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

Persistence diagrams (PDs) are used as signatures of point cloud data. Two clouds of points can be compared using the bottleneck distance \(\textrm{d}_\textrm{B}\) ${\text{d}}_{\text{B}}$ between their PDs. A potential drawback of this pipeline is that point clouds sampled from topologically similar manifolds can have arbitrarily large \(\textrm{d}_\textrm{B}\) ${\text{d}}_{\text{B}}$ 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, \(\textrm{d}_\textrm{N}\) ${\text{d}}_{\text{N}}$ . In defining \(\textrm{d}_\textrm{N}\) ${\text{d}}_{\text{N}}$ , 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 \(\textrm{d}_\textrm{N}\) ${\text{d}}_{\text{N}}$ 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 \(\textrm{d}_\textrm{N}\) ${\text{d}}_{\text{N}}$ . For a JL map \(f:X \rightarrow f(X)\) $f:X\to f(X)$ , we show that \(\textrm{d}_\textrm{N}({\text {dgm}}(X),{\text {dgm}}(f(X))) < \epsilon \) ${\text{d}}_{\text{N}}(\text{dgm}(X),\text{dgm}(f(X)))<ϵ$ where \({\text {dgm}}(X)\) $\text{dgm}(X)$ is the Vietoris–Rips PD of X, and pairwise distances are preserved by f up to the tolerance \(0< \epsilon < 1\) $0<ϵ<1$ . For mMDS, we present new bounds for \(\textrm{d}_\textrm{B}\) ${\text{d}}_{\text{B}}$ and \(\textrm{d}_\textrm{N}\) ${\text{d}}_{\text{N}}$ between PDs of X and its projection in terms of the eigenvalues of the covariance matrix. And for k-biLipschitz maps, we show that \(\textrm{d}_\textrm{N}\) ${\text{d}}_{\text{N}}$ is bounded by the product of \((k^2-1)/k\) $(k^2-1)/k$ 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.