Partially Observed Dynamic Tensor Response Regression

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.

Keywords:

• Alzheimer’s disease
• Digital advertising
• Neuroimaging analysis
• Nonconvex optimization
• Tensor completion
• Tensor regression

Supplementary Material

The supplementary materials collect all technical proofs and additional numerical results.

Acknowledgments

The authors thank to the editor Professor Ian McKeague, the associate editor and two anonymous reviewers for their valuable comments and suggestions which led to a much improved article. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Office of Naval Research, the National Science Foundation, or the National Institutes of Health.

Additional information

Funding

Will Wei Sun’s research was partially supported by ONR grant N00014-18-1-2759. Jingfei Zhang’s research was partially supported by NSF grant DMS-2015190. Lexin Li’s research was partially supported by NIH grants R01AG061303, R01AG062542, and R01AG034570.