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An Efficient Randomized Fixed-Precision Algorithm for Tensor Singular Value Decomposition

Salman Ahmadi-Asl

TL;DR

The paper tackles the challenge of automatically determining the tubal rank in tensor SVD by introducing a randomized fixed-precision algorithm that, given a third-order tensor and an error bound, adaptively discovers the optimal tubal rank and the corresponding low-rank approximation. It extends block-based randomized projection ideas to tensors, using power iterations and orthogonalization to construct a compact factorization with controlled error, from which the t-SVD of the original tensor is recovered. The approach achieves favorable computational complexity and is validated on synthetic and real datasets, delivering accurate tubal-rank estimation and significant speedups over standard truncated t-SVD while maintaining comparable reconstruction quality. The method provides a practical, scalable means to obtain fast, automatic low-tubal-rank tensor representations and can be extended to higher-order tensors and tensor completion tasks.

Abstract

The existing randomized algorithms need an initial estimation of the tubal rank to compute a tensor singular value decomposition. This paper proposes a new randomized fixedprecision algorithm which for a given third-order tensor and a prescribed approximation error bound, automatically finds an optimal tubal rank and the corresponding low tubal rank approximation. The algorithm is based on the random projection technique and equipped with the power iteration method for achieving a better accuracy. We conduct simulations on synthetic and real-world datasets to show the efficiency and performance of the proposed algorithm.

An Efficient Randomized Fixed-Precision Algorithm for Tensor Singular Value Decomposition

TL;DR

The paper tackles the challenge of automatically determining the tubal rank in tensor SVD by introducing a randomized fixed-precision algorithm that, given a third-order tensor and an error bound, adaptively discovers the optimal tubal rank and the corresponding low-rank approximation. It extends block-based randomized projection ideas to tensors, using power iterations and orthogonalization to construct a compact factorization with controlled error, from which the t-SVD of the original tensor is recovered. The approach achieves favorable computational complexity and is validated on synthetic and real datasets, delivering accurate tubal-rank estimation and significant speedups over standard truncated t-SVD while maintaining comparable reconstruction quality. The method provides a practical, scalable means to obtain fast, automatic low-tubal-rank tensor representations and can be extended to higher-order tensors and tensor completion tasks.

Abstract

The existing randomized algorithms need an initial estimation of the tubal rank to compute a tensor singular value decomposition. This paper proposes a new randomized fixedprecision algorithm which for a given third-order tensor and a prescribed approximation error bound, automatically finds an optimal tubal rank and the corresponding low tubal rank approximation. The algorithm is based on the random projection technique and equipped with the power iteration method for achieving a better accuracy. We conduct simulations on synthetic and real-world datasets to show the efficiency and performance of the proposed algorithm.
Paper Structure (8 sections, 3 theorems, 23 equations, 7 figures, 4 tables, 5 algorithms)

This paper contains 8 sections, 3 theorems, 23 equations, 7 figures, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor SVD (t-SVD) and Tensor QR (t-QR) decomposition
  4. Proposed approach
  5. Computational complexity
  6. Simulations
  7. Conclusion and future works
  8. Acknowledgements

Key Result

Lemma 1

For the linear operator defined as follow where $\mathcal{V}$ is the space of third-order tensors, the adjoint of $\mathcal{L}$ is ${\mathcal{L}^{adj}}\left( \underline{\bf X} \right)=\underline{\bf Q}^T*\underline{\bf X}.$

Figures (7)

  • Figure 1: Concatenation of tensors along the first mode (a) and the second mode (b) cichocki2016tensor.
  • Figure 2: Illustration of ( a) Tensor SVD (t-SVD) and ( b) truncated t-SVD for a third-order tensor.
  • Figure 3: Randomized t-SVD based on slice sampling tarzanagh2018fastahmadi2021cross.
  • Figure 4: Example \ref{['Ex1']}. Running times comparison against tensor dimensions for Cases I-III with tubal rank $R=10$.
  • Figure 5: Example \ref{['Ex2']}. Reconstructed images with the tubal rank $R=35$ and $\epsilon=0.05$ using the truncated t-SVD and the proposed algorithm.
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Lemma 1
  • proof
  • ...and 8 more