Bottleneck Distance Between Persistence Diagrams

Computes the bottleneck distance between two persistence diagrams. For finite pairs, the bottleneck distance is $$W_\infty(D_1, D_2) = \inf_{\gamma} \sup_{p \in D_1} \|p - \gamma(p)\|_\infty,$$ where \(\gamma\) ranges over bijections \(D_1 \cup \Delta \to D_2 \cup \Delta\) and \(\Delta = \{(x,x)\}\) is the diagonal. Each point may match a point in the other diagram or its projection onto the diagonal at cost \(|d - b|/2\). Computed via binary search on \(\varepsilon\) plus a Kuhn bipartite-matching feasibility check.

Essential classes (death = Inf in VR mode, or death = 0 in clique mode) are matched one-to-one within each dimension. If the diagrams have different numbers of essential classes in some dimension, the bottleneck distance for that dimension is Inf.

Usage

bottleneck_distance(d1, d2, dimension = NULL, tol = .Machine$double.eps^0.5)

Arguments

d1, d2

persistent_homology objects, or data.frames with columns dimension, birth, death.

dimension

Integer vector of dimensions to compare. NULL (default) compares all dimensions appearing in either diagram and returns a named numeric vector.

tol

Numerical tolerance for binary search (default .Machine$double.eps ^ 0.5).

Value

Named numeric vector. Names are "dim_<k>". Inf indicates a structural mismatch (different essential counts in that dimension); a self-distance is always 0.

References

Edelsbrunner, H. & Harer, J. (2010). Computational Topology: An Introduction. AMS. Section VIII.

Examples

mat1 <- matrix(c(0, .6, .5, .6, 0, .4, .5, .4, 0), 3, 3)
rownames(mat1) <- colnames(mat1) <- c("A","B","C")
ph1 <- persistent_homology(mat1, n_steps = 5)
bottleneck_distance(ph1, ph1)  # self-distance is 0
#> dim_0 
#>     0