Partially Observed Dynamic Tensor Response Regression
This work addresses a practical challenge in data science for applications like neuroimaging and advertising, but it is incremental as it builds on existing tensor methods by handling partial observations.
The authors tackled the problem of regression with partially observed dynamic tensor responses by developing a model that incorporates low-rank, sparsity, and fusion structures, and they derived finite-sample error bounds for their estimator, demonstrating efficacy through simulations and real applications in neuroimaging and digital advertising.
In modern data science, dynamic tensor data is prevailing in numerous applications. An important task is to characterize the relationship between such dynamic tensor and external covariates. However, the tensor data is often only partially observed, rendering many existing methods inapplicable. In this article, we develop a regression model with partially observed dynamic tensor as the response and external covariates as the predictor. We introduce the low-rank, sparsity and fusion structures on the regression coefficient tensor, and consider a loss function projected over the observed entries. We develop an efficient non-convex alternating updating algorithm, and derive the finite-sample error bound of the actual estimator from each step of our optimization algorithm. Unobserved entries in 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, a neuroimaging dementia study and a digital advertising study.