tensor-on-tensor regression: riemannian optimization, over-parameterization, statistical-computational gap and their interplay

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The purpose of the tensor-on-tensor regression, which we examine, is to relate tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without being aware of its intrinsic rank beforehand.

By examining the impact of rank over-parameterization, we suggest the Riemannian Gradient Descent (RGD) and Riemannian Gauss-Newton (RGN) methods to address the problem of unknown rank. By demonstrating that RGD and RGN, respectively, converge linearly and quadratically to a statistically optimal estimate in both rank correctly-parameterized and over-parameterized scenarios, we offer the first convergence guarantee for the generic tensor-on-tensor regression. According to our theory, Riemannian optimization techniques automatically adjust to over-parameterization without requiring implementation changes.

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A? ________ is an arrangement of r objects chosen from n distinct objects without repetition and without regard to order

aliya0001 [1]

If we do not care about order, then a combination is an arrangement of r objects chosen from n distinct objects. It can often be seen written as:

$^{n}C_r$ or $\left(\begin{array}{cc}n\\r\end{array}\right)$

Its factorial form is:

$^{n}C_r = \frac{n!}{r!(n - r)!}$