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Efficient Orthogonal Decomposition with Automatic Basis Extraction for Low-Rank Matrix Approximation

Weijie Shen, Weiwei Xu, Lei Zhu

TL;DR

This work proposes a low-rank approximation algorithm termed efficient orthogonal decomposition with automatic basis extraction (EOD-ABE) tailored for the scenario where the rank of the matrix is unknown, and introduces a randomized algorithm to automatically extract the basis that reveals the rank.

Abstract

Low-rank matrix approximation play a ubiquitous role in various applications such as image processing, signal processing, and data analysis. Recently, random algorithms of low-rank matrix approximation have gained widespread adoption due to their speed, accuracy, and robustness, particularly in their improved implementation on modern computer architectures. Existing low-rank approximation algorithms often require prior knowledge of the rank of the matrix, which is typically unknown. To address this bottleneck, we propose a low-rank approximation algorithm termed efficient orthogonal decomposition with automatic basis extraction (EOD-ABE) tailored for the scenario where the rank of the matrix is unknown. Notably, we introduce a randomized algorithm to automatically extract the basis that reveals the rank. The efficacy of the proposed algorithms is theoretically and numerically validated, demonstrating superior speed, accuracy, and robustness compared to existing methods. Furthermore, we apply the algorithms to image reconstruction, achieving remarkable results.

Efficient Orthogonal Decomposition with Automatic Basis Extraction for Low-Rank Matrix Approximation

TL;DR

This work proposes a low-rank approximation algorithm termed efficient orthogonal decomposition with automatic basis extraction (EOD-ABE) tailored for the scenario where the rank of the matrix is unknown, and introduces a randomized algorithm to automatically extract the basis that reveals the rank.

Abstract

Low-rank matrix approximation play a ubiquitous role in various applications such as image processing, signal processing, and data analysis. Recently, random algorithms of low-rank matrix approximation have gained widespread adoption due to their speed, accuracy, and robustness, particularly in their improved implementation on modern computer architectures. Existing low-rank approximation algorithms often require prior knowledge of the rank of the matrix, which is typically unknown. To address this bottleneck, we propose a low-rank approximation algorithm termed efficient orthogonal decomposition with automatic basis extraction (EOD-ABE) tailored for the scenario where the rank of the matrix is unknown. Notably, we introduce a randomized algorithm to automatically extract the basis that reveals the rank. The efficacy of the proposed algorithms is theoretically and numerically validated, demonstrating superior speed, accuracy, and robustness compared to existing methods. Furthermore, we apply the algorithms to image reconstruction, achieving remarkable results.
Paper Structure (16 sections, 10 theorems, 46 equations, 12 figures, 4 tables, 9 algorithms)

This paper contains 16 sections, 10 theorems, 46 equations, 12 figures, 4 tables, 9 algorithms.

Table of Contents

  1. Introduction
  2. Applications
  3. Contributions
  4. Notation
  5. Related works
  6. Efficient orthogonal decomposition with Automatic Basis Extraction
  7. Randomized algorithms for basis extraction
  8. Efficient orthogonal decomposition with automatic basis extraction
  9. Subspace iteration - EOD-ABE
  10. Computational cost
  11. Error analysis of EOD-ABE
  12. Numerical experiments
  13. Test Matrices With Randomly Generated Variables
  14. Comparison of low-rank approximation with existing methods
  15. Application: Image Reconstruction
  16. ...and 1 more sections

Key Result

Lemma 3.1

hmt11 Let $\boldsymbol{\Omega}\in \mathbb{R}^{m\times n}$ be standard Gaussian matrix. Let $\mathbf{U}=\left( \right) \in\mathbb{C}^{m\times m}$ and $\mathbf{V}=(\mathbf{V}_{1},\mathbf{V}_{2})\in\mathbb{C}^{n\times n}$ be orthonormal matrices. Let $\mathbf{U}\boldsymbol{\Omega}=\left( \right) =\le

Figures (12)

  • Figure 1: Singular value approximations for Matrix 1 with $\alpha=0.005$. Left: $\tau=0$. Right: $\tau=2$.
  • Figure 2: Singular value approximations for Matrix 1 with $\alpha=0.02$. Left: $\tau=0$. Right: $\tau=2$.
  • Figure 3: Singular value approximations for Matrix 2. Left: $\tau=0$. Right: $\tau=2$.
  • Figure 4: Singular value approximations for Matrix 3. Left: $\tau=0$. Right: $\tau=2$.
  • Figure 5: Singular value approximations for Matrix 4. Left: $\tau=0$. Right: $\tau=2$.
  • ...and 7 more figures

Theorems & Definitions (11)

  • Definition 2.1: UTV decomposition
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Lemma 4.4
  • Theorem 4.5
  • ...and 1 more