  Deep convolutional networks (DCNs) learn meaningful representations where
data that share the same abstract characteristics are positioned closer and
closer. Understanding these representations and how they are generated is of
unquestioned practical and theoretical interest. In this work we study the
evolution of the probability density of the ImageNet dataset across the hidden
layers in some state-of-the-art DCNs. We find that the initial layers generate
a unimodal probability density getting rid of any structure irrelevant for
classification. In subsequent layers density peaks arise in a hierarchical
fashion that mirrors the semantic hierarchy of the concepts. Density peaks
corresponding to single categories appear only close to the output and via a
very sharp transition which resembles the nucleation process of a heterogeneous
liquid. This process leaves a footprint in the probability density of the
output layer where the topography of the peaks allows reconstructing the
semantic relationships of the categories.
