一种稀疏图正则化的非负低秩矩阵分解算法

非负矩阵分解方法(non-negative matrix factorization，NMF)广泛应用于图像聚类、计算机视觉、信息检索等领域。但是，现有的NMF方法还存在一些不足之处：①NMF方法直接在高维原始图像数据集上计算它的低维表示，而实际上原始图像数据集的有效信息常常隐藏在它的低秩结构中；②NMF方法还存在对噪声敏感以及鲁棒性差的缺点。为了提高NMF算法的鲁棒性和可解释性，提出一种稀疏图正则化的非负低秩矩阵分解算法(sparse graph regularized non-negative low-rank matrix factorization,SGNLMF)。通过低秩约束和图正则化，SGNLMF算法同时利用了数据的几何信息和有效低秩结构；此外，SGNLMF算法还对基矩阵加以稀疏约束，使得其鲁棒性和可解释性均有一定的提升。还提出了一种求解SGNLMF的迭代算法，并从理论上分析了该求解算法的收敛性。通过在ORL和YaleB数据库上的实验结果表明SGNLMF算法的有效性。

A non-negative low-rank matrix factorization algorithm for regularization of sparse graphs

NMF method is widely used in image clustering,computer vision, information retrieval and other fields. However, the existing NMF methods still have some shortcomings: firstly, the NMF method calculates its low-dimensional representation directly on the high-dimensional original image data set, but in fact the effective information of the original image data set is often hidden in its low-rank structure; secondly,the NMF method also has the shortcomings of being sensitive to noise and poor robustness.In order to improve the robustness and interpretability of the NMF algorithm,a sparse graph regularized non-negative low-rank matrix factorization (SGNLMF) is proposed.Through low rank constraint and graph regularization,SGNLMF algorithm utilizes both geometric information of data and effective low rank structure.In addition, SGNLMF algorithm also constrains the base matrix sparsely, which improves its robustness and interpretability to a certain extent. In addition, an iterative algorithm for solving SGNLMF is proposed and its convergence is theoretically analyzed. The experimental results on ORL and YaleB databases show the effectiveness of SGNLMF.