Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tensor CUR Decomposition under the Linear-Map-Based Tensor-Tensor Multiplication

Susana Lopez-Moreno, June-Ho Lee, Taehyeong Kim

TL;DR

This work extends CUR decomposition to the linear-map-based tensor-tensor multiplication by introducing the $*_M$-CUR decomposition. It constructs the approximation by projecting along a chosen map $M$, performing per-slice CUR on the projected frontal slices, and reconstructing with the $*_M$-product, with exactness guarantees for invertible $M$ and perturbation bounds. The approach is validated on video foreground-background separation across several transforms (DCT, DFT, DST) and a data-driven map, showing competitive performance against Robust CUR, BM, and SS-SVD while providing a flexible framework for multiway data compression. Overall, the method offers a practical, theoretically grounded tool for tensor recovery tailored to the chosen linear map, with potential impact in video analysis and beyond.

Abstract

The factorization of three-dimensional data continues to gain attention due to its relevance in representing and compressing large-scale datasets. The linear-map-based tensor-tensor multiplication is a matrix-mimetic operation that extends the notion of matrix multiplication to higher order tensors, and which is a generalization of the T-product. Under this framework, we introduce the tensor CUR decomposition, show its performance in video foreground-background separation for different linear maps and compare it to a robust matrix CUR decomposition, another tensor approximation and the slice-based singular value decomposition (SS-SVD). We also provide a theoretical analysis of our tensor CUR decomposition, extending classical matrix results to establish exactness conditions and perturbation bounds.

Tensor CUR Decomposition under the Linear-Map-Based Tensor-Tensor Multiplication

TL;DR

This work extends CUR decomposition to the linear-map-based tensor-tensor multiplication by introducing the -CUR decomposition. It constructs the approximation by projecting along a chosen map , performing per-slice CUR on the projected frontal slices, and reconstructing with the -product, with exactness guarantees for invertible and perturbation bounds. The approach is validated on video foreground-background separation across several transforms (DCT, DFT, DST) and a data-driven map, showing competitive performance against Robust CUR, BM, and SS-SVD while providing a flexible framework for multiway data compression. Overall, the method offers a practical, theoretically grounded tool for tensor recovery tailored to the chosen linear map, with potential impact in video analysis and beyond.

Abstract

The factorization of three-dimensional data continues to gain attention due to its relevance in representing and compressing large-scale datasets. The linear-map-based tensor-tensor multiplication is a matrix-mimetic operation that extends the notion of matrix multiplication to higher order tensors, and which is a generalization of the T-product. Under this framework, we introduce the tensor CUR decomposition, show its performance in video foreground-background separation for different linear maps and compare it to a robust matrix CUR decomposition, another tensor approximation and the slice-based singular value decomposition (SS-SVD). We also provide a theoretical analysis of our tensor CUR decomposition, extending classical matrix results to establish exactness conditions and perturbation bounds.
Paper Structure (10 sections, 2 theorems, 14 equations, 1 figure, 2 tables)

This paper contains 10 sections, 2 theorems, 14 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Linear-map-based tensor-tensor multiplication
  5. Tensor CUR decomposition under the linear-map-based tensor-tensor multiplication
  6. Theoretical analysis
  7. Numerical Experiments
  8. Formulation of the problem
  9. Video foreground-background separation
  10. Conclusion

Key Result

Theorem 3.1

Let $\mathcal{A}\in\mathbb{K}^{m\times n\times p}$ and let $M\in\mathbb{K}^{p\times p}$ be invertible. Fix index sets $I\subset\{1,\dots,m\}$, $J\subset\{1,\dots,n\}$ such that $\operatorname{rank}_m(\mathcal{U})=\operatorname{rank}_m(\mathcal{A})$. Then, the $*_M$-CUR reconstruction is exact, i.e.,

Figures (1)

  • Figure 1: Visual comparison highlighting the robustness of the proposed $*_M$-CUR method against recent robust decomposition methods on the CDnet dataset. Our method (third column) effectively isolates the foreground with significantly reduced visual noise and sharper object boundaries compared to Robust CUR, Tensor BM, and SS-SVD. Selected frames: Highway (345), Office (1100), Pedestrians (175), and PETS2006 (620).

Theorems & Definitions (8)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 3.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof