A Normalized Bottleneck Distance on Persistence Diagrams and Homology
  Preservation under Dimension Reduction
Bala Krishnamoorthy Nathan H. May
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).