  Morphological neural networks, or layers, can be a powerful tool to boost the
progress in mathematical morphology, either on theoretical aspects such as the
representation of complete lattice operators, or in the development of image
processing pipelines. However, these architectures turn out to be difficult to
train when they count more than a few morphological layers, at least within
popular machine learning frameworks which use gradient descent based
optimization algorithms. In this paper we investigate the potential and
limitations of differentiation based approaches and back-propagation applied to
morphological networks, in light of the non-smooth optimization concept of
Bouligand derivative. We provide insights and first theoretical guidelines, in
particular regarding initialization and learning rates.
