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Randomized dual singular value decomposition

Mengyu Wang, Jingchun Zhou, Hanyu Li

TL;DR

The paper addresses efficient singular value decomposition for dual matrices by introducing CCDSVD, a concise real-Σ variant of CDSVD, and its randomized counterpart RCCDSVD to reduce computational cost. It provides theoretical guarantees, including Frobenius-norm and quasi-metric error bounds, and demonstrates through numerical experiments that CCDSVD maintains accuracy while offering speedups, with RCCDSVD delivering further computational gains. The work highlights the practical viability of dual-matrix SVD in large-scale settings and suggests extending randomized approaches to dual quaternions. Overall, it advances efficient, accurate dual-matrix decompositions with solid theoretical and empirical support.

Abstract

We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze the theoretical properties and illuminate the numerical performance of the randomized algorithm.

Randomized dual singular value decomposition

TL;DR

The paper addresses efficient singular value decomposition for dual matrices by introducing CCDSVD, a concise real-Σ variant of CDSVD, and its randomized counterpart RCCDSVD to reduce computational cost. It provides theoretical guarantees, including Frobenius-norm and quasi-metric error bounds, and demonstrates through numerical experiments that CCDSVD maintains accuracy while offering speedups, with RCCDSVD delivering further computational gains. The work highlights the practical viability of dual-matrix SVD in large-scale settings and suggests extending randomized approaches to dual quaternions. Overall, it advances efficient, accurate dual-matrix decompositions with solid theoretical and empirical support.

Abstract

We first propose a concise singular value decomposition of dual matrices. Then, the randomized version of the decomposition is presented. It can significantly reduce the computational cost while maintaining the similar accuracy. We analyze the theoretical properties and illuminate the numerical performance of the randomized algorithm.
Paper Structure (6 sections, 3 theorems, 12 equations, 1 figure, 1 table, 2 algorithms)

This paper contains 6 sections, 3 theorems, 12 equations, 1 figure, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. CCDSVD
  3. Randomized CCDSVD
  4. Error analysis
  5. Numerical experiments
  6. Concluding remarks

Key Result

Theorem 2.1

Let $\mathbf{A} = {\bf A}_s + {\bf A}_i\epsilon \in {\mathbb{DC}}^{m\times n}$ with $m\ge n$. Assume that ${\bf A}_s = \mathbf{U}_s {\bf \Sigma}_s \mathbf{V}_s^*$ is a compact SVD of ${\bf A}_s$. Then the CCDSVD of $\bf A$ exists if and only if Furthermore, if ${\bf A}$ has a CCDSVD, then there exists a particular pair $({\bf U}_s, {\bf V}_s)$ such that ${\bf A} = {\bf U} {\bf \Sigma} {\bf V}^*$,

Figures (1)

  • Figure 1: Errors incurred for different power schemes.

Theorems & Definitions (9)

  • Theorem 2.1
  • Remark 2.2
  • Remark 3.1
  • Definition 4.1: qi2022low
  • Theorem 4.2: Average Frobenius error
  • Proof 1
  • Definition 4.3: wei2024singular
  • Theorem 4.4: Average quasi-metric error
  • Proof 2