CP Degeneracy in Tensor Regression
This addresses a theoretical flaw in tensor regression methods for high-dimensional data analysis, though it is incremental as it builds on existing CP low-rank constraints.
The paper identifies that CP degeneracy can make tensor regression estimators ill-defined and proposes a penalized strategy to overcome this issue, with asymptotic properties studied and numerical experiments conducted.
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.