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Block Discrete Empirical Interpolation Methods

Perfect Y. Gidisu, Michiel E. Hochstenbach

TL;DR

This work addresses efficient, deterministic index selection for CUR decompositions by introducing block DEIM variants that select blocks of indices using either MaxVol or rank-revealing QR, plus an adaptive block DEIM variant. The methods aim to preserve the accuracy of standard DEIM while delivering computational speedups, particularly for large-scale data, and are supported by an established error framework tied to the singular value sequence via $\sigma_{k+1}$. Theoretical guarantees are provided through a CUR error bound involving constants $\eta_s$ and $\eta_p$, and empirical results demonstrate favorable accuracy–speed tradeoffs compared to existing deterministic index-selection methods. The proposed approaches expand the toolbox for interpretable, low-rank approximations with practical impact on large-scale data analysis, model order reduction, and related applications, with public software available.

Abstract

We present block variants of the discrete empirical interpolation method (DEIM); as a particular application, we will consider a CUR factorization. The block DEIM algorithms are based on the concept of the maximum volume of submatrices and a rank-revealing QR factorization. We also present a version of the block DEIM procedures, which allows for adaptive choice of block size. The results of the experiments indicate that the block DEIM algorithms exhibit comparable accuracy for low-rank matrix approximation compared to the standard DEIM procedure. However, the block DEIM algorithms also demonstrate potential computational advantages, showcasing increased efficiency in terms of computational time.

Block Discrete Empirical Interpolation Methods

TL;DR

This work addresses efficient, deterministic index selection for CUR decompositions by introducing block DEIM variants that select blocks of indices using either MaxVol or rank-revealing QR, plus an adaptive block DEIM variant. The methods aim to preserve the accuracy of standard DEIM while delivering computational speedups, particularly for large-scale data, and are supported by an established error framework tied to the singular value sequence via . Theoretical guarantees are provided through a CUR error bound involving constants and , and empirical results demonstrate favorable accuracy–speed tradeoffs compared to existing deterministic index-selection methods. The proposed approaches expand the toolbox for interpretable, low-rank approximations with practical impact on large-scale data analysis, model order reduction, and related applications, with public software available.

Abstract

We present block variants of the discrete empirical interpolation method (DEIM); as a particular application, we will consider a CUR factorization. The block DEIM algorithms are based on the concept of the maximum volume of submatrices and a rank-revealing QR factorization. We also present a version of the block DEIM procedures, which allows for adaptive choice of block size. The results of the experiments indicate that the block DEIM algorithms exhibit comparable accuracy for low-rank matrix approximation compared to the standard DEIM procedure. However, the block DEIM algorithms also demonstrate potential computational advantages, showcasing increased efficiency in terms of computational time.
Paper Structure (11 sections, 1 theorem, 12 equations, 6 figures, 4 tables, 4 algorithms)

This paper contains 11 sections, 1 theorem, 12 equations, 6 figures, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Review of CUR factorization algorithms
  3. MaxVol Algorithm
  4. Discrete Empirical Interpolation Method
  5. Pivoted QR factorization
  6. Block DEIM
  7. Block DEIM based on maximum volume/RRQR
  8. Adaptive block DEIM
  9. Error bounds
  10. Numerical Experiments
  11. Conclusions

Key Result

Proposition 3.1

Sorensen Given $A\in \mathbb R^{m \times n}$ and a target rank $k$, let $U\in \mathbb R^{m\times k}$ and $V\in \mathbb R^{n\times k}$ contain the leading $k$ left and right singular vectors of $A$, respectively. Suppose $C=AP$ and $R=S^T\!A$ are of full rank, and $V^T\!P$ and $S^TU$ are nonsingular. where $\eta_\mathbf p=\lVert(V^T\!P)^{-1}\rVert$, $\eta_\mathbf s=\lVert(S^TU)^{-1}\rVert$.

Figures (6)

  • Figure 1: Relative approximation errors (left) and runtimes (right) as a function of $k$ for the block DEIM CUR approximation algorithms compared with some standard CUR approximation algorithms using the Jester data set.
  • Figure 2: Relative approximation errors (left) and runtimes (right) as a function of $k$ for the block DEIM CUR approximation algorithms compared with some standard CUR approximation algorithms using the g7jac100 sparse matrix.
  • Figure 3: Relative approximation errors (left) and runtimes (right) as a function of $k$ for the block DEIM CUR approximation algorithms compared with some standard CUR approximation algorithms using the net100 sparse matrix.
  • Figure 4: Relative approximation errors (left) and runtimes (right) as a function of $k$ for the block DEIM CUR approximation algorithms compared with some standard CUR approximation algorithms using the Abacusa-shell-ud sparse matrix.
  • Figure 5: Relative approximation errors (left) and runtimes (right) as a function of $k$ for the block DEIM CUR approximation algorithms compared with some standard CUR approximation algorithms using the pkustk01 sparse matrix.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Proposition 3.1
  • Example 3.2