A Normalized Bottleneck Distance on Persistence Diagrams and Homology Preservation under Dimension Reduction (2306.06727v2)

Abstract: Persistence diagrams (PDs) are used as signatures of point cloud data. Two clouds of points can be compared using the bottleneck distance d_B between their PDs. A potential drawback of this pipeline is that point clouds sampled from topologically similar manifolds can have arbitrarily large d_B when there is a large scaling between them. This situation is typical in dimension reduction frameworks. We define, and study properties of, a new scale-invariant distance between PDs termed normalized bottleneck distance, d_N. In defining d_N, we develop a broader framework called metric decomposition for comparing finite metric spaces of equal cardinality with a bijection. We utilize metric decomposition to prove a stability result for d_N by deriving an explicit bound on the distortion of the bijective map. We then study two popular dimension reduction techniques, Johnson-Lindenstrauss (JL) projections and metric multidimensional scaling (mMDS), and a third class of general biLipschitz mappings. We provide new bounds on how well these dimension reduction techniques preserve homology with respect to d_N. For a JL map f that transforms input X to f(X), we show that d_N(dgm(X),dgm(f(X))) < e, where dgm(X) is the Vietoris-Rips PD of X, and pairwise distances are preserved by f up to the tolerance 0 < \epsilon < 1. For mMDS, we present new bounds for d_B and d_N between PDs of X and its projection in terms of the eigenvalues of the covariance matrix. And for k-biLipschitz maps, we show that d_N is bounded by the product of (k^2-1)/k and the ratio of diameters of X and f(X). Finally, we use computational experiments to demonstrate the increased effectiveness of using the normalized bottleneck distance for clustering sets of point clouds sampled from different shapes.