On Approximation Algorithm for Orthogonal Low-Rank Tensor Approximation

TL;DR

This paper improves an approximation algorithm for orthogonal low-rank tensor approximation by establishing bounds for multiple factors and introducing a flexible, more efficient method that reduces computational costs.

Contribution

It extends previous work by providing approximation bounds for multiple orthonormal factors and introduces a flexible algorithm that enhances efficiency through randomized or deterministic steps.

Findings

Established approximation lower bounds for multiple orthonormal factors.

Proposed a flexible algorithm reducing large SVD computations.

Numerical studies validate the algorithm's effectiveness.

Abstract

The goal of this work is to fill a gap in [Yang, SIAM J. Matrix Anal. Appl, 41 (2020), 1797--1825]. In that work, an approximation procedure was proposed for orthogonal low-rank tensor approximation; however, the approximation lower bound was only established when the number of orthonormal factors is one. To this end, by further exploring the multilinearity and orthogonality of the problem, we introduce a modified approximation algorithm. Approximation lower bound is established, either in deterministic or expected sense, no matter how many orthonormal factors there are. In addition, a major feature of the new algorithm is its flexibility to allow either deterministic or randomized procedures to solve a key step of each latent orthonormal factor involved in the algorithm. This feature can reduce the computation of large SVDs, making the algorithm more efficient. Some numerical studies are provided to validate the usefulness of the proposed algorithm.