Higher order Matching Pursuit for Low Rank Tensor Learning

TL;DR

This paper introduces higher order matching pursuit algorithms for low rank tensor learning that are scalable, efficient, and have proven linear convergence, applicable to large-scale problems with convex or nonconvex costs.

Contribution

It generalizes matching pursuit methods to tensor learning, providing scalable algorithms with low storage, efficient rank-one tensor computations, and convergence guarantees.

Findings

Algorithms are scalable to large datasets.

Proven linear convergence rate under various conditions.

Experimental results confirm efficiency and effectiveness.

Abstract

Low rank tensor learning, such as tensor completion and multilinear multitask learning, has received much attention in recent years. In this paper, we propose higher order matching pursuit for low rank tensor learning problems with a convex or a nonconvex cost function, which is a generalization of the matching pursuit type methods. At each iteration, the main cost of the proposed methods is only to compute a rank-one tensor, which can be done efficiently, making the proposed methods scalable to large scale problems. Moreover, storing the resulting rank-one tensors is of low storage requirement, which can help to break the curse of dimensionality. The linear convergence rate of the proposed methods is established in various circumstances. Along with the main methods, we also provide a method of low computational complexity for approximately computing the rank-one tensors, with provable approximation ratio, which helps to improve the efficiency of the main methods and to analyze the convergence rate. Experimental results on synthetic as well as real datasets verify the efficiency and effectiveness of the proposed methods.