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Determinant-Based Error Bounds for CUR Matrix Approximation: Oversampling and Volume Sampling

Frank de Hoog, Markus Hegland

TL;DR

The paper develops a determinant-based framework for CUR and Nyström matrix approximations by tying local bordered Gramian determinants to projection errors and employing volume sampling. It proves that oversampling reduces the expected Frobenius error in a controlled, linear fashion, with the error factor transitioning from (k+1)^2 to (k+1) as the sample size grows. A unified theory is established, connecting local determinant identities to global probabilistic bounds and expressing results in terms of the singular-value tail. These insights yield practical error guarantees and deepen the understanding of how determinant structure and oversampling govern the quality of CUR/Nyström low-rank approximations.

Abstract

We derive error bounds for CUR matrix approximation using determinant-based methods that relate local projection errors to global approximation quality. For general matrices, we establish determinant identities for bordered Gramian matrices that decompose CUR approximation errors into interpretable local components. These identities connect projection errors onto submatrix column spaces directly to determinants, providing geometric insight into approximation degradation. We develop a probabilistic framework based on volume sampling that yields interpolation-type error bounds quantifying the benefits of oversampling: when $r > k$ rows are selected for $k$ columns, the expected error factor transitions linearly from $(k+1)^2$ (no oversampling) to $(k+1)$ (full oversampling). Our analysis establishes that the expected squared error is bounded by this interpolation factor times the squared error of the best rank-$k$ approximation, directly connecting CUR approximation quality to the optimal low-rank approximation. The framework applies to both CUR decomposition for general matrices and the Nyström method for symmetric positive semi-definite matrices, providing a unified theoretical foundation for determinant-based low-rank approximation analysis.

Determinant-Based Error Bounds for CUR Matrix Approximation: Oversampling and Volume Sampling

TL;DR

The paper develops a determinant-based framework for CUR and Nyström matrix approximations by tying local bordered Gramian determinants to projection errors and employing volume sampling. It proves that oversampling reduces the expected Frobenius error in a controlled, linear fashion, with the error factor transitioning from (k+1)^2 to (k+1) as the sample size grows. A unified theory is established, connecting local determinant identities to global probabilistic bounds and expressing results in terms of the singular-value tail. These insights yield practical error guarantees and deepen the understanding of how determinant structure and oversampling govern the quality of CUR/Nyström low-rank approximations.

Abstract

We derive error bounds for CUR matrix approximation using determinant-based methods that relate local projection errors to global approximation quality. For general matrices, we establish determinant identities for bordered Gramian matrices that decompose CUR approximation errors into interpretable local components. These identities connect projection errors onto submatrix column spaces directly to determinants, providing geometric insight into approximation degradation. We develop a probabilistic framework based on volume sampling that yields interpolation-type error bounds quantifying the benefits of oversampling: when rows are selected for columns, the expected error factor transitions linearly from (no oversampling) to (full oversampling). Our analysis establishes that the expected squared error is bounded by this interpolation factor times the squared error of the best rank- approximation, directly connecting CUR approximation quality to the optimal low-rank approximation. The framework applies to both CUR decomposition for general matrices and the Nyström method for symmetric positive semi-definite matrices, providing a unified theoretical foundation for determinant-based low-rank approximation analysis.

Paper Structure

This paper contains 14 sections, 15 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Low-Rank Matrix Approximation and the CUR Framework
  3. Determinant Identities and Volume Sampling
  4. Historical Development and Recent Advances
  5. Contributions and Paper Organization
  6. Local error analysis for CUR
  7. Determinant identities for bordered Gramians
  8. Compound matrices and a geometric interpretation
  9. Local CUR error bounds
  10. Additional remark.
  11. Main results for general matrices
  12. Volume sampling and CUR approximation
  13. Error of $B$ component using CUR approximation of $M$
  14. Error of the CUR approximation of $M$

Key Result

Proposition 1

Let $A \in \mathbb{R}^{r\times k}$ with $r \ge k$ and full column rank $k$, and let $b \in \mathbb{R}^r$. Set $Y=[A\; b] \in\mathbb{R}^{r\times(k+1)}$. Then

Theorems & Definitions (32)

  • Proposition 1: Adding a column
  • proof
  • Proposition 2: Adding a row
  • proof
  • Proposition 3: Adding a row and a column
  • proof
  • Proposition 4: Cauchy--Binet identities
  • lemma 1
  • proof
  • Proposition 5
  • ...and 22 more