Continuous dictionaries meet low-rank tensor approximations

TL;DR

This paper establishes a connection between continuous sparse coding and low-rank tensor approximations, introducing a new optimization approach using BLasso and Frank-Wolfe algorithms for tensor data.

Contribution

It reveals that certain continuous dictionary-based regularizations are equivalent to BLasso problems and proposes a novel tensor rank selection method using Frank-Wolfe.

Findings

BLasso solutions match nuclear-norm regularized systems for specific dictionaries

Proposed Frank-Wolfe algorithm effectively performs tensor rank selection

New optimization framework for low-rank tensor approximation

Abstract

In this short paper we bridge two seemingly unrelated sparse approximation topics: continuous sparse coding and low-rank approximations. We show that for a specific choice of continuous dictionary, linear systems with nuclear-norm regularization have the same solutions as a BLasso problem. Although this fact was already partially understood in the matrix case, we further show that for tensor data, using BLasso solvers for the low-rank approximation problem leads to a new branch of optimization methods yet vastly unexplored. In particular, the proposed Frank-Wolfe algorithm is showcased on an automatic tensor rank selection problem.