Practical Approximation Algorithms for $\ell_1$-Regularized Sparse Rank-$1$ Approximation to Higher-Order Tensors

TL;DR

This paper introduces two scalable approximation algorithms for sparse rank-1 tensor approximation with $\,l_1$ regularization, supported by theoretical bounds and practical parameter selection methods.

Contribution

It presents novel scalable algorithms based on multilinear relaxation and sparsification for $\,l_1$-regularized tensor approximation, with theoretical analysis and parameter guidance.

Findings

Algorithms are easily implementable and scalable.

The second algorithm scales linearly with tensor size.

Numerical experiments confirm effectiveness of the methods.

Abstract

Two approximation algorithms are proposed for $\ell_1$-regularized sparse rank-1 approximation to higher-order tensors. The algorithms are based on multilinear relaxation and sparsification, which are easily implemented and well scalable. In particular, the second one scales linearly with the size of the input tensor. Based on a careful estimation of the $\ell_1$-regularized sparsification, theoretical approximation lower bounds are derived. Our theoretical results also suggest an explicit way of choosing the regularization parameters. Numerical examples are provided to verify the proposed algorithms.