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Deep embedded clustering has become a dominating approach to unsupervised categorization of objects with deep neural networks. The optimization of the most popular methods alternates between the training of a deep autoencoder and a k-means clustering of the autoencoder's embedding. The diachronic setting, however, prevents the former to benefit from valuable information acquired by the latter. In this paper, we present an alternative where the autoencoder and the clustering are learned simultaneously. This is achieved by providing novel theoretical insight, where we show that the objective function of a certain class of Gaussian mixture models (GMMs) can naturally be rephrased as the loss function of a one-hidden layer autoencoder thus inheriting the built-in clustering capabilities of the GMM. That simple neural network, referred to as the clustering module, can be integrated into a deep autoencoder resulting in a deep clustering model able to jointly learn a clustering and an embedding. Experiments confirm the equivalence between the clustering module and Gaussian mixture models. Further evaluations affirm the empirical relevance of our deep architecture as it outperforms related baselines on several data sets.