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In this work, we focus on connections between $K$-means clustering approachesand Orthogonal Nonnegative Matrix Factorization (ONMF) methods. We present anovel framework to extract the distance measure and the centroids of the$K$-means method based on first order conditions of the considered ONMFobjective function, which exploits the classical alternating minimizationschemes of Nonnegative Matrix Factorization (NMF) algorithms. While thistechnique is characterized by a simple derivation procedure, it can also beapplied to non-standard regularized ONMF models. Using this framework, weconsider in this work ONMF models with $\ell_1$ and standard $\ell_2$discrepancy terms with an additional elastic net regularization on bothfactorization matrices and derive the corresponding distance measures andcentroids of the generalized $K$-means clustering model. Furthermore, we givean intuitive view of the obtained results, examine special cases and comparethem to the findings described in the literature.

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