  In this article we propose a novel approach for comparing the persistent
homology representations of two spaces (filtrations). Commonly used methods are
based on numerical summaries such as persistence diagrams and persistence
landscapes, along with suitable metrics (e.g. Wasserstein). These summaries are
useful for computational purposes, but they are merely a marginal of the actual
topological information that persistent homology can provide. Instead, our
approach compares between two topological representations directly in the data
space. We do so by defining a correspondence relation between individual
persistent cycles of two different spaces, and devising a method for computing
this correspondence. Our matching of cycles is based on both the persistence
intervals and the spatial placement of each feature. We demonstrate our new
framework in the context of topological inference, where we use statistical
bootstrap methods in order to differentiate between real features and noise in
point cloud data.

Machine Learning;Computational Geometry;Algebraic Topology;Machine Learning