Riemannian preconditioned coordinate descent for low multi-linear rank approximation

2021/09/03 by Hamed, Mohammad, Hosseini, Reshad
#15A69 #58D17 #90C26 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2109.01632

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.

Related

• Low-rank tensor completion: a Riemannian manifold preconditioning approach
• A Riemannian rank-adaptive method for higher-order tensor completion in the tensor-train format
• Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
• A Single-Mode Quasi Riemannian Gradient Descent Algorithm for Low-Rank Tensor Recovery
• Structured Low-Rank Tensor Learning