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

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) (Moor et al., 2020) 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 ($\text{PH}^{0}$) of the Rips filtration. We also provide a counter example showing that this property no longer holds for a naive extension of TopoAE to $\text{PH}^{d}$ for $d\geq 1$. Based on this observation, we introduce a novel generalization of TopoAE to 1-dimensional persistent homology ($\text{PH}^{1}$), 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 $\text{PH}^{}$ 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.