Riemannian preconditioned coordinate descent for low multi-linear rank approximation

TL;DR

This paper introduces a memory-efficient Riemannian coordinate descent method for low multi-linear rank tensor approximation, leveraging second-order information and a novel metric to improve convergence and computational efficiency.

Contribution

It proposes a new Riemannian metric and coordinate descent approach for tensor approximation, with a global convergence guarantee and connections to orthogonal iteration.

Findings

Demonstrates computational advantages on high-dimensional tensors

Provides global convergence analysis for the proposed method

Shows the method's steps align with orthogonal iteration

Abstract

This paper presents a memory efficient, first-order method for low multi-linear rank approximation of high-order, high-dimensional tensors. In our method, we exploit the second-order information of the cost function and the constraints to suggest a new Riemannian metric on the Grassmann manifold. We use a Riemmanian coordinate descent method for solving the problem, and also provide a global convergence analysis matching that of the coordinate descent method in the Euclidean setting. We also show that each step of our method with the unit step-size is actually a step of the orthogonal iteration algorithm. Experimental results show the computational advantage of our method for high-dimensional tensors.