Low-rank tensor completion: a Riemannian manifold preconditioning approach

TL;DR

This paper introduces a Riemannian manifold preconditioning method for tensor completion that leverages a novel metric to improve optimization efficiency, outperforming existing algorithms on synthetic and real datasets.

Contribution

It develops a new Riemannian metric tailored for tensor completion, enabling preconditioned optimization algorithms with enhanced performance.

Findings

Algorithms outperform state-of-the-art methods

Effective in both synthetic and real-world datasets

Robust convergence across different setups

Abstract

We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms for batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.