Hyperspectral unmixing is an important step to learn the material categories and corresponding distributions in a scene. Over the past decade, nonnegative matrix factorization (NMF) has been utilized for this task, thanks to its good physical interpretation. The solution space of NMF is very huge due to its nonconvex objective function for both variables simultaneously. Many convex and nonconvex sparse regularizations are embedded into NMF to limit the number of trivial solutions. Unfortunately, they either produce biased sparse solutions or unbiased sparse solutions with the sacrifice of the convex objective function of NMF with respect to individual variable. In this article, we enhance NMF by introducing a generalized minimax concave (GMC) sparse regularization. The GMC regularization is nonconvex and nonseparable, enabling promotion of unbiased and sparser results while simultaneously preserving the convexity of NMF for each variable separately. Therefore, GMC-NMF better avoids being trapped into local minimals, and thereby produce physically meaningful and accurate results. Extensive experimental results on synthetic data and real-world data verify its utility when compared with several state-of-the-art approaches.

中文翻译：

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用于高光谱解混的非凸非可分稀疏非负矩阵分解

高光谱解混是学习场景中的材料类别和相应分布的重要步骤。在过去的十年中，由于其良好的物理解释，非负矩阵分解 (NMF) 已被用于此任务。NMF 的解空间非常巨大，因为它同时针对两个变量的非凸目标函数。许多凸和非凸稀疏正则化被嵌入到 NMF 中以限制平凡解的数量。不幸的是，它们要么产生有偏稀疏解，要么产生无偏稀疏解，同时牺牲了 NMF 相对于单个变量的凸目标函数。在本文中，我们通过引入广义极小极大凹（GMC）稀疏正则化来增强 NMF。GMC 正则化是非凸且不可分的，能够促进无偏和更稀疏的结果，同时分别为每个变量保留 NMF 的凸性。因此，GMC-NMF 更好地避免陷入局部极小值，从而产生物理上有意义且准确的结果。与几种最先进的方法相比，对合成数据和现实世界数据的大量实验结果验证了其实用性。