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Low-rank approximation of parameter-dependent matrices via CUR decomposition

Taejun Park, Yuji Nakatsukasa

TL;DR

This work tackles efficient low-rank approximation of parameter-dependent matrices $A(t)$ via CUR decompositions. It introduces AdaCUR, a rank-adaptive, error-controlled method, and FastAdaCUR, a faster variant with near-linear complexity after initialization; both exploit the insight that indices identified for nearby parameter values remain informative. The algorithms rely on randomized tools—random embeddings, pivoting on random sketches, and randomized rank/norm estimation—with oversampling and a buffer to robustly adjust indices as $t$ changes. Empirical results demonstrate AdaCUR’s accuracy guarantees and competitive speed, while FastAdaCUR delivers substantial speed gains though it lacks formal error control and can be vulnerable to adversarial inputs. The methods offer practical impact for dynamical systems, image and data series compression, and other scenarios where parameter-dependent matrices arise, enabling scalable, data-driven CUR approximations with controllable or high-throughput performance.

Abstract

A low-rank approximation of a parameter-dependent matrix $A(t)$ is an important task in the computational sciences appearing for example in dynamical systems and compression of a series of images. In this work, we introduce AdaCUR, an efficient algorithm for computing a low-rank approximation of parameter-dependent matrices via CUR decompositions. The key idea for this algorithm is that for nearby parameter values, the column and row indices for the CUR decomposition can often be reused. AdaCUR is rank-adaptive, provides error control, and has complexity that compares favorably against existing methods. A faster algorithm which we call FastAdaCUR that prioritizes speed over accuracy is also given, which is rank-adaptive and has complexity which is at most linear in the number of rows or columns, but without error control.

Low-rank approximation of parameter-dependent matrices via CUR decomposition

TL;DR

This work tackles efficient low-rank approximation of parameter-dependent matrices via CUR decompositions. It introduces AdaCUR, a rank-adaptive, error-controlled method, and FastAdaCUR, a faster variant with near-linear complexity after initialization; both exploit the insight that indices identified for nearby parameter values remain informative. The algorithms rely on randomized tools—random embeddings, pivoting on random sketches, and randomized rank/norm estimation—with oversampling and a buffer to robustly adjust indices as changes. Empirical results demonstrate AdaCUR’s accuracy guarantees and competitive speed, while FastAdaCUR delivers substantial speed gains though it lacks formal error control and can be vulnerable to adversarial inputs. The methods offer practical impact for dynamical systems, image and data series compression, and other scenarios where parameter-dependent matrices arise, enabling scalable, data-driven CUR approximations with controllable or high-throughput performance.

Abstract

A low-rank approximation of a parameter-dependent matrix is an important task in the computational sciences appearing for example in dynamical systems and compression of a series of images. In this work, we introduce AdaCUR, an efficient algorithm for computing a low-rank approximation of parameter-dependent matrices via CUR decompositions. The key idea for this algorithm is that for nearby parameter values, the column and row indices for the CUR decomposition can often be reused. AdaCUR is rank-adaptive, provides error control, and has complexity that compares favorably against existing methods. A faster algorithm which we call FastAdaCUR that prioritizes speed over accuracy is also given, which is rank-adaptive and has complexity which is at most linear in the number of rows or columns, but without error control.
Paper Structure (35 sections, 1 theorem, 24 equations, 9 figures, 3 tables, 7 algorithms)

This paper contains 35 sections, 1 theorem, 24 equations, 9 figures, 3 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Brief outline of AdaCUR and FastAdaCUR
  3. Existing methods
  4. Contributions
  5. Notation
  6. Preliminaries
  7. Random embeddings
  8. Pivoting on a random sketch
  9. Randomized rank estimation
  10. Combining pivoting on a random sketch and randomized rank estimation
  11. Randomized norm estimation
  12. Multiple error estimation
  13. Proposed method
  14. Computing indices from scratch
  15. AdaCUR algorithm for accuracy
  16. ...and 20 more sections

Key Result

Theorem 2.1

GrattonTitley-Peloquin2018 Let $A\in \mathbb{R}^{m\times n}$ be a matrix, $\Gamma\in \mathbb{R}^{s\times m}$ be a Gaussian matrix with i.i.d. entries $\Gamma_{ij} \sim \mathcal{N}(0,1)$ and set $\rho:= \left\lVert A\right\rVert_F^2/\left\lVert A\right\rVert_2^2$. For any $\tau > 1$ and $s\leq m$,

Figures (9)

  • Figure 1: An overview of AdaCUR
  • Figure 1: Testing the gap between theory and practice for the CUR bound in \ref{['eq:CAboundOS']} using the same set of indices for every parameter value. The parameter-dependent matrix in this example is the same as the one in Section \ref{['subsec:RoleTol']} (Equation \ref{['eq:synprob']}). The target rank is $r = 33$ in this experiment.
  • Figure 1: Testing tolerance parameter $\epsilon$ for AdaCUR and FastAdaCUR using the synthetic problem \ref{['eq:synprob']}.
  • Figure 2: An overview of FastAdaCUR
  • Figure 2: Testing the oversampling parameter $p$ for AdaCUR and FastAdaCUR using the parametric cookie problem \ref{['eq:PCprob']}.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem 2.1