A Method for Finding Structured Sparse Solutions to Non-negative Least Squares Problems with Applications

TL;DR

This paper introduces a novel method for solving structured sparse non-negative least squares problems, particularly in hyperspectral imaging and DOAS, using a scaled gradient projection algorithm with Hoyer measure-based penalties.

Contribution

It proposes a new nonconvex optimization approach with a scaled gradient projection algorithm for structured sparsity in NNLS problems, addressing issues with high dictionary coherence.

Findings

Effective in hyperspectral demixing and DOAS analysis

Outperforms traditional convex and greedy methods in coherence scenarios

Demonstrates promising numerical results on real data

Abstract

Demixing problems in many areas such as hyperspectral imaging and differential optical absorption spectroscopy (DOAS) often require finding sparse nonnegative linear combinations of dictionary elements that match observed data. We show how aspects of these problems, such as misalignment of DOAS references and uncertainty in hyperspectral endmembers, can be modeled by expanding the dictionary with grouped elements and imposing a structured sparsity assumption that the combinations within each group should be sparse or even 1-sparse. If the dictionary is highly coherent, it is difficult to obtain good solutions using convex or greedy methods, such as non-negative least squares (NNLS) or orthogonal matching pursuit. We use penalties related to the Hoyer measure, which is the ratio of the $l_1$ and $l_2$ norms, as sparsity penalties to be added to the objective in NNLS-type models. For solving the resulting nonconvex models, we propose a scaled gradient projection algorithm that requires solving a sequence of strongly convex quadratic programs. We discuss its close connections to convex splitting methods and difference of convex programming. We also present promising numerical results for example DOAS analysis and hyperspectral demixing problems.