Error Preserving Correction for CPD and Bounded-Norm CPD
For practitioners of tensor decomposition, this provides a solution to degeneracy issues that plague standard algorithms in challenging scenarios.
The paper addresses degeneracy in CANDECOMP/PARAFAC tensor decomposition, where rank-1 terms become large and cancel each other, causing algorithms to get stuck in local minima. They propose an error preservation correction method that minimizes norms of rank-1 components while preserving approximation error, and a bounded-norm CPD variant, enabling decomposition of tensors intractable by traditional algorithms.
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.