Abstract

In modern data science, dynamic tensor data prevail in numerous applications. An important task is to characterize the relationship between dynamic tensor datasets and external covariates. However, the tensor data are often only partially observed, rendering many existing methods inapplicable. In this article, we develop a regression model with a partially observed dynamic tensor as the response and external covariates as the predictor. We introduce the low-rankness, sparsity, and fusion structures on the regression coefficient tensor, and consider a loss function projected over the observed entries. We develop an efficient nonconvex alternating updating algorithm, and derive the finite-sample error bound of the actual estimator from each step of our optimization algorithm. Unobserved entries in the tensor response have imposed serious challenges. As a result, our proposal differs considerably in terms of estimation algorithm, regularity conditions, as well as theoretical properties, compared to the existing tensor completion or tensor response regression solutions. We illustrate the efficacy of our proposed method using simulations and two real applications, including a neuroimaging dementia study and a digital advertising study.

 抽象的

在现代数据科学中，动态张量数据在许多应用中盛行。一项重要的任务是表征动态张量数据集和外部协变量之间的关系。然而，张量数据通常只能被部分观察到，使得许多现有方法不适用。在本文中，我们开发了一个回归模型，其中部分观察到的动态张量作为响应，外部协变量作为预测变量。我们在回归系数张量上引入低秩、稀疏性和融合结构，并考虑在观察到的条目上投影的损失函数。我们开发了一种高效的非凸交替更新算法，并从优化算法的每一步推导出实际估计器的有限样本误差界。张量响应中未被观察到的条目带来了严峻的挑战。因此，与现有的张量完成或张量响应回归解决方案相比，我们的建议在估计算法、规律性条件以及理论属性方面有很大不同。我们使用模拟和两个实际应用（包括神经影像痴呆研究和数字广告研究）来说明我们提出的方法的有效性。