Nonparametric Tensor Regression with Low-Rank Decomposition

In this paper, we study the tensor-variate regression problems and propose its nonparametric extension, which we break a nonlinear relationship on high-dimensional tensors into simple local functions with incorporating low-rank tensor decompositions. Compared with the naive nonparametric approach, our formulation drastically improves the convergence rate of estimation while maintaining the consistency to the same function class under specific conditions. To estimate the regression function, we provide two estimation methods; the Bayesian estimator with the Gaussian process prior and the Nadaraya-Watson estimator. Experimental results confirm its theoretical properties and demonstrate the high performance for predicting a summary statistic of a real complex network.