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Effective algorithms for tensor train decomposition via the UTV framework

Yuchao Wang, Maolin Che, Yimin Wei

TL;DR

The paper tackles the high cost of constructing tensor-train representations by replacing SVD-based steps with rank-revealing UTV factorizations, forming the TT-UTV framework. It develops two variants, TT-ULV (left-to-right) and TT-URV (right-to-left), with rigorous error bounds and adaptive-rank, fixed-precision options. Theoretical analysis shows how local ULV/URV errors propagate to a global TT approximation, and practical guidance is provided for sweep direction and cost. Numerical experiments on video, image, and MRI data demonstrate competitive accuracy with significant computational savings, highlighting TT-UTV as a scalable alternative for large-scale tensor decompositions.

Abstract

The tensor-train (TT) decomposition is widely used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the well-known TT-SVD method, which performs singular value decompositions (SVDs) on the successive matrices sequentially. But in practical applications, it is often unnecessary to compute full SVDs. In this article, we propose a new method called the TT-UTV. It utilizes the virtues of rank-revealing UTV decomposition to compute the TT format for a large-scale tensor, resulting in lower computational cost. We analyze the error bounds on the accuracy of these algorithms in both the URV and ULV cases and then recommend different sweep patterns for these two cases. Based on the theoretical analysis, we also formulate the rank-adaptive algorithms with prescribed accuracy. Numerical experiments on various applications, including magnetic resonance imaging data completion, are performed to illustrate their good performance in practice.

Effective algorithms for tensor train decomposition via the UTV framework

TL;DR

The paper tackles the high cost of constructing tensor-train representations by replacing SVD-based steps with rank-revealing UTV factorizations, forming the TT-UTV framework. It develops two variants, TT-ULV (left-to-right) and TT-URV (right-to-left), with rigorous error bounds and adaptive-rank, fixed-precision options. Theoretical analysis shows how local ULV/URV errors propagate to a global TT approximation, and practical guidance is provided for sweep direction and cost. Numerical experiments on video, image, and MRI data demonstrate competitive accuracy with significant computational savings, highlighting TT-UTV as a scalable alternative for large-scale tensor decompositions.

Abstract

The tensor-train (TT) decomposition is widely used to compress large tensors into a more compact form by exploiting their inherent data structures. A fundamental approach for constructing the TT format is the well-known TT-SVD method, which performs singular value decompositions (SVDs) on the successive matrices sequentially. But in practical applications, it is often unnecessary to compute full SVDs. In this article, we propose a new method called the TT-UTV. It utilizes the virtues of rank-revealing UTV decomposition to compute the TT format for a large-scale tensor, resulting in lower computational cost. We analyze the error bounds on the accuracy of these algorithms in both the URV and ULV cases and then recommend different sweep patterns for these two cases. Based on the theoretical analysis, we also formulate the rank-adaptive algorithms with prescribed accuracy. Numerical experiments on various applications, including magnetic resonance imaging data completion, are performed to illustrate their good performance in practice.
Paper Structure (13 sections, 8 theorems, 32 equations, 5 figures, 2 tables, 4 algorithms)

This paper contains 13 sections, 8 theorems, 32 equations, 5 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tensor basics
  4. UTV decompositions
  5. The proposed methods
  6. TT-ULV case
  7. TT-URV case
  8. Computational cost
  9. Experiments
  10. Video data
  11. Image compression
  12. MRI data completion
  13. Conclusions

Key Result

Proposition 2.1

Let the tensor $\mathcal{A}$ be given in the Tucker format Tucker. Then the $k$th unfolding matrices $\mathbf{A}_k$ and $\mathbf{S}_k$ are related by the matrix equation where $\otimes$ denotes the matrix Kronecker product.

Figures (5)

  • Figure 1: Graphical illustration of the TT decomposition for a $d$th-order tensor.
  • Figure 2: Three color images from the Berkeley Segmentation Dataset.
  • Figure 3: The RSEs of TT compressions on these images are very close, and decline with the TT-rank $r$.
  • Figure 4: The reconstruction of one slice $\mathcal{M}_{::36}$ of the MRI brain data.
  • Figure 5: The RSEs decline and PSNRs increase with the iterations of the RGrad methods.

Theorems & Definitions (16)

  • Definition 2.1: The $k$th unfolding matrix oseledets2011tensor
  • Definition 2.2: Orthogonal TT-cores oseledets2011tensor
  • Definition 2.3: Mode product kolda2009tensor
  • Proposition 2.1
  • Proposition 2.2: golub2013matrix
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.1
  • ...and 6 more