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Fast randomized algorithms for computing the generalized tensor SVD based on the tubal product

Salman Ahmadi-Asl, Ugochukwu Ugwu

TL;DR

This work tackles the challenge of efficiently computing the Generalized Tensor SVD ($\mathrm{GTSVD}$) under the $T$-product for large-scale tensors. It introduces two fast randomized algorithms that use random projections to form small tensor sketches, on which deterministic $\mathrm{GTSVD}$ is performed, followed by reconstruction of the original $\mathrm{GTSVD}$. Theoretical error bounds are provided for the randomized methods, and extensive experiments (including synthetic tensors and image restoration) demonstrate substantial speedups with minimal loss of accuracy. The methods enable scalable, real-time tensor decompositions for big data and streaming applications, with potential extensions to streaming data and higher-order tensors discussed for future work.

Abstract

This work deals with developing two fast randomized algorithms for computing the generalized tensor singular value decomposition (GTSVD) based on the tubal product (t-product). The random projection method is utilized to compute the important actions of the underlying data tensors and use them to get small sketches of the original data tensors, which are easier to be handled. Due to the small size of the sketch tensors, deterministic approaches are applied to them to compute their GTSVDs. Then, from the GTSVD of the small sketch tensors, the GTSVD of the original large-scale data tensors is recovered. Some experiments are conducted to show the effectiveness of the proposed approach.

Fast randomized algorithms for computing the generalized tensor SVD based on the tubal product

TL;DR

This work tackles the challenge of efficiently computing the Generalized Tensor SVD () under the -product for large-scale tensors. It introduces two fast randomized algorithms that use random projections to form small tensor sketches, on which deterministic is performed, followed by reconstruction of the original . Theoretical error bounds are provided for the randomized methods, and extensive experiments (including synthetic tensors and image restoration) demonstrate substantial speedups with minimal loss of accuracy. The methods enable scalable, real-time tensor decompositions for big data and streaming applications, with potential extensions to streaming data and higher-order tensors discussed for future work.

Abstract

This work deals with developing two fast randomized algorithms for computing the generalized tensor singular value decomposition (GTSVD) based on the tubal product (t-product). The random projection method is utilized to compute the important actions of the underlying data tensors and use them to get small sketches of the original data tensors, which are easier to be handled. Due to the small size of the sketch tensors, deterministic approaches are applied to them to compute their GTSVDs. Then, from the GTSVD of the small sketch tensors, the GTSVD of the original large-scale data tensors is recovered. Some experiments are conducted to show the effectiveness of the proposed approach.
Paper Structure (10 sections, 3 theorems, 36 equations, 3 figures, 2 tables, 7 algorithms)

This paper contains 10 sections, 3 theorems, 36 equations, 3 figures, 2 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Basic definitions and concepts
  3. Tensor SVD (T-SVD) and Tensor QR (T-QR) decomposition
  4. Generalized singular value decomposition (GSVD) and its extension to tensors based on the T-product (GTSVD)
  5. Proposed fast randomized algorithms for computation of the GTSVD
  6. Error Analysis
  7. Experimental Results
  8. Conclusion and future works
  9. Acknowledgement
  10. Conflict of Interest Statement

Key Result

Theorem 1

paige1981towards Let two matrices ${\bf X}\in\mathbb{R}^{I_1\times I_2}$ and ${\bf Y}\in\mathbb{R}^{I_3\times I_2}$ be given and assume that the SVD of the matrix ${\bf C}=$ is with the unitary matrices ${\bf E}\in\mathbb{R}^{(I_1+I_3)\times( I_1+I_3)},\, {\bf Z}\in\mathbb{R}^{I_2\times I_2}$ and a diagonal matrix ${\bf \Gamma}\in\mathbb{R}^{k\times k}$. Here, $k={\rm rank}({\bf C})$. Then, there

Figures (3)

  • Figure 1: Running time comparison of different algorithms for the synthetic data tensors in Example \ref{['ex1']}.
  • Figure 2: Running time and relative error comparisons of different algorithms for the synthetic data tensors for Example \ref{['ex2']}.
  • Figure 3: (Top) The true benchmark images for Example \ref{['exa3']}.

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • ...and 4 more