dgh 0.1.1

Computing the Gromov–Hausdorff distance

dGH

Given the distance matrices of some metric spaces $X$ and $Y$, estimates 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 $R \subseteq X \times Y$.

The distance is estimated from above by solving its parametric relaxation whose solutions are guaranteed to deliver $d_\text{GH}(X, Y)$ for sufficiently large value of the parameter $c$. The quadratic relaxation with affine constraints is solved using the Frank–Wolfe algorithm in $O(n^3)$ time per its iteration, where $n = \max{|X|, |Y|}$ is the number of points in the larger space. Even if the algorithm fails to find $d_\text{GH}(X, Y)$ exactly, the resulting solution provides its upper bound.

A detailed description of the relaxation, its optimality guarantees and optimization landscape, and the approach to solving it can be found in Computing the Gromov–Hausdorff distance using first-order methods.

Quickstart

To install the package from Python Package Index:

$ pip install dgh

Consider an example in which $X$ is comprised by the vertices of a $1 \times 10$ rectangle and $Y$ — by the vertices of a unit equilateral triangle together with a point that is 10 away fom them (see illustration).

To create their distance matrices (in which the (i,j)-th entry contains the distance between the i-th and j-th points of the space):

import numpy as np

X = np.array([[0, 1, 10, 10],
              [0, 0, 10, 10],
              [0, 0, 0, 1],
              [0, 0, 0, 0]])
X += X.T

Y = np.array([[0, 1, 1, 10],
              [0, 0, 1, 10],
              [0, 0, 0, 10],
              [0, 0, 0, 0]])
Y += Y.T

To compute (an upper bound of) their Gromov--Hausdorff distance $d_\text{GH}(X, Y)$:

import dgh

dGH = dgh.upper(X, Y)

In this case, the distance is computed exactly as $d_\text{GH}(X, Y)=\frac{1}{2}$.

Basics

By default, the computational budget allocated for the search is 100 Frank–Wolfe iterations. Bigger budget means more random restarts (and/or better convergence) and therefore the accuracy. To set the budget:

dGH = dgh.upper(X, Y, iter_budget=my_budget)

To obtain 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:

dGH, f, g = dgh.upper(X, Y, return_fg=True)

Advanced

The performance can be improved by explicitly specifying the relaxation parameter $c \in (1, \infty)$. Small $c$ makes the relaxation easier to solve, but its solutions are more likely to deliver the Gromov–Hausdorff distance when $c$ is large.

By default, the method allocates half of the iteration budget to select the best value of $c$ for $(X, Y)$ from $1+10^{-4}, 1+10^{-2},\ldots,1+10^8$, and then spends the remaining half on refining the Gromov–Hausdorff distance using this $c$.

To reveal the value of $c$ selected after the search:

dgh.upper(X, Y, verbose=1)

To compute for specific value of $c$ (to avoid spending iterations on the search and/or to test for better accuracy):

dGH = dgh.upper(X, Y, c=my_c)

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.

Research

To cite dGH, you can use the following:

Oles, V. (2023). Computing the Gromov–Hausdorff distance using first-order methods. arXiv preprint arXiv:2307.13660.

@article{oles2023computing,
  title={Computing the Gromov--Hausdorff distance using first-order methods},
  author={Oles, Vladyslav},
  journal={arXiv preprint arXiv:2307.13660},
  year={2023}
}