Bottleneck distance user manual

Definition

_images/perturb_pd.png

Bottleneck distance is the length of the longest edge

Bottleneck distance measures the similarity between two persistence diagrams. It’s the shortest distance b for which there exists a perfect matching between the points of the two diagrams (+ all the diagonal points) such that any couple of matched points are at distance at most b, where the distance between points is the sup norm in R2.
Author

François Godi

Since

GUDHI 2.0.0

License

MIT (GPL v3)

Requires

CGAL ≥ 4.11.0

This implementation is based on ideas from “Geometry Helps in Bottleneck Matching and Related Problems” [18]. Another relevant publication, although it was not used is “Geometry Helps to Compare Persistence Diagrams” [20].

Function

gudhi.bottleneck_distance(diagram_1: numpy.ndarray[float64], diagram_2: numpy.ndarray[float64], e: float = 2.2250738585072014e-308) → float

This function returns the point corresponding to a given vertex.

Parameters

  • diagram_1 (numpy array of shape (m,2)) – The first diagram.

  • diagram_2 (numpy array of shape (n,2)) – The second diagram.

  • e (float) – If e is 0, this uses an expensive algorithm to compute the exact distance. If e is not 0, it asks for an additive e-approximation, and currently also allows a small multiplicative error (the last 2 or 3 bits of the mantissa may be wrong). This version of the algorithm takes advantage of the limited precision of double and is usually a lot faster to compute, whatever the value of e. Thus, by default, e is the smallest positive double.

Return type

float

Returns

the bottleneck distance.

Distance computation

The following example explains how the distance is computed:

import gudhi

message = "Bottleneck distance = " + '%.1f' % gudhi.bottleneck_distance([[0., 0.]], [[0., 13.]])
print(message)

Bottleneck distance = 6.5
_images/bottleneck_distance_example.png

The point (0, 13) is at distance 6.5 from the diagonal and more specifically from the point (6.5, 6.5)

Basic example

This other example computes the bottleneck distance from 2 persistence diagrams:

import gudhi

diag1 = [[2.7, 3.7],[9.6, 14.],[34.2, 34.974], [3.,float('Inf')]]
diag2 = [[2.8, 4.45],[9.5, 14.1],[3.2,float('Inf')]]

message = "Bottleneck distance approximation = " + '%.2f' % gudhi.bottleneck_distance(diag1, diag2, 0.1)
print(message)

message = "Bottleneck distance value = " + '%.2f' % gudhi.bottleneck_distance(diag1, diag2)
print(message)

The output is:

Bottleneck distance approximation = 0.81
Bottleneck distance value = 0.75