New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition

TL;DR

This paper introduces Riemannian preconditioned algorithms for low-rank tensor completion that leverage a novel non-Euclidean metric, improving efficiency and robustness over existing methods.

Contribution

The paper develops new Riemannian algorithms with a specialized metric based on Hessian approximation, offering global convergence guarantees and enhanced practical performance.

Findings

More memory- and time-efficient than state-of-the-art algorithms

Exhibit greater tolerance to overestimated rank parameters

Achieve effective tensor recovery on synthetic and real data

Abstract

We propose new Riemannian preconditioned algorithms for low-rank tensor completion via the polyadic decomposition of a tensor. These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. This new metric is designed using an approximation of the diagonal blocks of the Hessian of the tensor completion cost function, thus has a preconditioning effect on these algorithms. We prove that the proposed Riemannian gradient descent algorithm globally converges to a stationary point of the tensor completion problem, with convergence rate estimates using the $\L{}$ojasiewicz property. Numerical results on synthetic and real-world data suggest that the proposed algorithms are more efficient in memory and time compared to state-of-the-art algorithms. Moreover, the proposed algorithms display a greater tolerance for overestimated rank parameters in terms of the tensor recovery performance, thus enable a flexible choice of the rank parameter.