks (e.g. MTEB, BEIR, and AIR-Bench) have been established to evaluate embedding models accurately. Thanks to these benchmarks, we can use SOTA models; however, the deployment and application of these models in industry were hampered by their large vector dimensions and numerous parameters. To alleviate this problem, 1) we present a distillation technique that can enable a smaller student model to achieve good performance. 2) Inspired by MRL we present a training approach of reducing the vector dimensions based on its own vectors or its teacher vectors. 3) We do simple yet effective alignment training between images and text to make our model a multimodal encoder. We trained Stella and Jasper models using the technologies above and achieved high scores on the MTEB leaderboard. We release the model and data at Hugging Face Hub (https://huggingface.co/infgrad/jasper_en_vision_language_v1) and the training logs are at https://api.wandb.ai/links/dunnzhang0/z8jqoqpb.

Time series analysis is of immense importance in extensive applications, such as weather forecasting, anomaly detection, and action recognition. This paper focuses on temporal variation modeling, which is the common key problem of extensive analysis tasks. Previous methods attempt to accomplish this directly from the 1D time series, which is extremely challenging due to the intricate temporal patterns. Based on the observation of multi-periodicity in time series, we ravel out the complex temporal variations into the multiple intraperiod- and interperiod-variations. To tackle the limitations of 1D time series in representation capability, we extend the analysis of temporal variations into the 2D space by transforming the 1D time series into a set of 2D tensors based on multiple periods. This transformation can embed the intraperiod- and interperiod-variations into the columns and rows of the 2D tensors respectively, making the 2D-variations to be easily modeled by 2D kernels. Technically, we propose the TimesNet with TimesBlock as a task-general backbone for time series analysis. TimesBlock can discover the multi-periodicity adaptively and extract the complex temporal variations from transformed 2D tensors by a parameter-efficient inception block. Our proposed TimesNet achieves consistent state-of-the-art in five mainstream time series analysis tasks, including short- and long-term forecasting, imputation, classification, and anomaly detection. Code is available at this repository: https://github.com/thuml/TimesNet.

In this paper, we propose a highly parameter-efficient approach to scaling pre-trained language models (PLMs) to a deeper model depth. Unlike prior work that shares all parameters or uses extra blocks, we design a more capable parameter-sharing architecture based on matrix product operator (MPO). MPO decomposition can reorganize and factorize the information of a parameter matrix into two parts: the major part that contains the major information (central tensor) and the supplementary part that only has a small proportion of parameters (auxiliary tensors). Based on such a decomposition, our architecture shares the central tensor across all layers for reducing the model size and meanwhile keeps layer-specific auxiliary tensors (also using adapters) for enhancing the adaptation flexibility. To improve the model training, we further propose a stable initialization algorithm tailored for the MPO-based architecture. Extensive experiments have demonstrated the effectiveness of our proposed model in reducing the model size and achieving highly competitive performance.

The key-value (KV) cache accelerates LLMs decoding by storing KV tensors from previously generated tokens. It reduces redundant computation at the cost of increased memory usage. To mitigate this overhead, existing approaches compress KV tensors into lower-bit representations; however, quantization errors can accumulate as more tokens are generated, potentially resulting in undesired outputs. In this paper, we introduce SQuat (Subspace-orthogonal KV cache quantization). It first constructs a subspace spanned by query tensors to capture the most critical task-related information. During key tensor quantization, it enforces that the difference between the (de)quantized and original keys remains orthogonal to this subspace, minimizing the impact of quantization errors on the attention mechanism's outputs. SQuat requires no model fine-tuning, no additional calibration dataset for offline learning, and is grounded in a theoretical framework we develop. Through numerical experiments, we show that our method reduces peak memory by 2.17 to 2.82, improves throughput by 2.45 to 3.60, and achieves more favorable benchmark scores than existing KV cache quantization algorithms.

Implicit Neural Representation (INR) has emerged as a powerful tool for encoding discrete signals into continuous, differentiable functions using neural networks. However, these models often have an unfortunate reliance on monolithic architectures to represent high-dimensional data, leading to prohibitive computational costs as dimensionality grows. We propose F-INR, a framework that reformulates INR learning through functional tensor decomposition, breaking down high-dimensional tasks into lightweight, axis-specific sub-networks. Each sub-network learns a low-dimensional data component (e.g., spatial or temporal). Then, we combine these components via tensor operations, reducing forward pass complexity while improving accuracy through specialized learning. F-INR is modular and, therefore, architecture-agnostic, compatible with MLPs, SIREN, WIRE, or other state-of-the-art INR architecture. It is also decomposition-agnostic, supporting CP, TT, and Tucker modes with user-defined rank for speed-accuracy control. In our experiments, F-INR trains 100times faster than existing approaches on video tasks while achieving higher fidelity (+3.4 dB PSNR). Similar gains hold for image compression, physics simulations, and 3D geometry reconstruction. Through this, F-INR offers a new scalable, flexible solution for high-dimensional signal modeling.

A major open problem in neural machine translation (NMT) is the translation of idiomatic expressions, such as "under the weather". The meaning of these expressions is not composed by the meaning of their constituent words, and NMT models tend to translate them literally (i.e., word-by-word), which leads to confusing and nonsensical translations. Research on idioms in NMT is limited and obstructed by the absence of automatic methods for quantifying these errors. In this work, first, we propose a novel metric for automatically measuring the frequency of literal translation errors without human involvement. Equipped with this metric, we present controlled translation experiments with models trained in different conditions (with/without the test-set idioms) and across a wide range of (global and targeted) metrics and test sets. We explore the role of monolingual pretraining and find that it yields substantial targeted improvements, even without observing any translation examples of the test-set idioms. In our analysis, we probe the role of idiom context. We find that the randomly initialized models are more local or "myopic" as they are relatively unaffected by variations of the idiom context, unlike the pretrained ones.

Large Language Models (LLMs) have seen an impressive wave of advances recently, with models now excelling in a variety of tasks, such as mathematical reasoning and program synthesis. However, their potential to effectively use tools via API calls remains unfulfilled. This is a challenging task even for today's state-of-the-art LLMs such as GPT-4, largely due to their inability to generate accurate input arguments and their tendency to hallucinate the wrong usage of an API call. We release Gorilla, a finetuned LLaMA-based model that surpasses the performance of GPT-4 on writing API calls. When combined with a document retriever, Gorilla demonstrates a strong capability to adapt to test-time document changes, enabling flexible user updates or version changes. It also substantially mitigates the issue of hallucination, commonly encountered when prompting LLMs directly. To evaluate the model's ability, we introduce APIBench, a comprehensive dataset consisting of HuggingFace, TorchHub, and TensorHub APIs. The successful integration of the retrieval system with Gorilla demonstrates the potential for LLMs to use tools more accurately, keep up with frequently updated documentation, and consequently increase the reliability and applicability of their outputs. Gorilla's code, model, data, and demo are available at https://gorilla.cs.berkeley.edu

Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products bigotimes_{i in J} X_i come in two versions: a weaker but more general one for families of objects (X_i)_{i in J} in semicartesian symmetric monoidal categories, and a stronger but more specific one for families of objects in Markov categories. As a first application, we state and prove versions of the zero-one laws of Kolmogorov and Hewitt-Savage for Markov categories. This gives general versions of these results which can be instantiated not only in measure-theoretic probability, where they specialize to the standard ones in the setting of standard Borel spaces, but also in other contexts.

Dynamic shape computations have become critical in modern machine learning workloads, especially in emerging large language models. The success of these models has driven demand for deploying them to a diverse set of backend environments. In this paper, we present Relax, a compiler abstraction for optimizing end-to-end dynamic machine learning workloads. Relax introduces first-class symbolic shape annotations to track dynamic shape computations globally across the program. It also introduces a cross-level abstraction that encapsulates computational graphs, loop-level tensor programs, and library calls in a single representation to enable cross-level optimizations. We build an end-to-end compilation framework using the proposed approach to optimize dynamic shape models. Experimental results on large language models show that Relax delivers performance competitive with state-of-the-art hand-optimized systems across platforms and enables deployment of emerging dynamic models to a broader set of environments, including mobile phones, embedded devices, and web browsers.

We present Manify, an open-source Python library for non-Euclidean representation learning. Leveraging manifold learning techniques, Manify provides tools for learning embeddings in (products of) non-Euclidean spaces, performing classification and regression with data that lives in such spaces, and estimating the curvature of a manifold. Manify aims to advance research and applications in machine learning by offering a comprehensive suite of tools for manifold-based data analysis. Our source code, examples, datasets, results, and documentation are available at https://github.com/pchlenski/manify

In this work, we propose a concise neural operator architecture for operator learning. Drawing an analogy with a conventional fully connected neural network, we define the neural operator as follows: the output of the i-th neuron in a nonlinear operator layer is defined by mathcal O_i(u) = sigmaleft( sum_j mathcal W_{ij} u + mathcal B_{ij}right). Here, mathcal W_{ij} denotes the bounded linear operator connecting j-th input neuron to i-th output neuron, and the bias mathcal B_{ij} takes the form of a function rather than a scalar. Given its new universal approximation property, the efficient parameterization of the bounded linear operators between two neurons (Banach spaces) plays a critical role. As a result, we introduce MgNO, utilizing multigrid structures to parameterize these linear operators between neurons. This approach offers both mathematical rigor and practical expressivity. Additionally, MgNO obviates the need for conventional lifting and projecting operators typically required in previous neural operators. Moreover, it seamlessly accommodates diverse boundary conditions. Our empirical observations reveal that MgNO exhibits superior ease of training compared to other CNN-based models, while also displaying a reduced susceptibility to overfitting when contrasted with spectral-type neural operators. We demonstrate the efficiency and accuracy of our method with consistently state-of-the-art performance on different types of partial differential equations (PDEs).

Idiomatic expressions have always been a bottleneck for language comprehension and natural language understanding, specifically for tasks like Machine Translation(MT). MT systems predominantly produce literal translations of idiomatic expressions as they do not exhibit generic and linguistically deterministic patterns which can be exploited for comprehension of the non-compositional meaning of the expressions. These expressions occur in parallel corpora used for training, but due to the comparatively high occurrences of the constituent words of idiomatic expressions in literal context, the idiomatic meaning gets overpowered by the compositional meaning of the expression. State of the art Metaphor Detection Systems are able to detect non-compositional usage at word level but miss out on idiosyncratic phrasal idiomatic expressions. This creates a dire need for a dataset with a wider coverage and higher occurrence of commonly occurring idiomatic expressions, the spans of which can be used for Metaphor Detection. With this in mind, we present our English Possible Idiomatic Expressions(EPIE) corpus containing 25206 sentences labelled with lexical instances of 717 idiomatic expressions. These spans also cover literal usages for the given set of idiomatic expressions. We also present the utility of our dataset by using it to train a sequence labelling module and testing on three independent datasets with high accuracy, precision and recall scores.

Over the past years, Printed Mathematical Expression Recognition (PMER) has progressed rapidly. However, due to the insufficient context information captured by Convolutional Neural Networks, some mathematical symbols might be incorrectly recognized or missed. To tackle this problem, in this paper, a Dual Branch transformer-based Network (DBN) is proposed to learn both local and global context information for accurate PMER. In our DBN, local and global features are extracted simultaneously, and a Context Coupling Module (CCM) is developed to complement the features between the global and local contexts. CCM adopts an interactive manner so that the coupled context clues are highly correlated to each expression symbol. Additionally, we design a Dynamic Soft Target (DST) strategy to utilize the similarities among symbol categories for reasonable label generation. Our experimental results have demonstrated that DBN can accurately recognize mathematical expressions and has achieved state-of-the-art performance.

We propose Quantum-informed Tensor Adaptation (QuanTA), a novel, easy-to-implement, fine-tuning method with no inference overhead for large-scale pre-trained language models. By leveraging quantum-inspired methods derived from quantum circuit structures, QuanTA enables efficient high-rank fine-tuning, surpassing the limitations of Low-Rank Adaptation (LoRA)--low-rank approximation may fail for complicated downstream tasks. Our approach is theoretically supported by the universality theorem and the rank representation theorem to achieve efficient high-rank adaptations. Experiments demonstrate that QuanTA significantly enhances commonsense reasoning, arithmetic reasoning, and scalability compared to traditional methods. Furthermore, QuanTA shows superior performance with fewer trainable parameters compared to other approaches and can be designed to integrate with existing fine-tuning algorithms for further improvement, providing a scalable and efficient solution for fine-tuning large language models and advancing state-of-the-art in natural language processing.

Neural Radiance Fields (NeRF) methods have proved effective as compact, high-quality and versatile representations for 3D scenes, and enable downstream tasks such as editing, retrieval, navigation, etc. Various neural architectures are vying for the core structure of NeRF, including the plain Multi-Layer Perceptron (MLP), sparse tensors, low-rank tensors, hashtables and their compositions. Each of these representations has its particular set of trade-offs. For example, the hashtable-based representations admit faster training and rendering but their lack of clear geometric meaning hampers downstream tasks like spatial-relation-aware editing. In this paper, we propose Progressive Volume Distillation (PVD), a systematic distillation method that allows any-to-any conversions between different architectures, including MLP, sparse or low-rank tensors, hashtables and their compositions. PVD consequently empowers downstream applications to optimally adapt the neural representations for the task at hand in a post hoc fashion. The conversions are fast, as distillation is progressively performed on different levels of volume representations, from shallower to deeper. We also employ special treatment of density to deal with its specific numerical instability problem. Empirical evidence is presented to validate our method on the NeRF-Synthetic, LLFF and TanksAndTemples datasets. For example, with PVD, an MLP-based NeRF model can be distilled from a hashtable-based Instant-NGP model at a 10X~20X faster speed than being trained the original NeRF from scratch, while achieving a superior level of synthesis quality. Code is available at https://github.com/megvii-research/AAAI2023-PVD.

We introduce meta-factorization, a theory that describes matrix decompositions as solutions of linear matrix equations: the projector and the reconstruction equation. Meta-factorization reconstructs known factorizations, reveals their internal structures, and allows for introducing modifications, as illustrated with SVD, QR, and UTV factorizations. The prospect of meta-factorization also provides insights into computational aspects of generalized matrix inverses and randomized linear algebra algorithms. The relations between the Moore-Penrose pseudoinverse, generalized Nystr\"{o}m method, and the CUR decomposition are revealed here as an illustration. Finally, meta-factorization offers hints on the structure of new factorizations and provides the potential of creating them.

In the past decade, Deep Learning (DL) systems have been widely deployed in various domains to facilitate our daily life. Meanwhile, it is extremely challenging to ensure the correctness of DL systems (e.g., due to their intrinsic nondeterminism), and bugs in DL systems can cause serious consequences and may even threaten human lives. In the literature, researchers have explored various techniques to test, analyze, and verify DL models, since their quality directly affects the corresponding system behaviors. Recently, researchers have also proposed novel techniques for testing the underlying operator-level DL libraries (such as TensorFlow and PyTorch), which provide general binary implementations for each high-level DL operator for running various DL models on many platforms. However, there is still limited work targeting the reliability of the emerging tensor compilers, which aim to directly compile high-level tensor computation graphs into high-performance binaries for better efficiency, portability, and scalability. In this paper, we target the important problem of tensor compiler testing, and have proposed Tzer, a practical fuzzing technique for the widely used TVM tensor compiler. Tzer focuses on mutating the low-level Intermediate Representation (IR) for TVM due to the limited mutation space for the high-level IR. More specifically, Tzer leverages both general-purpose and tensor-compiler-specific mutators guided by coverage feedback for evolutionary IR mutation; furthermore, Tzer also performs pass mutation in tandem with IR mutation for more effective fuzzing. Our results show that Tzer substantially outperforms existing fuzzing techniques on tensor compiler testing, with 75% higher coverage and 50% more valuable tests than the 2nd-best technique. To date, Tzer has detected 49 previously unknown bugs for TVM, with 37 bugs confirmed and 25 bugs fixed (PR merged).

This paper presents the first comprehensive empirical study demonstrating the efficacy of the Brain Floating Point (BFLOAT16) half-precision format for Deep Learning training across image classification, speech recognition, language modeling, generative networks and industrial recommendation systems. BFLOAT16 is attractive for Deep Learning training for two reasons: the range of values it can represent is the same as that of IEEE 754 floating-point format (FP32) and conversion to/from FP32 is simple. Maintaining the same range as FP32 is important to ensure that no hyper-parameter tuning is required for convergence; e.g., IEEE 754 compliant half-precision floating point (FP16) requires hyper-parameter tuning. In this paper, we discuss the flow of tensors and various key operations in mixed precision training, and delve into details of operations, such as the rounding modes for converting FP32 tensors to BFLOAT16. We have implemented a method to emulate BFLOAT16 operations in Tensorflow, Caffe2, IntelCaffe, and Neon for our experiments. Our results show that deep learning training using BFLOAT16 tensors achieves the same state-of-the-art (SOTA) results across domains as FP32 tensors in the same number of iterations and with no changes to hyper-parameters.

Deep-learning (DL) compilers such as TVM and TensorRT are increasingly being used to optimize deep neural network (DNN) models to meet performance, resource utilization and other requirements. Bugs in these compilers can result in models whose semantics differ from the original ones, producing incorrect results that corrupt the correctness of downstream applications. However, finding bugs in these compilers is challenging due to their complexity. In this work, we propose a new fuzz testing approach for finding bugs in deep-learning compilers. Our core approach consists of (i) generating diverse yet valid DNN test models that can exercise a large part of the compiler's transformation logic using light-weight operator specifications; (ii) performing gradient-based search to find model inputs that avoid any floating-point exceptional values during model execution, reducing the chance of missed bugs or false alarms; and (iii) using differential testing to identify bugs. We implemented this approach in NNSmith which has found 72 new bugs for TVM, TensorRT, ONNXRuntime, and PyTorch to date. Of these 58 have been confirmed and 51 have been fixed by their respective project maintainers.

Vector Symbolic Architectures (VSAs) give a way to represent a complex object as a single fixed-length vector, so that similar objects have similar vector representations. These vector representations then become easy to use for machine learning or nearest-neighbor search. We review a previously proposed VSA method, MBAT (Matrix Binding of Additive Terms), which uses multiplication by random matrices for binding related terms. However, multiplying by such matrices introduces instabilities which can harm performance. Making the random matrices be orthogonal matrices provably fixes this problem. With respect to larger scale applications, we see how to apply MBAT vector representations for any data expressed in JSON. JSON is used in numerous programming languages to express complex data, but its native format appears highly unsuited for machine learning. Expressing JSON as a fixed-length vector makes it readily usable for machine learning and nearest-neighbor search. Creating such JSON vectors also shows that a VSA needs to employ binding operations that are non-commutative. VSAs are now ready to try with full-scale practical applications, including healthcare, pharmaceuticals, and genomics. Keywords: MBAT (Matrix Binding of Additive Terms), VSA (Vector Symbolic Architecture), HDC (Hyperdimensional Computing), Distributed Representations, Binding, Orthogonal Matrices, Recurrent Connections, Machine Learning, Search, JSON, VSA Applications

The task of recognising Handwritten Mathematical Expressions (HMER) is crucial in the fields of digital education and scholarly research. However, it is difficult to accurately determine the length and complex spatial relationships among symbols in handwritten mathematical expressions. In this study, we present a novel encoder-decoder architecture (DenseBAM-GI) for HMER, where the encoder has a Bottleneck Attention Module (BAM) to improve feature representation and the decoder has a Gated Input-GRU (GI-GRU) unit with an extra gate to make decoding long and complex expressions easier. The proposed model is an efficient and lightweight architecture with performance equivalent to state-of-the-art models in terms of Expression Recognition Rate (exprate). It also performs better in terms of top 1, 2, and 3 error accuracy across the CROHME 2014, 2016, and 2019 datasets. DenseBAM-GI achieves the best exprate among all models on the CROHME 2019 dataset. Importantly, these successes are accomplished with a drop in the complexity of the calculation and a reduction in the need for GPU memory.

Post-training quantization (PTQ) has been gaining popularity for the deployment of deep neural networks on resource-limited devices since unlike quantization-aware training, neither a full training dataset nor end-to-end training is required at all. As PTQ schemes based on reconstructing each layer or block output turn out to be effective to enhance quantized model performance, recent works have developed algorithms to devise and learn a new weight-rounding scheme so as to better reconstruct each layer or block output. In this work, we propose a simple yet effective new weight-rounding mechanism for PTQ, coined FlexRound, based on element-wise division instead of typical element-wise addition such that FlexRound enables jointly learning a common quantization grid size as well as a different scale for each pre-trained weight. Thanks to the reciprocal rule of derivatives induced by element-wise division, FlexRound is inherently able to exploit pre-trained weights when updating their corresponding scales, and thus, flexibly quantize pre-trained weights depending on their magnitudes. We empirically validate the efficacy of FlexRound on a wide range of models and tasks. To the best of our knowledge, our work is the first to carry out comprehensive experiments on not only image classification and natural language understanding but also natural language generation, assuming a per-tensor uniform PTQ setting. Moreover, we demonstrate, for the first time, that large language models can be efficiently quantized, with only a negligible impact on performance compared to half-precision baselines, achieved by reconstructing the output in a block-by-block manner.

We extend Textual Inversion to learn pseudo-words that represent a concept at different resolutions. This allows us to generate images that use the concept with different levels of detail and also to manipulate different resolutions using language. Once learned, the user can generate images at different levels of agreement to the original concept; "A photo of S^*(0)" produces the exact object while the prompt "A photo of S^*(0.8)" only matches the rough outlines and colors. Our framework allows us to generate images that use different resolutions of an image (e.g. details, textures, styles) as separate pseudo-words that can be composed in various ways. We open-soure our code in the following URL: https://github.com/giannisdaras/multires_textual_inversion

The objective of Classic Referring Expression Comprehension (REC) is to produce a bounding box corresponding to the object mentioned in a given textual description. Commonly, existing datasets and techniques in classic REC are tailored for expressions that pertain to a single target, meaning a sole expression is linked to one specific object. Expressions that refer to multiple targets or involve no specific target have not been taken into account. This constraint hinders the practical applicability of REC. This study introduces a new benchmark termed as Generalized Referring Expression Comprehension (GREC). This benchmark extends the classic REC by permitting expressions to describe any number of target objects. To achieve this goal, we have built the first large-scale GREC dataset named gRefCOCO. This dataset encompasses a range of expressions: those referring to multiple targets, expressions with no specific target, and the single-target expressions. The design of GREC and gRefCOCO ensures smooth compatibility with classic REC. The proposed gRefCOCO dataset, a GREC method implementation code, and GREC evaluation code are available at https://github.com/henghuiding/gRefCOCO.