Wasserstein Distance Between Persistence Diagrams — hg

Computes the finite-order Wasserstein distance between persistence diagrams. For order \(q\), it minimizes the sum of powered matching costs over bijections between diagram points and copies of the diagonal, then takes the \(q\)-th root. Point-to-point costs use an \(L_p\) ground metric; internal_p = Inf gives the usual \(L_\infty\) convention used by GUDHI.

Usage

hg_wasserstein(d1, d2, dimension = NULL, order = 1, internal_p = Inf)

Arguments

d1, d2: persistent_homology objects from hg_homology(), or data.frames with columns dimension, birth, death (finite births; deaths finite or Inf). An invalid diagram raises hypergraphs_bad_input.
dimension: Integer vector of dimensions to compare. NULL (default) compares all dimensions appearing in either diagram.
order: Finite Wasserstein order, a number greater than or equal to one. The common choices are 1 and 2.
internal_p: Ground-metric order, a number greater than or equal to one or Inf (the default).

Value

Named numeric vector, one value per requested dimension. Names are "dim_<k>".

Details

Finite points may match the diagonal. Essential classes are matched only to essential classes, and a dimension whose essential counts differ has distance Inf. An hg_homology() result marks its essential classes itself (death = Inf in Vietoris–Rips mode, death = 0 in clique mode and for a window complex); in a data.frame an essential class is written with death = Inf, and death = 0 is an ordinary finite death. The finite assignment is solved exactly with a native Hungarian algorithm, so no optional optimization package is required. Distances are computed on rescaled coordinates, so large orders and wide coordinate ranges do not overflow.

References

Kerber, M., Morozov, D., & Nigmetov, A. (2017). Geometry helps to compare persistence diagrams. Journal of Experimental Algorithmics, 22, 1–20. doi:10.1145/3064175

Examples

d1 <- data.frame(dimension = 0L, birth = 0, death = 2)
d2 <- data.frame(dimension = 0L, birth = 0, death = 3)
hg_wasserstein(d1, d2)
#> dim_0 
#>     1