An adaptive algebraic multigrid algorithm for low-rank canonical tensor decomposition

TL;DR

This paper introduces an adaptive multigrid algorithm for low-rank tensor decomposition that outperforms traditional ALS methods in accuracy and efficiency for certain problems.

Contribution

It develops a novel multilevel approach combining adaptive transfer operators with Bootstrap algebraic multigrid for tensor decomposition.

Findings

Multilevel method significantly outperforms ALS at high accuracy levels

Adaptive setup improves the construction of transfer operators and coarse tensors

Numerical tests demonstrate efficiency gains in specific test problems

Abstract

This paper presents a multigrid algorithm for the computation of the rank-R canonical decomposition of a tensor for low rank R. Standard alternating least squares (ALS) is used as the relaxation method. Transfer operators and coarse-level tensors are constructed in an adaptive setup phase based on multiplicative correction and on Bootstrap algebraic multigrid. An accurate solution is then computed by an additive solve phase based on the Full Approximation Scheme. Numerical tests show that for certain test problems the multilevel method significantly outperforms standalone ALS when a high level of accuracy is required.