Tensor data are multi-dimension arrays. Low-rank decomposition-based regression methods with tensor predictors exploit the structural information in tensor predictors while significantly reducing the number of parameters in tensor regression. We propose a method named NA$_0$CT$^2$ (Noise Augmentation for $\ell_0$ regularization on Core Tensor in Tucker decomposition) to regularize the parameters in tensor regression (TR), coupled with Tucker decomposition. We establish theoretically that NA$_0$CT$^2$ achieves exact $\ell_0$ regularization on the core tensor from the Tucker decomposition in linear TR and generalized linear TR. To our knowledge, NA$_0$CT$^2$ is the first Tucker decomposition-based regularization method in TR to achieve $\ell_0$ in core tensors. NA$_0$CT$^2$ is implemented through an iterative procedure and involves two straightforward steps in each iteration -- generating noisy data based on the core tensor from the Tucker decomposition of the updated parameter estimate and running a regular GLM on noise-augmented data on vectorized predictors. We demonstrate the implementation of NA$_0$CT$^2$ and its $\ell_0$ regularization effect in both simulation studies and real data applications. The results suggest that NA$_0$CT$^2$ can improve predictions compared to other decomposition-based TR approaches, with or without regularization and it identifies important predictors though not designed for that purpose.

翻译：张量数据是多维数组。基于低秩分解的张量预测变量回归方法在利用张量预测变量结构信息的同时，显著减少了张量回归中的参数数量。我们提出了一种名为NA$_0$CT$^2$（基于Tucker分解核心张量的$\ell_0$正则化噪声增强方法）的正则化方法，该方法结合Tucker分解对张量回归中的参数进行正则化。我们从理论上证明了NA$_0$CT$^2$在线性张量回归和广义线性张量回归中能够对Tucker分解得到的核心张量实现精确的$\ell_0$正则化。据我们所知，NA$_0$CT$^2$是首个在张量回归中基于Tucker分解实现对核心张量$\ell_0$正则化的方法。NA$_0$CT$^2$通过迭代过程实现，每次迭代包含两个简单步骤：基于当前参数估计的Tucker分解所得核心张量生成噪声数据，以及对向量化预测变量的噪声增强数据运行常规广义线性模型。我们通过模拟研究和实际数据应用展示了NA$_0$CT$^2$的实现过程及其$\ell_0$正则化效果。结果表明，与其它基于分解的张量回归方法（无论是否包含正则化）相比，NA$_0$CT$^2$能够提升预测性能，并且虽然该方法并非为此目的设计，但仍能识别重要预测变量。