  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.
