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Frontal Slice Approaches for Tensor Linear Systems

Hengrui Luo, Anna Ma

TL;DR

The paper tackles solving large-scale tensor linear systems of the form $\mathcal{A}*\mathcal{X}=\mathcal{B}$ under the $t$-product. It introduces Frontal Slice Descent (FSD) and its cyclic, block, and randomized variants, which update $\mathcal{X}$ using individual frontal slices and residual-based approximations, with convergence guarantees under conditions on $\kappa$ and $\mu$. The authors provide computational and storage analyses and validate the approach on synthetic data and real-world image/video deblurring tasks, showing favorable performance and consistency with theory. The work demonstrates scalable, structure-exploiting tensor solvers that leverage frontal-slice information, offering practical benefits for multi-dimensional data processing such as video and image restoration. These methods extend the toolbox for tensor computations with tunable trade-offs between memory, communication, and convergence speed, particularly in scenarios with localized frontal-slice information.

Abstract

Inspired by the row and column action methods for solving large-scale linear systems, in this work, we explore the use of frontal slices for solving tensor linear systems. In particular, this paper presents a novel approach for using frontal slices of a tensor $\mathcal{A}$ to solve tensor linear systems $\mathcal{A} * \mathcal{X} = \mathcal{B}$ where $*$ denotes the t-product. In addition, we consider variations of this method, including cyclic, block, and randomized approaches, each designed to optimize performance in different operational contexts. Our primary contribution lies in the development and convergence analysis of these methods. Experimental results on synthetically generated and real-world data, including applications such as image and video deblurring, demonstrate the efficacy of our proposed approaches and validate our theoretical findings.

Frontal Slice Approaches for Tensor Linear Systems

TL;DR

The paper tackles solving large-scale tensor linear systems of the form under the -product. It introduces Frontal Slice Descent (FSD) and its cyclic, block, and randomized variants, which update using individual frontal slices and residual-based approximations, with convergence guarantees under conditions on and . The authors provide computational and storage analyses and validate the approach on synthetic data and real-world image/video deblurring tasks, showing favorable performance and consistency with theory. The work demonstrates scalable, structure-exploiting tensor solvers that leverage frontal-slice information, offering practical benefits for multi-dimensional data processing such as video and image restoration. These methods extend the toolbox for tensor computations with tunable trade-offs between memory, communication, and convergence speed, particularly in scenarios with localized frontal-slice information.

Abstract

Inspired by the row and column action methods for solving large-scale linear systems, in this work, we explore the use of frontal slices for solving tensor linear systems. In particular, this paper presents a novel approach for using frontal slices of a tensor to solve tensor linear systems where denotes the t-product. In addition, we consider variations of this method, including cyclic, block, and randomized approaches, each designed to optimize performance in different operational contexts. Our primary contribution lies in the development and convergence analysis of these methods. Experimental results on synthetically generated and real-world data, including applications such as image and video deblurring, demonstrate the efficacy of our proposed approaches and validate our theoretical findings.
Paper Structure (21 sections, 12 theorems, 67 equations, 10 figures, 1 table, 3 algorithms)

This paper contains 21 sections, 12 theorems, 67 equations, 10 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Organization.
  3. Background
  4. Notation.
  5. The Tensor t-product
  6. Data perspectives.
  7. Related Works
  8. The Frontal Slice Descent Algorithm
  9. Using Frontal Slices
  10. Variations of Frontal Slice Descent
  11. Computational Considerations
  12. Convergence Analysis
  13. Experiments
  14. Synthetic Experiments
  15. Real World Application: Deblurring
  16. ...and 6 more sections

Key Result

Proposition 1

The computational complexity of t-product $\mathcal{A}*\mathcal{X}$ for $\mathcal{A}\in\mathbb{R}^{n_{1}\times n_{2}\times n}$ and $\mathcal{X}\in\mathbb{R}^{n_{2}\times n_{3}\times n}$. is $\mathcal{O}(n_1 n_2 n_3 n) \sim \mathcal{O}(m^3 n)$.

Figures (10)

  • Figure 1: The first row slice $\mathcal{A}_{1::}$ and frontal slice $\mathcal{A}_{1}$ of $\mathcal{A}$.
  • Figure 2: The original $512\times512$ pixel image of the Hubble Space Telescope https://github.com/jnagy1/IRtools/blob/master/Extra/test_data/HSTgray.jpg, its Gaussian blurred version and the frontal sketching (Algorithm \ref{['alg:tsolve_cyclic_slices']}) deblurred version. We defer the details of this deblurring experiment to Section \ref{['sec:Experiments']}.
  • Figure 3: Algorithms \ref{['alg:tsolve_full_gradient']}, \ref{['alg:tsolve_cyclic_slices']}, \ref{['alg:tsolve_random_slices']}$t=1,2,3,4,5,6$ with a tensor $\mathcal{A}$ of frontal size $n=5$ and block size $s=1$. We illustrate the 3-way tensor with 5 frontal slices and highlight the slice at a given iteration in blue.
  • Figure 4: The bound of learning rate $\alpha$ as shown in Theorem \ref{['thm:conv-cyclic']} for Algorithm \ref{['alg:tsolve_cyclic_slices']} to tensor $\mathcal{A}$ with (1) i.i.d Gaussian entries (2) i.i.d. Unif[0,1] entries, and (3) an additive mixture between Gaussian and uniform entries. We also provide synthetic cases where $\mathcal{A}$ has (4) frontal slices, which are each diagonal matrices (5) mutually orthogonal frontal slices (w.r.t. matrix inner product) (6) mutual inner products between frontal slices are bounded by 0.1 (7) mutual inner products between frontal slices are bounded by 0.01 (8) mutually orthogonal frontal slices with Gaussian noise.
  • Figure 5: Generate $\mathcal{A},\mathcal{X}$ and compute consistent $\mathcal{B}=\mathcal{A}*\mathcal{X}$ for the synthetic system for varying dimension sizes. In each panel, we show the final approximation error $\mathcal{E}(T)$ using different methods (Frontal: Algorithm \ref{['alg:tsolve_cyclic_slices']}; Sample (Random): Algorithm \ref{['alg:tsolve_random_slices']} with uniform random sampling of slices; Sample (Leverage): Algorithm \ref{['alg:tsolve_random_slices']} with leverage score sampling of slices; TRK: ma2022randomized) and learning rate $\alpha=0.001$ and the maximal number of iterations of 1000 steps. Comparison of performance of different iterative methods for solving tensor systems when $n_{1}=100$, $n_{2}=20$, $n_{3}=10$, $n=10$, and $\alpha=\{0.5,1,2,4$} for frontal descent methods
  • ...and 5 more figures

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Corollary 4
  • ...and 11 more