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Tensors are becoming prevalent in modern applications such as medical imaging and digital marketing. In this paper, we propose a sparse tensor additive regression (STAR) that models a scalar response as a flexible nonparametric function of tensor covariates. The proposed model effectively exploits the sparse and low-rank structures in the tensor additive regression. We formulate the parameter estimation as a non-convex optimization problem, and propose an efficient penalized alternating minimization algorithm. We establish a non-asymptotic error bound for the estimator obtained from each iteration of the proposed algorithm, which reveals an interplay between the optimization error and the statistical rate of convergence. We demonstrate the efficacy of STAR through extensive comparative simulation studies, and an application to the click-through-rate prediction in online advertising. 

We propose a general framework of using a multi-level log-Gaussian Cox process to model repeatedly observed point processes with complex structures; such type of data has become increasingly available in various areas including medical research, social sciences, economics, and finance due to technological advances. A novel nonparametric approach is developed to efficiently and consistently estimate the covariance functions of the latent Gaussian processes at all levels. To predict the functional principal component scores, we propose a consistent estimation procedure by maximizing the conditional likelihood of super-positions of point processes. We further extend our procedure to the bivariate point process case in which potential correlations between the processes can be assessed. Asymptotic properties of the proposed estimators are investigated, and the effectiveness of our procedures is illustrated through a simulation study and an application to a stock trading dataset.