# A Fast Gradient Method for Nonnegative Sparse Regression with Self Dictionary

arXiv: https://arxiv.org/abs/1610.01349 · HTML: https://mlsanity.com/paper/1610.01349/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Oct 2016.

## Built on by (most cited first, 8 of 8)

- [Memory-Efficient Convex Optimization for Self-Dictionary Separable Nonnegative Matrix Factorization: A Frank-Wolfe Approach](https://mlsanity.com/paper/2109.11135/) (2021)
- [Refinement of Hottopixx Method for Nonnegative Matrix Factorization Under Noisy Separability](https://mlsanity.com/paper/2109.02863/) (2021)
- [Endmember Extraction from Hyperspectral Images Using Self-Dictionary Approach with Linear Programming](https://mlsanity.com/paper/2404.13098/) (2024)
- [Co-Separable Nonnegative Matrix Factorization](https://mlsanity.com/paper/2109.00749/) (2021)
- [Coseparable Nonnegative Tensor Factorization With T-CUR Decomposition](https://mlsanity.com/paper/2401.16836/) (2024)
- [A Provably-Correct and Robust Convex Model for Smooth Separable NMF](https://mlsanity.com/paper/2511.07109/) (2025)
- [Hyperspectral Image Data Reduction for Endmember Extraction](https://mlsanity.com/paper/2512.10506/) (2025)
- [Separable Nonnegative Matrix Factorization Using Powered Ratio-of-Norms Regularization](https://mlsanity.com/paper/2608.28799/) (2026)

## Abstract (arXiv)

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.