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

arXiv: https://arxiv.org/abs/1301.0413 · HTML: https://mlsanity.com/paper/1301.0413/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jan 2013.

## Built on by (most cited first, 5 of 5)

- [Learning to Optimize by Differentiable Programming](https://mlsanity.com/paper/2601.16510/) (2026)
- [l1-2 GLasso: L1-2 Regularized Multi-task Graphical Lasso for Joint Estimation of eQTL Mapping and Gene Network](https://mlsanity.com/paper/2301.02225/) (2023)
- [Orthogonally weighted ℓ2,1 regularization for rank-aware joint sparse recovery: algorithm and analysis](https://mlsanity.com/paper/2311.12282/) (2023)
- [A General Framework for Group Sparsity in Hyperspectral Unmixing Using Endmember Bundles](https://mlsanity.com/paper/2505.14634/) (2025)
- [A proximal subgradient method for nonconvex stochastic optimization under the Kurdyka-Lojasiewicz condition](https://mlsanity.com/paper/2608.05460/) (2026)

## Abstract (arXiv)

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 l1 and l2 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.