Abstract:
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Motivated by structural and functional neuroimaging analysis, we propose a new class of tensor response regression models. The model embeds two key low-dimensional structures: sparsity and low-rankness, and can handle both a general and a symmetric tensor response. We formulate parameter estimation of our model as a non-convex optimization problem, and develop an efficient alternating updating algorithm. Theoretically, we establish a non-asymptotic estimation error bound for the actual estimator obtained from our algorithm. This error bound reveals an interesting interaction between the computational efficiency and the statistical rate of convergence. Based on this general error bound, we further obtain an optimal estimation error rate when the distribution of the error tensor is Gaussian. Our technical analysis is based upon exploitation of the bi-convex structure of the objective function and a careful characterization of the impact of an intermediate sparse tensor decomposition step. Simulations and an analysis of the autism spectrum disorder imaging data further illustrate the efficacy of our method.
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