Abstract
In this thesis, we systematically address two important research aims in tensor data analysis. The first aim is to relate partially observed dynamic tensor response to external covariates and understand how covariates affect tensor data. It is motivated by a neuroimaging study of dementia and a digital advertising placement application. The second aim is to interactively complete a partially observed tensor with bandit feedback. This aim is motivated by multi-dimensional online decision-making problems such as online advertisement recommendation. In the first part of the thesis, 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 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 posed 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 which include a neuroimaging dementia study and a digital advertising study. In the second part of the thesis, we introduce a new stochastic low-rank tensor bandits model, which considers a class of bandits whose mean rewards can be represented as a low-rank tensor. We propose three novel algorithms including tensor epoch-greedy, tensor elimination and tensor ensemble sampling, where the first two methods are for tensor bandits without contexts information and the last method is for contextual tensor bandits. We illustrate that our algorithms outperform various state-of-the-art approaches that ignore the low-rank tensor structure.