Applied Mathematics Letters ( IF 3.7 ) Pub Date : 2020-02-24 , DOI: 10.1016/j.aml.2020.106300 Guang-Jing Song , Michael K. Ng
This paper describes a new algorithm for computing Nonnegative Low Rank Matrix (NLRM) approximation for nonnegative matrices. Our approach is completely different from classical nonnegative matrix factorization (NMF) which has been studied for more than twenty five years. For a given nonnegative matrix, the usual NMF approach is to determine two nonnegative low rank matrices such that the distance between their product and the given nonnegative matrix is as small as possible. However, the proposed NLRM approach is to determine a nonnegative low rank matrix such that the distance between such matrix and the given nonnegative matrix is as small as possible. There are two advantages. (i) The minimized distance by the proposed NLRM method can be smaller than that by the NMF method, and it implies that the proposed NLRM method can obtain a better low rank matrix approximation. (ii) Our low rank matrix admits a matrix singular value decomposition automatically which provides a significant index based on singular values that can be used to identify important singular basis vectors, while this information cannot be obtained in the classical NMF. The proposed NLRM approximation algorithm was derived using the alternating projection on the low rank matrix manifold and the non-negativity property. Experimental results are presented to demonstrate the above mentioned advantages of the proposed NLRM method compared the NMF method.
中文翻译:
非负矩阵的非负低秩矩阵逼近
本文介绍了一种用于计算非负矩阵的非负低秩矩阵(NLRM)近似的新算法。我们的方法与经典非负矩阵分解(NMF)完全不同,后者已经研究了25年以上。对于给定的非负矩阵,通常的NMF方法是确定两个非负的低秩矩阵,以使它们的乘积与给定的非负矩阵之间的距离尽可能小。然而,提出的NLRM方法是确定非负的低秩矩阵,使得该矩阵与给定的非负矩阵之间的距离尽可能小。有两个优点。(i)提议的NLRM方法的最小距离可以小于NMF方法的最小距离,这意味着所提出的NLRM方法可以获得更好的低秩矩阵近似。(ii)我们的低秩矩阵会自动接受矩阵奇异值分解,该分解基于奇异值提供可用于识别重要奇异基矢量的有效索引,而传统NMF无法获得此信息。利用在低秩矩阵流形上的交替投影和非负性,推导了所提出的NLRM近似算法。实验结果表明,与NMF方法相比,所提出的NLRM方法具有上述优点。(ii)我们的低秩矩阵会自动接受矩阵奇异值分解,该分解基于奇异值提供可用于识别重要奇异基矢量的有效索引,而传统NMF无法获得此信息。利用在低秩矩阵流形上的交替投影和非负性,推导了所提出的NLRM近似算法。实验结果表明,与NMF方法相比,所提出的NLRM方法具有上述优点。(ii)我们的低秩矩阵会自动接受矩阵奇异值分解,该分解基于奇异值提供可用于识别重要奇异基矢量的有效索引,而传统NMF无法获得此信息。利用在低秩矩阵流形上的交替投影和非负性,推导了所提出的NLRM近似算法。实验结果表明,与NMF方法相比,所提出的NLRM方法具有上述优点。