In this work, we focus on connections between $K$-means clustering approaches and Orthogonal Nonnegative Matrix Factorization (ONMF) methods. We present a novel framework to extract the distance measure and the centroids of the $K$-means method based on first order conditions of the considered ONMF objective function, which exploits the classical alternating minimization schemes of Nonnegative Matrix Factorization (NMF) algorithms. While this technique is characterized by a simple derivation procedure, it can also be applied to non-standard regularized ONMF models. Using this framework, we consider in this work ONMF models with $\ell_1$ and standard $\ell_2$ discrepancy terms with an additional elastic net regularization on both factorization matrices and derive the corresponding distance measures and centroids of the generalized $K$-means clustering model. Furthermore, we give an intuitive view of the obtained results, examine special cases and compare them to the findings described in the literature.
翻译:在这项工作中,我们注重以K$为单位的组合法与正统非负式矩阵系数法(ONMF)方法之间的联系,我们根据考虑的ONMF客观功能的第一阶条件,提出了一个新的框架,以提取以K$为单位的距离计量法和以美元为单位的中间体计算法的中间体,这一功能利用了传统的非负式矩阵系数法(NMF)的交替最小化办法,虽然这一技术的特点是简单的衍生程序,但它也可以适用于非标准的正规化的ONMF模型。我们利用这个框架,在这项工作中考虑以$/ell_1和标准$/ell_2美元为单位的ONMF模型,在两个系数矩阵上增加一个弹性净固定条件,并得出相应的距离计量法和通用的以K$美元为单位的组合模型的中间体。此外,我们从直观的角度审视所获得的结果,研究特殊案例,并将它们与文献中描述的调查结果进行比较。