A Normalized Bottleneck Distance on Persistence Diagrams and Homology Preservation under Dimension Reduction
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Persistence diagrams are used as signatures of point cloud data assumed to be sampled from manifolds, and represent their topology in a compact fashion. Further, two given clouds of points can be compared by directly comparing their persistence diagrams using the bottleneck distance, d_B. But one potential drawback of this pipeline is that point clouds sampled from topologically similar manifolds can have arbitrarily large d_B values when there is a large degree of scaling between them. This situation is typical in dimension reduction frameworks that are also aiming to preserve topology. We define a new scale-invariant distance between persistence diagrams termed normalized bottleneck distance, d_N, and study its properties. In defining d_N, we also 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 associated 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 persistence diagram of X, and 0 < e < 1 is the tolerance up to which pairwise distances are preserved by f. For mMDS, we present new bounds for both d_B and d_N between persistence diagrams 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).
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