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.
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