dgh 0.0.6

Computing the Gromov–Hausdorff distance

dGH

Computes the Gromov–Hausdorff distance $$d_\text{GH}(X, Y) = \frac{1}{2}\inf_{f:X\to Y, g:Y\to X} \text{dis}\Big(\{(x, f(x)): x \in X\} \cup \{(g(y), y): y \in Y\}\Big),$$ where $\text{dis}(R) = \sup_{(x, y), (x', y') \in R} |d_X(x, x') - d_Y(y, y')|$ for $R \subseteq X \times Y,$ by solving (a parametric family of) indefinite quadratic minimizations with affine constraints, whose solutions are guaranteed to deliver $d_\text{GH}(X, Y)$ for sufficiently large value of the parameter $c$. The minimizations are solved using the Frank-Wolfe algorithm in $O(n^3)$ time per its iteration, where $n = |X| + |Y|$ is the total number of points. Even when the algorithm fails to find a global minimum, the resulting solution provides an upper bound for $d_\text{GH}(X, Y)$.

A manuscript describing the underlying theory is currently in preparation.

Quickstart

Installing the package from Python Package Index:

$ pip install dgh

Computing $d_\text{GH}(X, Y)$ where $X$ is the vertices of a regular 4-simplex of diameter $\frac{3}{2}$ and $Y$ is (a shortest path-based metric representation of) the star graph of order 6:

import numpy as np
from dgh import dgh

# Set distance matrix for the simplex.
X = np.array([[0, 1.5, 1.5, 1.5, 1.5],
              [0, 0, 1.5, 1.5, 1.5],
              [0, 0, 0, 1.5, 1.5],
              [0, 0, 0, 0, 1.5],
              [0, 0, 0, 0, 0]])
X += X.T

# Set distance matrix for the star graph.
Y = np.array([[0, 1, 1, 1, 1, 1],
              [0, 0, 2, 2, 2, 2],
              [0, 0, 0, 2, 2, 2],
              [0, 0, 0, 0, 2, 2],
              [0, 0, 0, 0, 0, 2],
              [0, 0, 0, 0, 0, 0]])
Y += Y.T

# Find an upper bound of the Gromov–Hausdorff distance.
dgh.ub(X, Y)

Increasing the budget of Frank-Wolfe iterations, and thus the number of restarts, allocated for the search (the default budget is 100):

d = dgh.ub(X, Y, iter_budget=1000)

Obtaining the mappings $f:X\to Y, g:Y\to X$ (as arrays s.t. $f_i = j \Leftrightarrow f(x_i) = y_j$) that deliver the found minimum:

d, f, g = dgh.ub(X, Y, return_fg=True)

Contributing

If you found a bug or want to suggest an enhancement, you can create a GitHub Issue. Alternatively, you can email vlad.oles (at) proton (dot) me.

License

dGH is released under the MIT license.