This paper addresses the issue of building a part-based representation of a dataset of images. More precisely, we look for a non-negative, sparse decomposition of the images on a reduced set of atoms, in order to unveil a morphological and explainable structure of the data. Additionally, we want this decomposition to be computed online for any new sample that is not part of the initial dataset. Therefore, our solution relies on a sparse, non-negative auto-encoder, where the encoder is deep (for accuracy) and the decoder shallow (for explainability). This method compares favorably to the state-of-the-art online methods on two benchmark datasets (MNIST and Fashion MNIST) and on a hyperspectral image, according to classical evaluation measures and to a new one we introduce, based on the equivariance of the representation to morphological operators.