  We propose a novel approach for preserving topological structures of the
input space in latent representations of autoencoders. Using persistent
homology, a technique from topological data analysis, we calculate topological
signatures of both the input and latent space to derive a topological loss
term. Under weak theoretical assumptions, we construct this loss in a
differentiable manner, such that the encoding learns to retain multi-scale
connectivity information. We show that our approach is theoretically
well-founded and that it exhibits favourable latent representations on a
synthetic manifold as well as on real-world image data sets, while preserving
low reconstruction errors.
