Computing sparse solutions to overdetermined linear systems is a ubiquitous problem in several fields such as regression analysis, signal and image processing, information theory and machine learning. Additional non-negativity constraints in the solution are useful for interpretability. Most of the previous research efforts aimed at approximating the sparsity constrained linear least squares problem, and/or finding local solutions by means of descent algorithms. The objective of the present paper is to report on an efficient and modular implicit enumeration algorithm to find provably optimal solutions to the NP-hard problem of sparsity-constrained non-negative least squares. We focus on the problem where the system is assumed to be over-determined where the matrix has full column rank. Numerical results with real test data as well as comparisons of competing methods and an application to hyperspectral imaging are reported. Finally, we present a Python library implementation of our algorithm.

计算超定线性系统的稀疏解是回归分析、信号和图像处理、信息论和机器学习等多个领域中普遍存在的问题。解决方案中的其他非负约束对于可解释性很有用。以前的大部分研究工作旨在近似稀疏约束线性最小二乘问题，和/或通过下降算法寻找局部解决方案。本文的目的是报告一种高效的模块化隐式枚举算法，以找到可证明的最优解来解决稀疏约束非负最小二乘法的 NP 难题。我们专注于在矩阵具有完整列秩的情况下假设系统过度确定的问题。报告了具有真实测试数据的数值结果以及竞争方法的比较和高光谱成像的应用。最后，我们展示了我们算法的 Python 库实现。