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A unified error analysis for randomized low-rank approximation with application to data assimilation

Alexandre Scotto Di Perrotolo, Youssef Diouane, Selime Gürol, Xavier Vasseur

TL;DR

This paper introduces a unified stochastic framework for the Frobenius-norm low-rank approximation error of randomized methods, valid for Z = A(A^T A)^q L G with Gaussian G and arbitrary L satisfying minimal conditions. It derives sharp bounds in expectation and probability that generalize and tighten prior results, showing how the choice of L and q recovers RSVD, power iterations, and generalized RSVD analyses. The framework exposes how problem structure, via covariance design, improves approximation quality, and numerical experiments in data assimilation confirm practical gains from covariance-aware range-finders and hybrid information strategies. The work lays a foundation for covariance-informed randomized algorithms, with potential impact on preconditioning and large-scale data assimilation workflows.

Abstract

Randomized algorithms have proven to perform well on a large class of numerical linear algebra problems. Their theoretical analysis is critical to provide guarantees on their behaviour, and in this sense, the stochastic analysis of the randomized low-rank approximation error plays a central role. Indeed, several randomized methods for the approximation of dominant eigen- or singular modes can be rewritten as low-rank approximation methods. However, despite the large variety of algorithms, the existing theoretical frameworks for their analysis rely on a specific structure for the covariance matrix that is not adapted to all the algorithms. We propose a unified framework for the stochastic analysis of the low-rank approximation error in Frobenius norm for centered and non-standard Gaussian matrices. Under minimal assumptions on the covariance matrix, we derive accurate bounds both in expectation and probability. Our bounds have clear interpretations that enable us to derive properties and motivate practical choices for the covariance matrix resulting in efficient low-rank approximation algorithms. The most commonly used bounds in the literature have been demonstrated as a specific instance of the bounds proposed here, with the additional contribution of being tighter. Numerical experiments related to data assimilation further illustrate that exploiting the problem structure to select the covariance matrix improves the performance as suggested by our bounds.

A unified error analysis for randomized low-rank approximation with application to data assimilation

TL;DR

This paper introduces a unified stochastic framework for the Frobenius-norm low-rank approximation error of randomized methods, valid for Z = A(A^T A)^q L G with Gaussian G and arbitrary L satisfying minimal conditions. It derives sharp bounds in expectation and probability that generalize and tighten prior results, showing how the choice of L and q recovers RSVD, power iterations, and generalized RSVD analyses. The framework exposes how problem structure, via covariance design, improves approximation quality, and numerical experiments in data assimilation confirm practical gains from covariance-aware range-finders and hybrid information strategies. The work lays a foundation for covariance-informed randomized algorithms, with potential impact on preconditioning and large-scale data assimilation workflows.

Abstract

Randomized algorithms have proven to perform well on a large class of numerical linear algebra problems. Their theoretical analysis is critical to provide guarantees on their behaviour, and in this sense, the stochastic analysis of the randomized low-rank approximation error plays a central role. Indeed, several randomized methods for the approximation of dominant eigen- or singular modes can be rewritten as low-rank approximation methods. However, despite the large variety of algorithms, the existing theoretical frameworks for their analysis rely on a specific structure for the covariance matrix that is not adapted to all the algorithms. We propose a unified framework for the stochastic analysis of the low-rank approximation error in Frobenius norm for centered and non-standard Gaussian matrices. Under minimal assumptions on the covariance matrix, we derive accurate bounds both in expectation and probability. Our bounds have clear interpretations that enable us to derive properties and motivate practical choices for the covariance matrix resulting in efficient low-rank approximation algorithms. The most commonly used bounds in the literature have been demonstrated as a specific instance of the bounds proposed here, with the additional contribution of being tighter. Numerical experiments related to data assimilation further illustrate that exploiting the problem structure to select the covariance matrix improves the performance as suggested by our bounds.
Paper Structure (24 sections, 8 theorems, 85 equations, 11 figures, 1 table, 1 algorithm)

This paper contains 24 sections, 8 theorems, 85 equations, 11 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Submultiplicativity
  4. Partial ordering on the set of symmetric matrices
  5. Projection matrices
  6. Sherman-Morrison formula
  7. Principal angles between subspaces
  8. Tangent matrix of the principal angles
  9. General randomized low-rank approximation error bounds
  10. Particular choices for the covariance matrix
  11. The RSVD ($\mathbf{L} = \mathbf{I}_n$ and $q=0$)
  12. The randomized power iteration ($\mathbf{L} = \mathbf{I}_n$ and $q \geq 0$)
  13. The generalized RSVD ($q=0$)
  14. $\mathbf{L}$ as a low-rank matrix update
  15. Numerical experiments
  16. ...and 9 more sections

Key Result

Theorem 3.1

Let $\mathbf{A} \in \mathbb{R}^{m \times n}$ be an arbitrary matrix, and $\mathbf{U}_k$, $\bm{\Sigma}_k$ and $\overline{\bm{\Sigma}}_k$ the matrices related to the singular value decomposition of $\mathbf{A}$ and defined in eq:partitioning_svd. Let $\mathbf{D} \in \mathbb{R}^{m \times r}$ be a matri Moreover, if $\ell \geq k + 4$, then for all $u, t \geq 1$, holds with probability at least $1 - e

Figures (11)

  • Figure 1: Eigenvalue distribution of $\mathbf{A}$ in the LowObs and HighObs scenarios.
  • Figure 2: Bounds for the low-rank approximation error versus the target rank $k$ with $p=10$.
  • Figure 3: Bounds for the low-rank approximation error versus the oversampling parameter $p = \ell - k$ with $k=20$.
  • Figure 4: Bounds for the low-rank approximation error versus the target rank $k$ with $p=10$ and $\mathbf{L} = \mathbf{W}$.
  • Figure 5: Bounds for the low-rank approximation error versus the oversampling parameter $p = \ell - k$ with $k=20$ and $\mathbf{L} = \mathbf{W}$.
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Corollary 4.1
  • proof
  • Corollary 4.2
  • proof
  • Corollary 4.3
  • proof
  • Lemma A.1
  • ...and 6 more