SIAM Journal on Imaging Sciences, Volume 13, Issue 1, Page 381-421, January 2020.
We consider the nonnegative matrix factorization (NMF) problem with sparsity constraints formulated as a nonconvex composite minimization problem. We introduce four novel proximal gradient based algorithms proven globally convergent to a critical point and which are applicable to sparsity constrained NMF models. Our approach builds on recent results allowing one to lift the classical global Lipschitz continuity requirement through the use of a non-Euclidean Bregman based distance. Since under the proposed framework we are not restricted by the gradient Lipschitz continuity assumption, we can consider new decomposition settings of the NMF problem. Two of the derived schemes are genuine non-Euclidean proximal methods that tackle nonstandard decompositions of the NMF problem. The two other schemes are novel extensions of the well-known state-of-the-art methods (the multiplicative and hierarchical alternating least squares), thus allowing one to significantly broaden the scope of these algorithms. Numerical experiments illustrate the performance of the proposed methods.

中文翻译：

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具有稀疏约束的非负矩阵分解的新近邻梯度方法

SIAM影像科学杂志，第13卷，第1期，第381-421页，2020年1月。
我们考虑稀疏约束的非负矩阵分解（NMF）问题，该问题被构造为非凸复合最小化问题。我们介绍了四种新颖的基于近端梯度的算法，这些算法已被全局证明收敛到临界点，并且适用于稀疏约束的NMF模型。我们的方法基于最近的结果，允许通过使用基于非欧几里德Bregman的距离来满足经典的全球Lipschitz连续性要求。由于在提出的框架下我们不受梯度Lipschitz连续性假设的限制，因此我们可以考虑NMF问题的新分解设置。其中两个推导方案是真正的非欧几里得近端方法，可以解决NMF问题的非标准分解。另外两个方案是对众所周知的最新方法（乘法和分层交替最小二乘）的新颖扩展，因此可以大大扩展这些算法的范围。数值实验说明了所提出方法的性能。