Recently, tensor data (or multidimensional array) have been generated in many modern applications, such as functional magnetic resonance imaging (fMRI) in neuroscience and videos in video analysis. Many efforts are made in recent years to predict the relationship between tensor features and univariate responses. However, previously proposed methods either lose structural information within tensor data or have prohibitively expensive time costs, especially for large-scale data with high-order structures. To address such problems, we propose the Sparse and Low-rank Tensor Regression (SLTR) model. Our model enforces sparsity and low-rankness of the tensor coefficient by directly applying $\ell_1$ norm and tensor nuclear norm, such that it preserves structural information of the tensor. To make the solving procedure scalable and efficient, SLTR makes use of the proximal gradient method, which can be easily implemented parallelly. We evaluate SLTR on several simulated datasets and one video action recognition dataset. Experiment results show that, compared with previous models, SLTR can obtain a better solution with much fewer time costs. Moreover, our model's predictions exhibit meaningful interpretations on the video dataset.

翻译：近年来，许多现代应用生成了张量数据（或多维数组），例如神经科学中的功能性磁共振成像（fMRI）和视频分析中的视频数据。为预测张量特征与单变量响应之间的关系，近年已有多项研究投入努力。然而，现有方法要么丢失张量数据的结构信息，要么面临高昂的时间成本，尤其对于具有高阶结构的大规模数据。为解决这些问题，我们提出稀疏低秩张量回归（SLTR）模型。该模型通过直接应用ℓ1范数和张量核范数强制张量系数的稀疏性与低秩性，从而保留张量的结构信息。为使求解过程可扩展且高效，SLTR采用近端梯度方法，该方法易于并行实现。我们在多个模拟数据集和一个人体动作识别视频数据集上评估了SLTR。实验结果表明，与先前模型相比，SLTR能以更短的时间成本获得更优解。此外，我们的模型预测在视频数据集上展现出具有意义的可解释性。