Nonnegative matrix factorization (NMF) has been widely studied in recent years due to its effectiveness in representing nonnegative data with parts-based representations. For NMF, a sparser solution implies better parts-based representation.However, current NMF methods do not always generate sparse solutions.In this paper, we propose a new NMF method with log-norm imposed on the factor matrices to enhance the sparseness.Moreover, we propose a novel column-wisely sparse norm, named $\ell_{2,\log}$-(pseudo) norm to enhance the robustness of the proposed method.The $\ell_{2,\log}$-(pseudo) norm is invariant, continuous, and differentiable.For the $\ell_{2,\log}$ regularized shrinkage problem, we derive a closed-form solution, which can be used for other general problems.Efficient multiplicative updating rules are developed for the optimization, which theoretically guarantees the convergence of the objective value sequence.Extensive experimental results confirm the effectiveness of the proposed method, as well as the enhanced sparseness and robustness.

翻译：近年来,对非负矩阵因子化(NMF)进行了广泛研究,因为其有效代表了非负值数据以部分代表形式。对于NMF, 稀释式的解决方案意味着更好地以部分为代表。 但是,目前的NMF方法并不总是产生稀释式的解决方案。 在本文中,我们提议了一种新的NMF方法,在要素矩阵上加对正对-北线以强化稀释性。 之后,我们提议了一种新颖的分栏式稀释规范,名为$\ell ⁇ 2,\log}-(假币),以加强拟议方法的稳健性。 $\ell ⁇ 2,\log}-(假币)的规范是不变的、持续的和不同的。 对于$=2,\log}正规化的萎缩问题,我们提出了一种封闭式解决方案,可用于解决其他一般性问题。我们为优化制定了有效的多版本更新规则,从理论上保证了客观价值序列的趋同。广泛的实验结果证实了拟议方法的有效性,以及强化的稀缺性和稳健性。