  We study the problem of learning representations with controllable
connectivity properties. This is beneficial in situations when the imposed
structure can be leveraged upstream. In particular, we control the connectivity
of an autoencoder's latent space via a novel type of loss, operating on
information from persistent homology. Under mild conditions, this loss is
differentiable and we present a theoretical analysis of the properties induced
by the loss. We choose one-class learning as our upstream task and demonstrate
that the imposed structure enables informed parameter selection for modeling
the in-class distribution via kernel density estimators. Evaluated on computer
vision data, these one-class models exhibit competitive performance and, in a
low sample size regime, outperform other methods by a large margin. Notably,
our results indicate that a single autoencoder, trained on auxiliary
(unlabeled) data, yields a mapping into latent space that can be reused across
datasets for one-class learning.
