arXiv subjects

Haim Avron

Publications and source records attributed to Haim Avron.

At least 19 recordsLinked to original sources

arXiv2026

Exact Symmetry as Algebra: A Machine-Verified Tensor Calculus that Enforces Physical Selection Rules

Symmetry is central to the physical sciences, yet machine learning usually captures it only approximately, leaving a residual per-step equivariance error $\varepsilon$ that compounds with depth $M$ as $M\varepsilon$, whereas exact equivariance holds at unbounded depth; we demonstrate this divergence at fourteen orders of magnitude. We show that a symmetry can be made exact by construction, as the multiplication rule of a tensor algebra. In the resulting $\starG$ algebra, defined by any finite group $G$, the group-Fourier transform block-diagonalizes every tensor into irreducible-representation blocks, making equivariance intrinsic; requiring equivariance conversely \emph{forces} this suitably normalized transform, so the algebra is determined by $G$ rather than chosen. The standard matrix toolbox, including a Frobenius-optimal low-rank factorization, transfers blockwise, machine-checked in Lean~4 under an explicit axiom budget, and extends unchanged to band-limited compact groups and, under periodic boundary conditions, to all 230 crystallographic space groups and the compact little-group fibers of Euclidean and Poincaré symmetry. This exactness is an applied capability: on inorganic-crystal elastic tensors the algebra enforces point-group selection rules exactly on the output of \emph{any} predictor, driving a trained graph network's forbidden-channel leakage from $10^{-2}$ to machine zero, eliminating mechanically unstable predictions, and recovering viable materials that an unconstrained screen discards; on molecular data, with no quantum-mechanical input, it exposes octahedral selection-rule signatures consistent with the Wigner--Eckart theorem. Matched networks lead on pooled molecular accuracy, which we report plainly: the contribution is a complementary algebraic calculus, structural and diagnostic, exact at any depth.

cs.LG↗

arXiv2026

Demystifying Tubal Tensor Algebra

Developed in a series of seminal papers in the early 2010s, the tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features such as a tensor Singular Value Decomposition and Eckart-Young-like optimality results. It has proven to be a powerful tool for analyzing inherently multilinear data arising in hyperspectral imaging, medical imaging, neural dynamics, scientific simulations, and more. At the heart of tubal tensor algebra lies a special tensor-tensor product: originally the t-product, later generalized into a full family of products via the $\star_M$-product. Though initially defined through the multiplication of a block-circulant unfolding of one tensor by a matricization of another, it was soon observed that the t-product can be interpreted as standard matrix multiplication where the scalars are tubes-i.e., real vectors twisted ``inward.'' Yet, a fundamental question remains: why is this the ``right'' way to define a tensor-tensor product in the tubal setting? In this paper, we show that the t-product and its $\star_M$ generalization arise naturally when viewing third-order tensors as matrices of tubes, together with a small set of desired algebraic properties. Furthermore, we prove that the $\star_M$-product is, in fact, the only way to define a tubal product satisfying these properties. Thus, while partly expository in nature - aimed at presenting the foundations of tubal tensor algebra in a cohesive and accessible way - this paper also addresses theoretical gaps in the tubal tensor framework, proves new results, and provides justification for the tubal tensor framework central constructions, thereby shedding new light on it.

math.NA↗

arXiv2025

On Encoding Matrices using Quantum Circuits

Over a decade ago, it was demonstrated that quantum computing has the potential to revolutionize numerical linear algebra by enabling algorithms with complexity superior to what is classically achievable, e.g., the seminal HHL algorithm for solving linear systems. Efficient execution of such algorithms critically depends on representing inputs (matrices and vectors) as quantum circuits that encode or implement these inputs. For that task, two common circuit representations emerged in the literature: block encodings and state preparation circuits. In this paper, we systematically study encodings matrices in the form of block encodings and state preparation circuits. We examine methods for constructing these representations from matrices given in classical form, as well as quantum two-way conversions between circuit representations. Two key results we establish (among others) are: (a) a general method for efficiently constructing a block encoding of an arbitrary matrix given in classical form (entries stored in classical random access memory); and (b) low-overhead, bidirectional conversion algorithms between block encodings and state preparation circuits, showing that these models are essentially equivalent. From a technical perspective, two central components of our constructions are: (i) a special constant-depth multiplexer that simultaneously multiplexes all higher-order Pauli matrices of a given size, and (ii) an algorithm for performing a quantum conversion between a matrix's expansion in the standard basis and its expansion in the basis of higher-order Pauli matrices.

quant-ph↗

arXiv2025

Flatness After All?

Recent literature generalization in deep learning has examined the relationship between the curvature of the loss function at minima and generalization, mainly in the context of overparameterized neural networks. A key observation is that "flat" minima tend to generalize better than "sharp" minima. While this idea is supported by empirical evidence, it has also been shown that deep networks can generalize even with arbitrary sharpness, as measured by either the trace or the spectral norm of the Hessian. In this paper, we argue that generalization could be assessed by measuring flatness using a soft rank measure of the Hessian. We show that when an exponential family neural network model is exactly calibrated, and its prediction error and its confidence on the prediction are not correlated with the first and the second derivative of the network's output, our measure accurately captures the asymptotic expected generalization gap. For non-calibrated models, we connect a soft rank based flatness measure to the well-known Takeuchi Information Criterion and show that it still provides reliable estimates of generalization gaps for models that are not overly confident. Experimental results indicate that our approach offers a robust estimate of the generalization gap compared to baselines.

cs.LG↗

arXiv2025

Unbiased Stochastic Optimization for Gaussian Processes on Finite Dimensional RKHS

Current methods for stochastic hyperparameter learning in Gaussian Processes (GPs) rely on approximations, such as computing biased stochastic gradients or using inducing points in stochastic variational inference. However, when using such methods we are not guaranteed to converge to a stationary point of the true marginal likelihood. In this work, we propose algorithms for exact stochastic inference of GPs with kernels that induce a Reproducing Kernel Hilbert Space (RKHS) of moderate finite dimension. Our approach can also be extended to infinite dimensional RKHSs at the cost of forgoing exactness. Both for finite and infinite dimensional RKHSs, our method achieves better experimental results than existing methods when memory resources limit the feasible batch size and the possible number of inducing points.

cs.LG↗

arXiv2025

PCENet: High Dimensional Surrogate Modeling for Learning Uncertainty

Learning data representations under uncertainty is an important task that emerges in numerous scientific computing and data analysis applications. However, uncertainty quantification techniques are computationally intensive and become prohibitively expensive for high-dimensional data. In this study, we introduce a dimensionality reduction surrogate modeling (DRSM) approach for representation learning and uncertainty quantification that aims to deal with data of moderate to high dimensions. The approach involves a two-stage learning process: 1) employing a variational autoencoder to learn a low-dimensional representation of the input data distribution; and 2) harnessing polynomial chaos expansion (PCE) formulation to map the low dimensional distribution to the output target. The model enables us to (a) capture the system dynamics efficiently in the low-dimensional latent space, (b) learn under uncertainty, a representation of the data and a mapping between input and output distributions, (c) estimate this uncertainty in the high-dimensional data system, and (d) match high-order moments of the output distribution; without any prior statistical assumptions on the data. Numerical results are presented to illustrate the performance of the proposed method.

cs.LG↗

arXiv2025

Quasitubal Tensor Algebra Over Separable Hilbert Spaces

The tubal tensor framework provides a clean and effective algebraic setting for tensor computations, supporting matrix-mimetic features like Singular Value Decomposition and Eckart-Young-like optimality results. Underlying the tubal tensor framework is a view of a tensor as a matrix of finite sized tubes. In this work, we lay the mathematical and computational foundations for working with tensors with infinite size tubes: matrices whose elements are elements from a separable Hilbert space. A key challenge is that existence of important desired matrix-mimetic features of tubal tensors rely on the existence of a unit element in the ring of tubes. Such unit element cannot exist for tubes which are elements of an infinite-dimensional Hilbert space. We sidestep this issue by embedding the tubal space in a commutative unital C*-algebra of bounded operators. The resulting quasitubal algebra recovers the structural properties needed for decomposition and low-rank approximation. In addition to laying the theoretical groundwork for working with tubal tensors with infinite dimensional tubes, we discuss computational aspects of our construction, and provide a numerical illustration where we compute a finite dimensional approximation to a infinitely-sized synthetic tensor using our theory. We believe our theory opens new exciting avenues for applying matrix mimetic tensor framework in the context of inherently infinite dimensional problems.

math.NA↗

arXiv2025

Higher Order Reduced Rank Regression

Reduced Rank Regression (RRR) is a widely used method for multi-response regression. However, RRR assumes a linear relationship between features and responses. While linear models are useful and often provide a good approximation, many real-world problems involve more complex relationships that cannot be adequately captured by simple linear interactions. One way to model such relationships is via multilinear transformations. This paper introduces Higher Order Reduced Rank Regression (HORRR), an extension of RRR that leverages multi-linear transformations, and as such is capable of capturing nonlinear interactions in multi-response regression. HORRR employs tensor representations for the coefficients and a Tucker decomposition to impose multilinear rank constraints as regularization akin to the rank constraints in RRR. Encoding these constraints as a manifold allows us to use Riemannian optimization to solve this HORRR problems. We theoretically and empirically analyze the use of Riemannian optimization for solving HORRR problems.

stat.ML↗

arXiv2025

Hutchinson's Estimator is Bad at Kronecker-Trace-Estimation

We study the problem of estimating the trace of a matrix $\mathbf{A}$ that can only be accessed through Kronecker-matrix-vector products. That is, for any Kronecker-structured vector $\mathrm{x} = \otimes_{i=1}^k \mathrm{x}_i$, we can compute $\mathbf{A}\mathrm{x}$. We focus on the natural generalization of Hutchinson's Estimator to this setting, proving tight rates for the number of matrix-vector products this estimator needs to find a $(1\pm\varepsilon)$ approximation to the trace of $\mathbf{A}$. We find an exact equation for the variance of the estimator when using a Kronecker of Gaussian vectors, revealing an intimate relationship between Hutchinson's Estimator, the partial trace operator, and the partial transpose operator. Using this equation, we show that when using real vectors, in the worst case, this estimator needs $O(\frac{3^k}{\varepsilon^2})$ products to recover a $(1\pm\varepsilon)$ approximation of the trace of any PSD $\mathbf{A}$, and a matching lower bound for certain PSD $\mathbf{A}$. However, when using complex vectors, this can be exponentially improved to $Θ(\frac{2^k}{\varepsilon^2})$. Further, if the $\mathrm{x}_i$ vectors are low-dimensional and if we instead build $\mathrm{x}$ as the Kronecker product of (scaled) random unit vectors on the complex sphere, then as few as $\frac{1.33^k}{\varepsilon^2}$ samples suffice. We show that Hutchinson's Estimator converges slowest when $\mathbf{A}$ itself also has Kronecker structure. We conclude with some theoretical evidence suggesting that, by combining Hutchinson's Estimator with other techniques, it may be possible to avoid the exponential dependence on $k$.

cs.DS↗

arXiv2024

Faster Randomized Methods for Orthogonality Constrained Problems

Recent literature has advocated the use of randomized methods for accelerating the solution of various matrix problems arising throughout data science and computational science. One popular strategy for leveraging randomization is to use it as a way to reduce problem size. However, methods based on this strategy lack sufficient accuracy for some applications. Randomized preconditioning is another approach for leveraging randomization, which provides higher accuracy. The main challenge in using randomized preconditioning is the need for an underlying iterative method, thus randomized preconditioning so far have been applied almost exclusively to solving regression problems and linear systems. In this article, we show how to expand the application of randomized preconditioning to another important set of problems prevalent across data science: optimization problems with (generalized) orthogonality constraints. We demonstrate our approach, which is based on the framework of Riemannian optimization and Riemannian preconditioning, on the problem of computing the dominant canonical correlations and on the Fisher linear discriminant analysis problem. For both problems, we evaluate the effect of preconditioning on the computational costs and asymptotic convergence, and demonstrate empirically the utility of our approach.

math.NA↗

arXiv2024

Multivariate trace estimation using quantum state space linear algebra

In this paper, we present a quantum algorithm for approximating multivariate traces, i.e. the traces of matrix products. Our research is motivated by the extensive utility of multivariate traces in elucidating spectral characteristics of matrices, as well as by recent advancements in leveraging quantum computing for faster numerical linear algebra. Central to our approach is a direct translation of a multivariate trace formula into a quantum circuit, achieved through a sequence of low-level circuit construction operations. To facilitate this translation, we introduce \emph{quantum Matrix States Linear Algebra} (qMSLA), a framework tailored for the efficient generation of state preparation circuits via primitive matrix algebra operations. Our algorithm relies on sets of state preparation circuits for input matrices as its primary inputs and yields two state preparation circuits encoding the multivariate trace as output. These circuits are constructed utilizing qMSLA operations, which enact the aforementioned multivariate trace formula. We emphasize that our algorithm's inputs consist solely of state preparation circuits, eschewing harder to synthesize constructs such as Block Encodings. Furthermore, our approach operates independently of the availability of specialized hardware like QRAM, underscoring its versatility and practicality.

quant-ph↗

arXiv2024

On the Role of Initialization on the Implicit Bias in Deep Linear Networks

Despite Deep Learning's (DL) empirical success, our theoretical understanding of its efficacy remains limited. One notable paradox is that while conventional wisdom discourages perfect data fitting, deep neural networks are designed to do just that, yet they generalize effectively. This study focuses on exploring this phenomenon attributed to the implicit bias at play. Various sources of implicit bias have been identified, such as step size, weight initialization, optimization algorithm, and number of parameters. In this work, we focus on investigating the implicit bias originating from weight initialization. To this end, we examine the problem of solving underdetermined linear systems in various contexts, scrutinizing the impact of initialization on the implicit regularization when using deep networks to solve such systems. Our findings elucidate the role of initialization in the optimization and generalization paradoxes, contributing to a more comprehensive understanding of DL's performance characteristics.

cs.LG↗

arXiv2024

Multi-Function Multi-Way Analog Technology for Sustainable Machine Intelligence Computation

Numerical computation is essential to many areas of artificial intelligence (AI), whose computing demands continue to grow dramatically, yet their continued scaling is jeopardized by the slowdown in Moore's law. Multi-function multi-way analog (MFMWA) technology, a computing architecture comprising arrays of memristors supporting in-memory computation of matrix operations, can offer tremendous improvements in computation and energy, but at the expense of inherent unpredictability and noise. We devise novel randomized algorithms tailored to MFMWA architectures that mitigate the detrimental impact of imperfect analog computations while realizing their potential benefits across various areas of AI, such as applications in computer vision. Through analysis, measurements from analog devices, and simulations of larger systems, we demonstrate orders of magnitude reduction in both computation and energy with accuracy similar to digital computers.

math.NA↗

arXiv2023

Low-Rank Updates of Matrix Square Roots

Models in which the covariance matrix has the structure of a sparse matrix plus a low rank perturbation are ubiquitous in data science applications. It is often desirable for algorithms to take advantage of such structures, avoiding costly matrix computations that often require cubic time and quadratic storage. This is often accomplished by performing operations that maintain such structures, e.g. matrix inversion via the Sherman-Morrison-Woodbury formula. In this paper we consider the matrix square root and inverse square root operations. Given a low rank perturbation to a matrix, we argue that a low-rank approximate correction to the (inverse) square root exists. We do so by establishing a geometric decay bound on the true correction's eigenvalues. We then proceed to frame the correction as the solution of an algebraic Riccati equation, and discuss how a low-rank solution to that equation can be computed. We analyze the approximation error incurred when approximately solving the algebraic Riccati equation, providing spectral and Frobenius norm forward and backward error bounds. Finally, we describe several applications of our algorithms, and demonstrate their utility in numerical experiments.

math.NA↗

arXiv2022

Riemannian optimization with a preconditioning scheme on the generalized Stiefel manifold

Optimization problems on the generalized Stiefel manifold (and products of it) are prevalent across science and engineering. For example, in computational science they arise in symmetric (generalized) eigenvalue problems, in nonlinear eigenvalue problems, and in electronic structures computations, to name a few problems. In statistics and machine learning, they arise, for example, in various dimensionality reduction techniques such as canonical correlation analysis. In deep learning, regularization and improved stability can be obtained by constraining some layers to have parameter matrices that belong to the Stiefel manifold. Solving problems on the generalized Stiefel manifold can be approached via the tools of Riemannian optimization. However, using the standard geometric components for the generalized Stiefel manifold has two possible shortcomings: computing some of the geometric components can be too expensive and convergence can be rather slow in certain cases. Both shortcomings can be addressed using a technique called Riemannian preconditioning, which amounts to using geometric components derived by a precoditioner that defines a Riemannian metric on the constraint manifold. In this paper we develop the geometric components required to perform Riemannian optimization on the generalized Stiefel manifold equipped with a non-standard metric, and illustrate theoretically and numerically the use of those components and the effect of Riemannian preconditioning for solving optimization problems on the generalized Stiefel manifold.

math.NA↗

arXiv2022

Faster Randomized Interior Point Methods for Tall/Wide Linear Programs

Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such as combinatorics. It is also used in many machine learning applications, such as $\ell_1$-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider both feasible and infeasible IPMs for the special case where the number of variables is much larger than the number of constraints. Using tools from Randomized Linear Algebra, we present a preconditioning technique that, when combined with the iterative solvers such as Conjugate Gradient or Chebyshev Iteration, provably guarantees that IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real-world and synthetic data.

cs.DS↗

arXiv2022

Manifold Free Riemannian Optimization

Riemannian optimization is a principled framework for solving optimization problems where the desired optimum is constrained to a smooth manifold $\mathcal{M}$. Algorithms designed in this framework usually require some geometrical description of the manifold, which typically includes tangent spaces, retractions, and gradients of the cost function. However, in many cases, only a subset (or none at all) of these elements can be accessed due to lack of information or intractability. In this paper, we propose a novel approach that can perform approximate Riemannian optimization in such cases, where the constraining manifold is a submanifold of $\R^{D}$. At the bare minimum, our method requires only a noiseless sample set of the cost function $(\x_{i}, y_{i})\in {\mathcal{M}} \times \mathbb{R}$ and the intrinsic dimension of the manifold $\mathcal{M}$. Using the samples, and utilizing the Manifold-MLS framework (Sober and Levin 2020), we construct approximations of the missing components entertaining provable guarantees and analyze their computational costs. In case some of the components are given analytically (e.g., if the cost function and its gradient are given explicitly, or if the tangent spaces can be computed), the algorithm can be easily adapted to use the accurate expressions instead of the approximations. We analyze the global convergence of Riemannian gradient-based methods using our approach, and we demonstrate empirically the strength of this method, together with a conjugate-gradients type method based upon similar principles.

math.OC↗

arXiv2022

Solving Trust Region Subproblems Using Riemannian Optimization

The Trust Region Subproblem is a fundamental optimization problem that takes a pivotal role in Trust Region Methods. However, the problem, and variants of it, also arise in quite a few other applications. In this article, we present a family of iterative Riemannian optimization algorithms for a variant of the Trust Region Subproblem that replaces the inequality constraint with an equality constraint, and converge to a global optimum. Our approach uses either a trivial or a non-trivial Riemannian geometry of the search-space, and requires only minimal spectral information about the quadratic component of the objective function. We further show how the theory of Riemannian optimization promotes a deeper understanding of the Trust Region Subproblem and its difficulties, e.g., a deep connection between the Trust Region Subproblem and the problem of finding affine eigenvectors, and a new examination of the so-called hard case in light of the condition number of the Riemannian Hessian operator at a global optimum. Finally, we propose to incorporate preconditioning via a careful selection of a variable Riemannian metric, and establish bounds on the asymptotic convergence rate in terms of how well the preconditioner approximates the input matrix.

math.OC↗