Several Approximation Algorithms for Sparse Best Rank-1 Approximation to Higher-Order Tensors

TL;DR

This paper introduces four efficient approximation algorithms for sparse best rank-1 tensor approximation, providing theoretical guarantees and demonstrating effectiveness through experiments on synthetic and real data.

Contribution

It proposes novel low-complexity algorithms for sparse tensor BR1Approx with proven worst-case approximation bounds.

Findings

Algorithms are easy to implement and computationally efficient.

Theoretical worst-case approximation bounds are established.

Numerical experiments confirm the algorithms' effectiveness.

Abstract

Sparse tensor best rank-1 approximation (BR1Approx), which is a sparsity generalization of the dense tensor BR1Approx, and is a higher-order extension of the sparse matrix BR1Approx, is one of the most important problems in sparse tensor decomposition and related problems arising from statistics and machine learning. By exploiting the multilinearity as well as the sparsity structure of the problem, four approximation algorithms are proposed, which are easily implemented, of low computational complexity, and can serve as initial procedures for iterative algorithms. In addition, theoretically guaranteed worst-case approximation lower bounds are proved for all the algorithms. We provide numerical experiments on synthetic and real data to illustrate the effectiveness of the proposed algorithms.