ABSTRACT 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.

中文翻译：

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张量对张量的回归

摘要 我提出了一个框架，使用收缩的张量积从另一个任意维度的多路数组线性预测一个多路数组（即张量）。该框架概括了几种现有方法，包括从张量预测标量结果、从矩阵预测矩阵或从标量预测张量的方法。我描述了一种方法，该方法通过限制系数来降低 PARAFAC/CANDECOMP 等级，从而利用预测变量和结果的多路结构。我提出了一种用于惩罚最小二乘估计的通用且有效的算法，该算法允许对系数进行岭（L2）惩罚。该目标给出了贝叶斯后验的模式，这激发了吉布斯采样算法的推理。我通过面部图像数据的应用来说明该方法。R 包可从 https://github.com/lockEF/MultiwayRegression 获取。