Fast adaptive tubal rank-revealing algorithm for t-product based tensor approximation

TL;DR

This paper presents a fast, adaptive randomized algorithm for low tubal-rank tensor approximation that reduces computational costs and does not require prior rank estimation, with proven theoretical guarantees and practical validation.

Contribution

It introduces an adaptive randomized method for tubal rank detection in tensors, eliminating the need for prior rank or singular value estimations, and provides theoretical guarantees.

Findings

Efficient tensor approximation in image processing tasks.

Theoretical guarantees for low tubal-rank approximation accuracy.

Validated effectiveness in real-world applications like background modeling.

Abstract

Color images and video sequences can be modeled as three-way tensors, which admit low tubal-rank approximations via convex surrogate minimization. This optimization problem is efficiently addressed by tensor singular value thresholding (t-SVT). To mitigate the computational burden of tensor singular value decomposition (t-SVD) in each iteration, this paper introduces an adaptive randomized algorithm for tubal rank revelation in data tensors \(\mathcal{A}\). Our method selectively captures the principal information from frontal slices in the Fourier domain using a predefined threshold, obviating the need for priori tubal-rank and Fourier-domain singular values estimations while providing an explicit tensor approximation. Leveraging optimality results from matrix randomized SVD, we establish theoretical guarantees demonstrating that the proposed algorithm computes low tubal-rank approximations within constants dependent on data dimensions and the Fourier-domain singular value gap. Empirical evaluations validate its efficacy in image processing and background modeling tasks.