Error Preserving Correction for CPD and Bounded-Norm CPD

TL;DR

This paper introduces an error preservation correction method for CPD that minimizes rank-1 component norms while maintaining approximation accuracy, addressing degeneracy issues in tensor decomposition.

Contribution

It proposes a novel correction approach and a bounded-norm CPD method to improve tensor decomposition in challenging scenarios with degeneracy.

Findings

Effective in handling degeneracy in CPD

Reduces large rank-1 component norms

Applicable to tensors like matrix multiplication tensors

Abstract

In CANDECOMP/PARAFAC tensor decomposition, degeneracy often occurs in some difficult scenarios, e.g., when the rank exceeds the tensor dimension, or when the loading components are highly collinear in several or all modes, or when CPD does not have an optimal solution. In such the cases, norms of some rank-1 terms become significantly large and cancel each other. This makes algorithms getting stuck in local minima while running a huge number of iterations does not improve the decomposition. In this paper, we propose an error preservation correction method to deal with such problem. Our aim is to seek a new tensor whose norms of rank-1 tensor components are minimised in an optimization problem, while it preserves the approximation error. An alternating correction algorithm and an all-atone algorithm have been developed for the problem. In addition, we propose a novel CPD with a bound constraint on a norm of the rank-one tensors. The method can be useful for decomposing tensors that cannot be analyzed by traditional algorithms, such as tensors corresponding to the matrix multiplication.