Multi-subspace power method for decomposing partially symmetric tensors

TL;DR

This paper introduces a novel algorithm for low-rank tensor decomposition that works across various symmetry types, utilizing a transformed tensor approach and a shifted power method with proven convergence.

Contribution

It proposes a new tensor decomposition method that overcomes previous limitations by transforming tensors and employing a shifted power method with global convergence proof.

Findings

Achieves higher accuracy than existing methods

Faster runtime in numerical experiments

Effective across all symmetry types of tensors

Abstract

We present an algorithm for low rank decomposition of tensors of any symmetry type, from fully asymmetric to fully symmetric. It recovers the decomposition one summand at a time via the higher-order power method. This approach is known to fail in general: there need not be a relationship between the summands of a decomposition and the (partially symmetric) singular vector tuples (pSVTs) of the tensor. Our approach overcomes this problem by transforming the input to a tensor with orthonormal slices, via orthogonalization of a flattening. The summands of the decomposition of the original tensor can be recovered from the pSVTs of this new transformed tensor. We introduce a shifted power method for computing pSVTs and prove its global convergence. Numerical experiments demonstrate that our algorithm achieves higher accuracy and faster runtime than existing methods.