arXiv subjects

Bart Vandereycken

Publications and source records attributed to Bart Vandereycken.

At least 19 recordsLinked to original sources

arXiv2026

A counterexample to the symmetric-maximizer conjecture for Lyapunov operators

It has been conjectured that the operator norm of the Lyapunov operator induced by the Frobenius norm is always attained at a symmetric matrix. The conjecture is known to hold for all matrices of order at most five. We give an integer matrix of order seven for which the skew-symmetric restricted norm is strictly larger than the symmetric restricted norm. A rational separator and exact-arithmetic certificates establish the strict inequality without relying on floating-point computations. A direct-sum construction yields counterexamples in every order $n \geq 7$; the case $n = 6$ remains open.

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arXiv2025

Interpolatory Dynamical Low-Rank Approximation: Theoretical Foundations and Algorithms

Dynamical low-rank approximation (DLRA) is a widely used paradigm for solving large-scale matrix differential equations, as they arise, for example, from the discretization of time-dependent partial differential equations on tensorized domains. Through orthogonally projecting the dynamics onto the tangent space of a low-dimensional manifold, DLRA achieves a significant reduction of the storage required to represent the solution. However, the need for evaluating the velocity field can make it challenging to attain a corresponding reduction of computational cost in the presence of nonlinearities. In this work, we address this challenge by replacing orthogonal tangent space projections with oblique, data-sparse projections selected by a discrete empirical interpolation method (DEIM). At the continuous-time level, this leads to DLRA-DEIM, a well-posed differential inclusion (in the Filippov sense) that captures the discontinuities induced by changes in the indices selected by DEIM. We establish an existence result, exactness property and error bound for DLRA-DEIM that match existing results for DLRA. For the particular case of QDEIM, a popular variant of DEIM, we provide an explicit convex-polytope characterization of the differential inclusion. Building on DLRA-DEIM, we propose a new class of projected integrators, called PRK-DEIM, that combines explicit Runge--Kutta methods with DEIM-based projections. We analyze the convergence order of PRK-DEIM and show that it matches the accuracy of previously proposed projected Runge-Kutta methods, while being significantly cheaper. Extensions to exponential Runge--Kutta methods and low-order tensor differential equations demonstrate the versatility of our framework.

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arXiv2025

Gauss-Southwell type descent methods for low-rank matrix optimization

We consider gradient-related methods for low-rank matrix optimization with a smooth cost function. The methods operate on single factors of the low-rank factorization and share aspects of both alternating and Riemannian optimization. Two possible choices for the search directions based on Gauss-Southwell type selection rules are compared: one using the gradient of a factorized non-convex formulation, the other using the Riemannian gradient. While both methods provide gradient convergence guarantees that are similar to the unconstrained case, numerical experiments on a quadratic cost function indicate that the version based on the Riemannian gradient is significantly more robust with respect to small singular values and the condition number of the cost function. As a side result of our approach, we also obtain new convergence results for the alternating least squares method.

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arXiv2025

Riemannian optimization using three different metrics for Hermitian PSD fixed-rank constraints: an extended version

For smooth optimization problems with a Hermitian positive semi-definite fixed-rank constraint, we consider three existing approaches including the simple Burer--Monteiro method, and Riemannian optimization over quotient geometry and the embedded geometry. These three methods can be all represented via quotient geometry with three Riemannian metrics $g^i(\cdot, \cdot)$ $(i=1,2,3)$. By taking the nonlinear conjugate gradient method (CG) as an example, we show that CG in the factor-based Burer--Monteiro approach is equivalent to Riemannian CG on the quotient geometry with the Bures-Wasserstein metric $g^1$. Riemannian CG on the quotient geometry with the metric $g^3$ is equivalent to Riemannian CG on the embedded geometry. For comparing the three approaches, we analyze the condition number of the Riemannian Hessian near the minimizer under the three different metrics. Under certain assumptions, the condition number from the Bures-Wasserstein metric $g^1$ is significantly different from the other two metrics. Numerical experiments show that the Burer--Monteiro CG method has obviously slower asymptotic convergence rate when the minimizer is rank deficient, which is consistent with the condition number analysis.

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arXiv2025

A geodesic convexity-like structure for the polar decomposition of a square matrix

We make a full landscape analysis of the (generally non-convex) orthogonal Procrustes problem. This problem is equivalent to computing the polar factor of a square matrix. We reveal a convexity-like structure, which explains the already established tractability of the problem and show that gradient descent in the orthogonal group computes the polar factor of a square matrix with linear convergence rate if the matrix is invertible and with an algebraic one if the matrix is singular. These results are similar to the ones of Alimisis and Vandereycken (2024) for the symmetric eigenvalue problem.

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arXiv2024

A preconditioned inverse iteration with an improved convergence guarantee

Preconditioned eigenvalue solvers offer the possibility to incorporate preconditioners for the solution of large-scale eigenvalue problems, as they arise from the discretization of partial differential equations. The convergence analysis of such methods is intricate. Even for the relatively simple preconditioned inverse iteration (PINVIT), which targets the smallest eigenvalue of a symmetric positive definite matrix, the celebrated analysis by Neymeyr is highly nontrivial and only yields convergence if the starting vector is fairly close to the desired eigenvector. In this work, we prove a new non-asymptotic convergence result for a variant of PINVIT. Our proof proceeds by analyzing an equivalent Riemannian steepest descent method and leveraging convexity-like properties. We show a convergence rate that nearly matches the one of PINVIT. As a major benefit, we require a condition on the starting vector that tends to be less stringent. This improved global convergence property is demonstrated for two classes of preconditioners with theoretical bounds and a range of numerical experiments.

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arXiv2024

Randomized sketching of nonlinear eigenvalue problems

Rational approximation is a powerful tool to obtain accurate surrogates for nonlinear functions that are easy to evaluate and linearize. The interpolatory adaptive Antoulas--Anderson (AAA) method is one approach to construct such approximants numerically. For large-scale vector- and matrix-valued functions, however, the direct application of the set-valued variant of AAA becomes inefficient. We propose and analyze a new sketching approach for such functions called sketchAAA that, with high probability, leads to much better approximants than previously suggested approaches while retaining efficiency. The sketching approach works in a black-box fashion where only evaluations of the nonlinear function at sampling points are needed. Numerical tests with nonlinear eigenvalue problems illustrate the efficacy of our approach, with speedups above 200 for sampling large-scale black-box functions without sacrificing on accuracy.

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arXiv2024

A Nesterov-style Accelerated Gradient Descent Algorithm for the Symmetric Eigenvalue Problem

We develop an accelerated gradient descent algorithm on the Grassmann manifold to compute the subspace spanned by a number of leading eigenvectors of a symmetric positive semi-definite matrix. This has a constant cost per iteration and a provable iteration complexity of $\tilde{\mathcal{O}}(1/\sqrtδ)$, where $δ$ is the spectral gap and $\tilde{\mathcal{O}}$ hides logarithmic factors. This improves over the $\tilde{\mathcal{O}}(1/δ)$ complexity achieved by subspace iteration and standard gradient descent, in cases that the spectral gap is tiny. It also matches the iteration complexity of the Lanczos method that has however a growing cost per iteration. On the theoretical part, we rely on the formulation of Riemannian accelerated gradient descent by [26] and new characterizations of the geodesic convexity of the symmetric eigenvalue problem by [8]. On the empirical part, we test our algorithm in synthetic and real matrices and compare with other popular methods.

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arXiv2024

Projected exponential methods for stiff dynamical low-rank approximation problems

The numerical integration of stiff equations is a challenging problem that needs to be approached by specialized numerical methods. Exponential integrators form a popular class of such methods since they are provably robust to stiffness and have been successfully applied to a variety of problems. The dynamical low- \rank approximation is a recent technique for solving high-dimensional differential equations by means of low-rank approximations. However, the domain is lacking numerical methods for stiff equations since existing methods are either not robust-to-stiffness or have unreasonably large hidden constants. In this paper, we focus on solving large-scale stiff matrix differential equations with a Sylvester-like structure, that admit good low-rank approximations. We propose two new methods that have good convergence properties, small memory footprint and that are fast to compute. The theoretical analysis shows that the new methods have order one and two, respectively. We also propose a practical implementation based on Krylov techniques. The approximation error is analyzed, leading to a priori error bounds and, therefore, a mean for choosing the size of the Krylov space. Numerical experiments are performed on several examples, confirming the theory and showing good speedup in comparison to existing techniques.

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arXiv2024

Gradient-type subspace iteration methods for the symmetric eigenvalue problem

This paper explores variants of the subspace iteration algorithm for computing approximate invariant subspaces. The standard subspace iteration approach is revisited and new variants that exploit gradient-type techniques combined with a Grassmann manifold viewpoint are developed. A gradient method as well as a nonlinear conjugate gradient technique are described. Convergence of the gradient-based algorithm is analyzed and a few numerical experiments are reported, indicating that the proposed algorithms are sometimes superior to standard algorithms. This includes the Chebyshev-based subspace iteration and the locally optimal block conjugate gradient method, when compared in terms of number of matrix vector products and computational time, resp. The new methods, on the other hand, do not require estimating optimal parameters. An important contribution of this paper to achieve this good performance is the accurate and efficient implementation of an exact line search. In addition, new convergence proofs are presented for the non-accelerated gradient method that includes a locally exponential convergence if started in a $\mathcal{O(\sqrtδ)}$ neighbourhood of the dominant subspace with spectral gap $δ$.

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arXiv2023

Geodesic Convexity of the Symmetric Eigenvalue Problem and Convergence of Riemannian Steepest Descent

We study the convergence of the Riemannian steepest descent algorithm on the Grassmann manifold for minimizing the block version of the Rayleigh quotient of a symmetric matrix. Even though this problem is non-convex in the Euclidean sense and only very locally convex in the Riemannian sense, we discover a structure for this problem that is similar to geodesic strong convexity, namely, weak-strong convexity. This allows us to apply similar arguments from convex optimization when studying the convergence of the steepest descent algorithm but with initialization conditions that do not depend on the eigengap $δ$. When $δ>0$, we prove exponential convergence rates, while otherwise the convergence is algebraic. Additionally, we prove that this problem is geodesically convex in a neighbourhood of the global minimizer of radius $\mathcal{O}(\sqrtδ)$.

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arXiv2023

Implicit low-rank Riemannian schemes for the time integration of stiff partial differential equations

We propose two implicit numerical schemes for the low-rank time integration of stiff nonlinear partial differential equations. Our approach uses the preconditioned Riemannian trust-region method of Absil, Baker, and Gallivan, 2007. We demonstrate the efficiency of our method for solving the Allen-Cahn and the Fisher-KPP equation on the manifold of fixed-rank matrices. Furthermore, our approach allows us to avoid the restriction on the time step typical of methods that use the fixed-point iteration to solve the inner nonlinear equations. Finally, we demonstrate the efficiency of the preconditioner on the same variational problems presented in Sutti and Vandereycken, 2021.

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arXiv2023

The leapfrog algorithm as nonlinear Gauss-Seidel

Several applications in optimization, image, and signal processing deal with data that belong to the Stiefel manifold St(n,p), that is, the set of n-by-p matrices with orthonormal columns. Some applications, like the Riemannian center of mass, require evaluating the Riemannian distance between two arbitrary points on St(n,p). This can be done by explicitly constructing the geodesic connecting these two points. An existing method for finding geodesics is the leapfrog algorithm of J. L. Noakes. This algorithm is related to the Gauss-Seidel method, a classical iterative method for solving a linear system of equations that can be extended to nonlinear systems. We propose a convergence proof of leapfrog as a nonlinear Gauss-Seidel method. Our discussion is limited to the case of the Stiefel manifold, however, it may be generalized to other embedded submanifolds. We discuss other aspects of leapfrog and present some numerical experiments.

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arXiv2022

Low-rank Parareal: a low-rank parallel-in-time integrator

In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge--Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.

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arXiv2022

Streaming Tensor Train Approximation

Tensor trains are a versatile tool to compress and work with high-dimensional data and functions. In this work we introduce the Streaming Tensor Train Approximation (STTA), a new class of algorithms for approximating a given tensor $\mathcal T$ in the tensor train format. STTA accesses $\mathcal T$ exclusively via two-sided random sketches of the original data, making it streamable and easy to implement in parallel -- unlike existing deterministic and randomized tensor train approximations. This property also allows STTA to conveniently leverage structure in $\mathcal T$, such as sparsity and various low-rank tensor formats, as well as linear combinations thereof. When Gaussian random matrices are used for sketching, STTA is admissible to an analysis that builds and extends upon existing results on the generalized Nyström approximation for matrices. Our results show that STTA can be expected to attain a nearly optimal approximation error if the sizes of the sketches are suitably chosen. A range of numerical experiments illustrates the performance of STTA compared to existing deterministic and randomized approaches.

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arXiv2022

A note on the optimal convergence rate of descent methods with fixed step sizes for smooth strongly convex functions

Based on a result by Taylor, Hendrickx, and Glineur (J. Optim. Theory Appl., 178(2):455--476, 2018) on the attainable convergence rate of gradient descent for smooth and strongly convex functions in terms of function values, an elementary convergence analysis for general descent methods with fixed step sizes is presented. It covers general variable metric methods, gradient related search directions under angle and scaling conditions, as well as inexact gradient methods. In all cases, optimal rates are obtained.

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arXiv2022

TTML: tensor trains for general supervised machine learning

This work proposes a novel general-purpose estimator for supervised machine learning (ML) based on tensor trains (TT). The estimator uses TTs to parametrize discretized functions, which are then optimized using Riemannian gradient descent under the form of a tensor completion problem. Since this optimization is sensitive to initialization, it turns out that the use of other ML estimators for initialization is crucial. This results in a competitive, fast ML estimator with lower memory usage than many other ML estimators, like the ones used for the initialization.

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arXiv2021

Distributed Principal Component Analysis with Limited Communication

We study efficient distributed algorithms for the fundamental problem of principal component analysis and leading eigenvector computation on the sphere, when the data are randomly distributed among a set of computational nodes. We propose a new quantized variant of Riemannian gradient descent to solve this problem, and prove that the algorithm converges with high probability under a set of necessary spherical-convexity properties. We give bounds on the number of bits transmitted by the algorithm under common initialization schemes, and investigate the dependency on the problem dimension in each case.

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