Wedderburn rank reduction and Krylov subspace method for tensor approximation. Part 1: Tucker case

TL;DR

This paper introduces new algorithms for Tucker tensor approximation that leverage Wedderburn rank reduction and Krylov subspace methods, offering faster and more accurate results than existing approaches.

Contribution

It extends Krylov subspace methods and Wedderburn rank reduction techniques from matrices to tensors for efficient approximation.

Findings

Proposed algorithms outperform minimal Krylov recursion in speed.

New methods achieve higher accuracy in tensor approximation.

Numerical experiments validate the effectiveness of the algorithms.

Abstract

New algorithms are proposed for the Tucker approximation of a 3-tensor, that access it using only the tensor-by-vector-by-vector multiplication subroutine. In the matrix case, Krylov methods are methods of choice to approximate the dominant column and row subspaces of a sparse or structured matrix given through the matrix-by-vector multiplication subroutine. Using the Wedderburn rank reduction formula, we propose an algorithm of matrix approximation that computes Krylov subspaces and allows generalization to the tensor case. Several variants of proposed tensor algorithms differ by pivoting strategies, overall cost and quality of approximation. By convincing numerical experiments we show that the proposed methods are faster and more accurate than the minimal Krylov recursion, proposed recently by Elden and Savas.