Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay. (arXiv:2206.08756v1 [math.ST])

We study the tensor-on-tensor regression, where the goal is to connect tensor
responses to tensor covariates with a low Tucker rank parameter tensor/matrix
without the prior knowledge of its intrinsic rank. We propose the Riemannian
gradient descent (RGD) and Riemannian Gauss-Newton (RGN) methods and cope with
the challenge of unknown rank by studying the effect of rank
over-parameterization. We provide the first convergence guarantee for the
general tensor-on-tensor regression by showing that RGD and RGN respectively
converge linearly and quadratically …

arxiv computational gap math optimization regression statistical tensor