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Maximal Volume Matrix Cross Approximation for Image Compression and Least Squares Solution

Kenneth Allen, Ming-Jun Lai, Zhaiming Shen

TL;DR

This work advances matrix cross approximation by refining error bounds for CUR-type skeleton representations using maximal-volume submatrices and introducing a family of greedy maximum volume algorithms (GMVA) to efficiently locate near-optimal submatrices. The authors prove an improved residual bound, connect it with dominant-submatrix theory, and provide convergent algorithms that speed up computation. They validate the approach with numerical experiments, demonstrating effective image compression with low storage and fast, accurate least-squares solves via reduced subproblems and pivotal rows. The methods offer interpretable, low-rank approximations suitable for large-scale, partially observed, or structured data, with practical impact in image processing and data-fitting tasks.

Abstract

We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms to improve the computational efficiency of matrix cross approximation. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: image compression and the least squares approximation of continuous functions. Our numerical results at the end of the paper demonstrate the effective performance of our approach.

Maximal Volume Matrix Cross Approximation for Image Compression and Least Squares Solution

TL;DR

This work advances matrix cross approximation by refining error bounds for CUR-type skeleton representations using maximal-volume submatrices and introducing a family of greedy maximum volume algorithms (GMVA) to efficiently locate near-optimal submatrices. The authors prove an improved residual bound, connect it with dominant-submatrix theory, and provide convergent algorithms that speed up computation. They validate the approach with numerical experiments, demonstrating effective image compression with low storage and fast, accurate least-squares solves via reduced subproblems and pivotal rows. The methods offer interpretable, low-rank approximations suitable for large-scale, partially observed, or structured data, with practical impact in image processing and data-fitting tasks.

Abstract

We study the classic matrix cross approximation based on the maximal volume submatrices. Our main results consist of an improvement of the classic estimate for matrix cross approximation and a greedy approach for finding the maximal volume submatrices. More precisely, we present a new proof of the classic estimate of the inequality with an improved constant. Also, we present a family of greedy maximal volume algorithms to improve the computational efficiency of matrix cross approximation. The proposed algorithms are shown to have theoretical guarantees of convergence. Finally, we present two applications: image compression and the least squares approximation of continuous functions. Our numerical results at the end of the paper demonstrate the effective performance of our approach.
Paper Structure (12 sections, 14 theorems, 34 equations, 2 figures, 4 tables, 5 algorithms)

This paper contains 12 sections, 14 theorems, 34 equations, 2 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Improved error bounds
  3. Proof of Theorems \ref{['mjlai06062021']} and \ref{['mjlai06082021']}
  4. Greedy Maximum Volume Algorithms
  5. Numerical Experiments on GMVA and Applications
  6. Performance of GMVA
  7. Application to image compression
  8. Application to least squares approximation
  9. Conclusion and future research
  10. Funding
  11. Data Availability
  12. Conflict of Interest

Key Result

Theorem 1

Let $A$ be a matrix of size $m\times n$. Fix $r \in [1, \min\{m,n\})$. Suppose that $I \subset \{1,\cdots,m\}$ with $\#(I)=r$ and $J\subset \{1, \cdots, n\}$ with $\#(J)=r$ such that $A_{I,J}$ has the maximal volume among all $r \times r$ submatrices of $A$. Then the Chebyshev norm of the residual m where $A_r= A_{:,J} A_{I,J}^{-1} A_{I,:}$ is the $r$-cross approximation of $A$, $\sigma_{r+1}(A)$

Figures (2)

  • Figure 1: Comparison of original images, subsampled images, and reconstructed images
  • Figure 2: Left: Plot of some testing functions (top two: Franke and Ackley. bottom two: Rastrigin and wavy). Right: The entire data locations (red) and the pivotal data locations (blue).

Theorems & Definitions (17)

  • Theorem 1: cf. GT01
  • Theorem 2: cf. GH17
  • Theorem 3
  • Definition 1
  • Lemma 1
  • Lemma 2
  • Theorem 4
  • Lemma 3
  • Remark 1
  • Remark 2
  • ...and 7 more