Kullback–Leibler divergence, it is in fact equivalent to another instance of multinomial PCA, probabilistic latent semantic analysis, trained by maximum likelihood estimation. That method is commonly used for analyzing and clustering textual data and is also related to the latent class model. NMF with the least-squares objective is equivalent to a relaxed form of K-means clustering: the matrix factor contains cluster centroids and contains cluster membership indicators. This provides a theoretical foundation for using NMF for data clustering. However, k-means does not enforce non-negativity on its centroids, so the closest analogy is in fact with "semi-NMF". NMF can be seen as a two-layer directed graphical model with one layer of observed random variables and one layer of hidden random variables. NMF extends beyond matrices to tensors of arbitrary order. This extension may be viewed as a non-negative counterpart to, e.g., the PARAFAC model. Other extensions of NMF include joint factorization of several data matrices and tensors where some factors are shared. Such models are useful for sensor fusion and relational learning. NMF is an instance of nonnegative quadratic programming (NQP), just like the support vector machine (SVM). However, SVM and NMF are related at a more intimate level than that of NQP, which allows direct application of the solution algorithms developed for either of the two methods to problems in both domains. Uniqueness The factorization is not unique: A matrix and its inverse can be used to transform the two factorization matrices by, e.g., If the two new matrices and are non-negative they form another parametrization of the factorization. The non-negativity of and applies at least if is a non-negative monomial matrix. In this simple case it will just correspond to a scaling and a permutation. More control over the non-uniqueness of NMF is obtained with sparsity constraints. Applications Astronomy In astronomy, NMF is a promising method for dimension reduction in the sense that astrophysical signals are non-negative. NMF has been applied to the spectroscopic observations and the direct imaging observations as a method to study the common properties of astronomical objects and post-process the astronomical observations. The advances in the spectroscopic observations by Blanton & Roweis (2007) takes into account of the uncertainties of astronomical observations, which is later improved by Zhu (2016) where missing data are also considered and parallel computing is enabled. Their method is then adopted by Ren et al. (2018) to the direct imaging field