Regularized Orthogonal Nonnegative Matrix Factorization and $K$-means Clustering

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.