Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pass-efficient Randomized Algorithms for Low-rank Approximation of Quaternion Matrices

Salman Ahmadi-Asl, Malihe Nobakht Kooshkghazi, Valentin Leplat

Abstract

Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input matrix-which is critical in modern computing environments dominated by communication costs. We address this gap by proposing a suite of pass-efficient randomized algorithms that let users directly trade pass budget for approximation accuracy. Our contributions include: (i) a family of arbitrary-pass randomized algorithms for low-rank approximation of quaternion matrices that operate under a user-specified number of matrix views, and (ii) a pass-efficient extension of block Krylov subspace methods that accelerates convergence for matrices with slowly decaying spectra. Furthermore, we establish spectral norm error bounds showing that the expected approximation error decays exponentially with the number of passes. Finally, we validate our framework through extensive numerical experiments and demonstrate its practical relevance across multiple applications, including quaternionic data compression, matrix completion, image super-resolution, and deep learning.

Pass-efficient Randomized Algorithms for Low-rank Approximation of Quaternion Matrices

Abstract

Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input matrix-which is critical in modern computing environments dominated by communication costs. We address this gap by proposing a suite of pass-efficient randomized algorithms that let users directly trade pass budget for approximation accuracy. Our contributions include: (i) a family of arbitrary-pass randomized algorithms for low-rank approximation of quaternion matrices that operate under a user-specified number of matrix views, and (ii) a pass-efficient extension of block Krylov subspace methods that accelerates convergence for matrices with slowly decaying spectra. Furthermore, we establish spectral norm error bounds showing that the expected approximation error decays exponentially with the number of passes. Finally, we validate our framework through extensive numerical experiments and demonstrate its practical relevance across multiple applications, including quaternionic data compression, matrix completion, image super-resolution, and deep learning.

Paper Structure

This paper contains 15 sections, 6 theorems, 56 equations, 10 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proposed pass-efficient algorithms for quaternion matrices
  4. Theoretical results
  5. Proposed pass-efficient block Krylov algorithms for quaternion matrices
  6. Classical block Krylov scheme (even views).
  7. Flexible block Krylov (arbitrary views).
  8. Execution cost
  9. Pass-efficient quaternion randomization for fast image inpainting, image super-resolution and deep learning
  10. Quaternion Matrix Completion as a Core Tool
  11. Image Inpainting
  12. Image Super-Resolution
  13. Improving Deep Learning Robustness
  14. Experimental results
  15. Conclusion and future work

Key Result

Lemma 1

liu2022randomized (Deviation bound for approximation errors of Algorithm ALg_1.) Let the QSVD of the $I_{1}\times I_{2},\,(I_{1}\geq I_{2})$ quaternion matrix ${\bf X}$ be where the singular value matrix ${\bf \Sigma}={\rm diag}(\sigma_1,\sigma_2,\ldots,\sigma_{I_{2}})$ with $\sigma_1\geq \sigma_2 \geq \dots \geq \sigma_{I_{2}} \geq 0$, $k$ is the target rank. For oversampling parameter $p \geq 1

Figures (10)

  • Figure 1: Benchmark images used in our simulation for image compression.
  • Figure 2: Compression results for four benchmark images and different passes.
  • Figure 3: The time and PSNR comparisons for different benchmark images using different passes.
  • Figure 4: Recovered images with 70% missing pixels. Original image (left), image with missing pixels (middle image) and recovered image (right image).
  • Figure 5: Super resolution results for four images .
  • ...and 5 more figures

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4: Corrected block-Krylov bound via range inclusion
  • proof
  • Corollary 5: Block case $q=0$
  • ...and 7 more