A Normalized Bottleneck Distance on Persistence Diagrams and Homology Preservation under Dimension Reduction

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).