# Riemannian preconditioned algorithms for tensor completion via tensor ring decomposition

## TL;DR

This paper introduces Riemannian preconditioned algorithms for tensor completion using tensor ring decomposition, improving efficiency and accuracy through a novel metric and optimization methods with proven convergence.

## Contribution

The paper develops a new Riemannian metric on tensor ring decomposition space and proposes gradient-based algorithms with convergence guarantees for tensor completion.

## Key findings

- Algorithms outperform existing methods in efficiency.
- Proposed methods achieve better reconstruction accuracy.
- Numerical experiments validate effectiveness on real datasets.

## Abstract

We propose Riemannian preconditioned algorithms for the tensor completion problem via tensor ring decomposition. A new Riemannian metric is developed on the product space of the mode-2 unfolding matrices of the core tensors in tensor ring decomposition. The construction of this metric aims to approximate the Hessian of the cost function by its diagonal blocks, paving the way for various Riemannian optimization methods. Specifically, we propose the Riemannian gradient descent and Riemannian conjugate gradient algorithms. We prove that both algorithms globally converge to a stationary point. In the implementation, we exploit the tensor structure and adopt an economical procedure to avoid large matrix formulation and computation in gradients, which significantly reduces the computational cost. Numerical experiments on various synthetic and real-world datasets -- movie ratings, hyperspectral images, and high-dimensional functions -- suggest that the proposed algorithms are more efficient and have better reconstruction ability than other candidates.