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

TL;DR

This paper introduces a tensor CUR decomposition method based on linear-map tensor-tensor multiplication, demonstrating its effectiveness in video separation tasks and providing theoretical guarantees for its performance.

Contribution

It presents the first tensor CUR decomposition under linear-map tensor multiplication, extending matrix CUR concepts to tensors with theoretical analysis.

Findings

Effective in video foreground-background separation

Outperforms some existing tensor approximation methods

Provides theoretical bounds and exactness conditions

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.