  In the context of recent deep clustering studies, discriminative models
dominate the literature and report the most competitive performances. These
models learn a deep discriminative neural network classifier in which the
labels are latent. Typically, they use multinomial logistic regression
posteriors and parameter regularization, as is very common in supervised
learning. It is generally acknowledged that discriminative objective functions
(e.g., those based on the mutual information or the KL divergence) are more
flexible than generative approaches (e.g., K-means) in the sense that they make
fewer assumptions about the data distributions and, typically, yield much
better unsupervised deep learning results. On the surface, several recent
discriminative models may seem unrelated to K-means. This study shows that
these models are, in fact, equivalent to K-means under mild conditions and
common posterior models and parameter regularization. We prove that, for the
commonly used logistic regression posteriors, maximizing the $L_2$ regularized
mutual information via an approximate alternating direction method (ADM) is
equivalent to a soft and regularized K-means loss. Our theoretical analysis not
only connects directly several recent state-of-the-art discriminative models to
K-means, but also leads to a new soft and regularized deep K-means algorithm,
which yields competitive performance on several image clustering benchmarks.
