Topological Autoencoders
Authors: Michael Moor, Max Horn, Bastian Rieck, Karsten Borgwardt
Keywords: Topology, Deep Learning, Autoencoders, Persistent Homology, Representation Learning, Dimensionality Reduction, Topological Machine Learning, Topological Data Analysis
Abstract: 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 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.