论文标题
张量回归中的CP变性
CP Degeneracy in Tensor Regression
论文作者
论文摘要
张量线性回归是分析张量数据的重要且有用的工具。为了处理高维度，CandeComp/Parafac（CP）低级别约束通常是在（受惩罚）$ m $估计中的系数张量参数上施加的。但是，我们表明可能无法实现相应的优化，并且当发生这种情况时，估算器的定义不明确。这与低级张量近似问题中的一种现象（称为CP变性）密切相关。在本文中，我们在张量回归问题中提供了CP退化的有用结果。此外，我们提供了一般的惩罚策略，以解决CP退化的解决方案。还研究了所得估计的渐近特性。进行数值实验以说明我们的发现。
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.
