A Fast Gradient Method for Nonnegative Sparse Regression with Self Dictionary

TL;DR

This paper introduces a fast gradient method for nonnegative sparse regression with self dictionaries, enabling efficient computation of nonnegative matrix factorizations under the separability assumption, with applications to hyperspectral imaging.

Contribution

The paper proposes a novel smooth optimization model with linear constraints for nonnegative sparse regression and develops a fast gradient algorithm for it.

Findings

The proposed method efficiently solves large-scale convex problems.

It outperforms existing methods on synthetic and real data.

The approach is effective for hyperspectral image analysis.

Abstract

A nonnegative matrix factorization (NMF) can be computed efficiently under the separability assumption, which asserts that all the columns of the given input data matrix belong to the cone generated by a (small) subset of them. The provably most robust methods to identify these conic basis columns are based on nonnegative sparse regression and self dictionaries, and require the solution of large-scale convex optimization problems. In this paper we study a particular nonnegative sparse regression model with self dictionary. As opposed to previously proposed models, this model yields a smooth optimization problem where the sparsity is enforced through linear constraints. We show that the Euclidean projection on the polyhedron defined by these constraints can be computed efficiently, and propose a fast gradient method to solve our model. We compare our algorithm with several state-of-the-art methods on synthetic data sets and real-world hyperspectral images.