  Learning task-specific representations of persistence diagrams is an
important problem in topological data analysis and machine learning. However,
current state of the art methods are restricted in terms of their expressivity
as they are focused on Euclidean representations. Persistence diagrams often
contain features of infinite persistence (i.e., essential features) and
Euclidean spaces shrink their importance relative to non-essential features
because they cannot assign infinite distance to finite points. To deal with
this issue, we propose a method to learn representations of persistence
diagrams on hyperbolic spaces, more specifically on the Poincare ball. By
representing features of infinite persistence infinitesimally close to the
boundary of the ball, their distance to non-essential features approaches
infinity, thereby their relative importance is preserved. This is achieved
without utilizing extremely high values for the learnable parameters, thus the
representation can be fed into downstream optimization methods and trained
efficiently in an end-to-end fashion. We present experimental results on graph
and image classification tasks and show that the performance of our method is
on par with or exceeds the performance of other state of the art methods.
