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Randomized block Krylov method for approximation of truncated tensor SVD

Malihe Nobakht Kooshkghazi, Salman Ahmadi-Asl, Andre L. F. de Almeida

Abstract

This paper is devoted to studying the application of the block Krylov subspace method for approximation of the truncated tensor SVD (T-SVD). The theoretical results of the proposed randomized approach are presented. Several experimental experiments using synthetics and real-world data are conducted to verify the efficiency and feasibility of the proposed randomized approach, and the numerical results show that the proposed method provides promising results. Applications of the proposed approach to data completion and data compression are presented.

Randomized block Krylov method for approximation of truncated tensor SVD

Abstract

This paper is devoted to studying the application of the block Krylov subspace method for approximation of the truncated tensor SVD (T-SVD). The theoretical results of the proposed randomized approach are presented. Several experimental experiments using synthetics and real-world data are conducted to verify the efficiency and feasibility of the proposed randomized approach, and the numerical results show that the proposed method provides promising results. Applications of the proposed approach to data completion and data compression are presented.

Paper Structure

This paper contains 7 sections, 16 theorems, 68 equations, 10 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary concepts
  3. Proposed randomized block Krylov method for approximate truncated T-SVD
  4. Theoretical results
  5. Experimental results
  6. Conclusion and future works
  7. Acknowledgment

Key Result

Lemma 1

Let $\mathcal{A}, \mathcal{A}_{1} \in \Bbb R^{n\times s \times n_{3}}, \mathcal{B}, \mathcal{B}_{1} \in \Bbb R^{n\times p \times n_{3}}, \mathcal{A}_{2} \in \Bbb R^{l \times s \times n_{3}}, \mathcal{B}_{2} \in \Bbb R^{l\times p \times n_{3}}, \mathcal{C} \in \Bbb R^{s \times n \times n_{3}}, \mathc

Figures (10)

  • Figure 1: Relative error and computing time for Algorithms \ref{['ALg_1']} and \ref{['ALg_2']} for Example \ref{['EX:1']}, Case 1.
  • Figure 2: Relative error and computing time for Algorithms \ref{['ALg_1']} and \ref{['ALg_2']} for Example \ref{['EX:1']}, Case 2.
  • Figure 3: Relative error and computing time for Algorithms \ref{['ALg_1']} and \ref{['ALg_2']} for Example \ref{['EX:1']}, Case 3.
  • Figure 4: Comparing the PSNR of compressed images using Algorithms \ref{['ALg_1']} and \ref{['ALg_2']} for Example \ref{['EX:2']}.
  • Figure 5: Some random samples of the kodak dataset
  • ...and 5 more figures

Theorems & Definitions (45)

  • Definition 1: kilmer2011factorization
  • Definition 2: kilmer2011factorization
  • Definition 3
  • Definition 4: Ichi,nobakht2023
  • Lemma 1: Ichi,nobakht2023
  • Definition 5
  • Lemma 2
  • proof
  • Definition 6
  • Lemma 3
  • ...and 35 more