Topological Autoencoders

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 can 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, while exhibiting favourable latent representations on synthetic manifold data sets. Moreover, on real-world data sets, introducing our topological loss leads to more meaningful latent representations while preserving low reconstruction errors.