Topological Autoencoders++: Fast and Accurate Cycle-Aware Dimensionality Reduction

Abstract

This paper presents a novel topology-aware dimensionality reduction approach aiming at accurately visualizing the cyclic patterns present in high dimensional data. To that end, we build on the Topological Autoencoders (TopoAE) formulation. First, we provide a novel theoretical analysis of its associated loss and show that a zero loss indeed induces identical persistence pairs (in high and low dimensions) for the $0$-dimensional persistent homology (PH$^0$) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to PH$^d$ for $d\ge 1$. Based on this observation, we introduce a novel generalization of TopoAE to $1$-dimensional persistent homology (PH$^1$), called TopoAE++, for the accurate generation of cycle-aware planar embeddings, addressing the above failure case. This generalization is based on the notion of cascade distortion, a new penalty term favoring an isometric embedding of the $2$-chains filling persistent $1$-cycles, hence resulting in more faithful geometrical reconstructions of the $1$-cycles in the plane. We further introduce a novel, fast algorithm for the exact computation of PH for Rips filtrations in the plane, yielding improved runtimes over previously documented topology-aware methods. Our method also achieves a better balance between the topological accuracy, as measured by the Wasserstein distance, and the visual preservation of the cycles in low dimensions. Our C++ implementation is available at https://github.com/MClemot/TopologicalAutoencodersPlusPlus.