  Representations learnt through deep neural networks tend to be highly
informative, but opaque in terms of what information they learn to encode. We
introduce an approach to probabilistic modelling that learns to represent data
with two separate deep representations: an invariant representation that
encodes the information of the class from which the data belongs, and an
equivariant representation that encodes the symmetry transformation defining
the particular data point within the class manifold (equivariant in the sense
that the representation varies naturally with symmetry transformations). This
approach is based primarily on the strategic routing of data through the two
latent variables, and thus is conceptually transparent, easy to implement, and
in-principle generally applicable to any data comprised of discrete classes of
continuous distributions (e.g. objects in images, topics in language,
individuals in behavioural data). We demonstrate qualitatively compelling
representation learning and competitive quantitative performance, in both
supervised and semi-supervised settings, versus comparable modelling approaches
in the literature with little fine tuning.
