  Deep Nonnegative Matrix Factorization (deep NMF) has recently emerged as a
valuable technique for extracting multiple layers of features across different
scales. However, all existing deep NMF models and algorithms have primarily
centered their evaluation on the least squares error, which may not be the most
appropriate metric for assessing the quality of approximations on diverse
datasets. For instance, when dealing with data types such as audio signals and
documents, it is widely acknowledged that $\beta$-divergences offer a more
suitable alternative. In this paper, we develop new models and algorithms for
deep NMF using some $\beta$-divergences, with a focus on the Kullback-Leibler
divergence. Subsequently, we apply these techniques to the extraction of facial
features, the identification of topics within document collections, and the
identification of materials within hyperspectral images.
