CP Degeneracy in Tensor Regression

TL;DR

This paper investigates the issue of CP degeneracy in tensor regression, demonstrating its impact on estimator well-definedness and proposing penalized strategies to address it, supported by theoretical analysis and numerical experiments.

Contribution

It introduces a general penalized approach to overcome CP degeneracy in tensor regression and studies its asymptotic properties.

Findings

CP degeneracy can cause non-attainability of the optimization

A penalized strategy effectively mitigates CP degeneracy

Numerical experiments validate the proposed methods

Abstract

Tensor linear regression is an important and useful tool for analyzing tensor data. To deal with high dimensionality, CANDECOMP/PARAFAC (CP) low-rank constraints are often imposed on the coefficient tensor parameter in the (penalized) $M$-estimation. However, we show that the corresponding optimization may not be attainable, and when this happens, the estimator is not well-defined. This is closely related to a phenomenon, called CP degeneracy, in low-rank tensor approximation problems. In this article, we provide useful results of CP degeneracy in tensor regression problems. In addition, we provide a general penalized strategy as a solution to overcome CP degeneracy. The asymptotic properties of the resulting estimation are also studied. Numerical experiments are conducted to illustrate our findings.