Tensor-on-Tensor Regression
提出一种用收缩张量积从任意维度的多路数组（张量）预测另一个多路数组的线性回归框架，通过低秩约束和岭惩罚实现高效估计，并用吉布斯采样进行推断，在面部图像数据上展示了应用。
I propose a framework for the linear prediction of a multiway array (i.e., a tensor) from another multiway array of arbitrary dimension, using the contracted tensor product. This framework generalizes several existing approaches, including methods to predict a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar. I describe an approach that exploits the multiway structure of both the predictors and the outcomes by restricting the coefficients to have reduced PARAFAC/CANDECOMP rank. I propose a general and efficient algorithm for penalized least-squares estimation, which allows for a ridge (L2) penalty on the coefficients. The objective is shown to give the mode of a Bayesian posterior, which motivates a Gibbs sampling algorithm for inference. I illustrate the approach with an application to facial image data. An R package is available at https://github.com/lockEF/MultiwayRegression.