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Efficient error estimators for Generalized Nyström

Lorenzo Lazzarino, Katherine J. Pearce, Nathaniel Pritchard

TL;DR

This work extends the leave-one-out framework to the generalized Nystr\"om decomposition, an approach that can be applied to general rectangular matrices and derives three new leave-one-out error estimators and validates their effectiveness through numerical experiments.

Abstract

Randomized algorithms in numerical linear algebra have proven to be effective in ameliorating issues of scalability when working with large matrices, efficiently producing accurate low-rank approximations. A key remaining challenge, however, is to efficiently assess the approximation accuracy of randomized methods without additional expensive matrix accesses. Recent work has addressed this issue by deriving fast leave-one-out error estimators for the randomized SVD and Nyström decomposition, enabling accurate error estimation with no additional matrix accesses. In this work, we extend the leave-one-out framework to the generalized Nyström decomposition, an approach that can be applied to general rectangular matrices. We do this by deriving three new leave-one-out error estimators and validating their effectiveness through numerical experiments.

Efficient error estimators for Generalized Nyström

TL;DR

This work extends the leave-one-out framework to the generalized Nystr\"om decomposition, an approach that can be applied to general rectangular matrices and derives three new leave-one-out error estimators and validates their effectiveness through numerical experiments.

Abstract

Randomized algorithms in numerical linear algebra have proven to be effective in ameliorating issues of scalability when working with large matrices, efficiently producing accurate low-rank approximations. A key remaining challenge, however, is to efficiently assess the approximation accuracy of randomized methods without additional expensive matrix accesses. Recent work has addressed this issue by deriving fast leave-one-out error estimators for the randomized SVD and Nyström decomposition, enabling accurate error estimation with no additional matrix accesses. In this work, we extend the leave-one-out framework to the generalized Nyström decomposition, an approach that can be applied to general rectangular matrices. We do this by deriving three new leave-one-out error estimators and validating their effectiveness through numerical experiments.
Paper Structure (19 sections, 30 equations, 4 figures)

This paper contains 19 sections, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Matrix decompositions
  5. The QR Decomposition
  6. The Singular Value Decomposition
  7. Randomized rangefinding
  8. Nyström and generalized Nyström approximation s
  9. Leave-one-out error estimation
  10. Fast estimation of approximation error for generalized Nyström
  11. The LPO and LTO formulas
  12. Leave-Pair-out (LPO) replicate formula
  13. Leave-Twins-out (LTO) replicate formula
  14. The LRO formula
  15. Numerical Experiments
  16. ...and 4 more sections

Figures (4)

  • Figure 1: Non-discrepant case for exponentially decaying singular values. (Left) Error from the generalized Nyström approximation, with the naïve (brute) computations (solid) lines and fast replicate computations (dashed). (Right) Runtime plot.
  • Figure 2: Discrepant case for exponentially decaying singular values. (Left) Error from the generalized Nyström approximation with the naïve (brute) computations (solid) lines and fast replicate computations (dashed). (Right) Runtime plot.
  • Figure 3: Error for the non-discrepant case (left) and discrepant case (right) of the generalized Nyström approximation of the Chan matrix.
  • Figure 4: Non-discrepant case for the advection-diffusion equations. (Left) Error from the generalized Nyström approximation, with the naïve (brute) computations (solid) lines and fast replicate computations (dashed). (Right) Runtime plot.