Tensor-on-Tensor Regression

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 (<i>L</i>₂) 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 <i>https://github.com/lockEF/MultiwayRegression</i>.

本文提出一种基于收缩张量积（contracted tensor product）的线性预测框架，可通过另一任意维度的多阵列（multiway array，即张量tensor）对目标多阵列开展线性预测。该框架推广了多种现有方法，涵盖从张量预测标量结果、从矩阵预测矩阵，以及从标量预测张量等场景。本文描述了一种方法，通过将系数约束为低秩PARAFAC/CANDECOMP（并行因子分析/规范分解）形式，充分利用预测变量与响应变量的多阵列结构。本文提出一种通用高效的惩罚最小二乘估计算法，可对系数施加岭（ridge，L₂）惩罚。研究表明，该目标函数对应贝叶斯后验分布的众数，这一性质为吉布斯采样（Gibbs sampling）推断算法提供了理论依据。本文通过面部图像数据集的应用案例展示了该方法的有效性。相关R包（R package）可在https://github.com/lockEF/MultiwayRegression获取。

搜集汇总

背景与挑战

背景概述

该数据集名为'Tensor-on-Tensor Regression'，是一个多路数组（张量）线性预测框架的数据集，使用收缩张量积方法推广了现有预测方法，并支持基于PARAFAC/CANDECOMP秩降低和岭惩罚的算法。数据集包含R包实现，并应用于面部图像数据示例，发布于2018年，受NIH资助，采用CC BY 4.0许可证。