An Efficient Two-Sided Sketching Method for Large-Scale Tensor Decomposition Based on Transformed Domains

TL;DR

This paper introduces a novel two-sided sketching technique utilizing transformed domains for efficient large-scale tensor decomposition, significantly reducing computation time while maintaining high approximation accuracy.

Contribution

The paper proposes a new two-sided sketching method based on the $igstar_{L}$-product and transformed domains, with theoretical error analysis and improvements via power iteration.

Findings

Reduces CPU time for tensor approximation

Achieves high accuracy in synthetic and real-world data

Outperforms existing methods in efficiency and precision

Abstract

Large tensors are frequently encountered in various fields such as computer vision, scientific simulations, sensor networks, and data mining. However, these tensors are often too large for convenient processing, transfer, or storage. Fortunately, they typically exhibit a low-rank structure that can be leveraged through tensor decomposition. However, performing large-scale tensor decomposition can be time-consuming. Sketching is a useful technique to reduce the dimensionality of the data. In this paper, we propose a novel two-sided sketching method based on the $\star_{L}$-product decomposition and transformed domains like the discrete cosine transformation. A rigorous theoretical analysis is also conducted to assess the approximation error of the proposed method. Specifically, we improve our method with power iteration to achieve more precise approximate solutions. Extensive numerical experiments and comparisons on low-rank approximation of synthetic large tensors and real-world data like color images and grayscale videos illustrate the efficiency and effectiveness of the proposed approach in terms of both CPU time and approximation accuracy.