With this post I delve into a topic which is somehow new to me, although I planned to look deeper into this for quite some time already. I stumbled upon the paper Gromov-Wasserstein distances and the metric approach to object matching by Facundo Mémoli which was a pleasure to read and motivated this post.

1. Comparing measures with norms and metrics

There are different notions in mathematics to compare two objects, think of the size of real numbers, the cardinality of sets or the length of the difference of two vectors. Here we will deal with not only comparison of objects but with “measures of similarity”. Two fundamental notions for this are norms in vector spaces and metrics. The norm is the stronger concept in that it uses more structure than a metric and also, every norm induces a metric but not the other way round. There are occasions in which both a norm and a metric are available but lead to different concepts of similarity. One of these instances occurs in sparse recovery, especially in the continuous formulation, e.g. as described in a previous post. Consider the unit interval ${I = [0,1]}$ and two Radon measures ${\mu_1}$ and ${\mu_2}$ on ${I}$ (${I}$ could also be an aritrary metric space). On the space of Radon measures ${\mathfrak{M}(I)}$ there is the variation norm

$\displaystyle \|\mu\|_{\mathfrak{M}}= \sup_\Pi\sum_{A\in\Pi}|\mu(A)|$

where the supremum is taken over all partitions ${\Pi}$ of ${I}$ into a finite number of measurable sets. Moreover, there are different metrics one can put on the space of Radon measures, e.g. the Prokhorov metric which is defined for two probability measures (e.g. non-negative ones with unit total mass)

$\displaystyle \begin{array}{rcl} d_P(\mu_1,\mu_2) & = & \inf\{\epsilon>0\ :\ \mu_1(A)\leq \mu_2(A^\epsilon) + \epsilon,\nonumber\\ & & \qquad \mu_2(A)\leq \mu_1(A^\epsilon) + \epsilon\ \text{for all measurable}\ A\} \end{array}$

where ${A^\epsilon}$ denotes the ${\epsilon}$-neighborhood of ${A}$. Another familiy of metrics are the Wasserstein metrics: For ${p\geq 1}$ define

$\displaystyle d_{W,p}(\mu_1,\mu_2) = \Big(\inf_\nu\int_{I\times I} |x-y|^p d\nu(x,y)\Big)^{1/p} \ \ \ \ \ (1)$

where the infimum is taken over all measure couplings of ${\mu_1}$ and ${\mu_2}$, that is, all measures ${\nu}$ on ${I\times I}$ such that for measurable ${A}$ it holds that

$\displaystyle \nu(A\times I) = \mu_1(A)\ \text{and}\ \nu(I\times A) = \mu_2(A).$

Example 1 We compare two Dirac measures ${\mu_1 = \delta_{x_1}}$ and ${\mu_2 = \delta_{x_2}}$ located at distinct points ${x_1\neq x_2}$ in ${I}$ as seen here:

The variation norm measures their distance as

$\displaystyle \|\mu_1-\mu_2\|_{\mathfrak{M}} = \sup_\Pi\sum_{A\in\Pi}|\delta_{x_1}(A) - \delta_{x_2}(A)| = 2$

(choose ${\Pi}$ such that it contains ${A_1}$ and ${A_2}$ small enough that ${x_1\in A_1}$, ${x_2\in A_2}$ but ${x_1\notin A_2}$ and ${x_2\notin A_1}$). The calculate the Prokhorov metric note that you only need to consider ${A}$‘s which contain only one of the points ${x_{1/2}}$ and hence, it evaluates to

$\displaystyle d_P(\mu_1,\mu_2) = |x_1-x_2|.$

For the Wasserstein metric we observe that there is only one possible measure coupling of ${\delta_{x_1}}$ and ${\delta_{x_2}}$, namely the measure ${\nu = \delta_{(x_1,x_2)}}$. Hence, we have

$\displaystyle d_{W,p}(\mu_1,\mu_2) = \Big(\int_{I\times I}|x-y|^pd\delta_{(x_1,x_2)}(x,y)\Big)^{1/p} = |x_1-x_2|.$

The variation norm distinguishes the two Diracs but is not able to grasp the distance of their supports. On the other hand, both metrics return the geometric distance of the supports in the underlying space ${I}$ as distance of the Diracs. Put in pictures: The variation norm of the difference measures the size ob this object

while both metrics capture the distance of the measures like here

It should not stay unnoted that convergence in both the Prokhorov metric and the Wasserstein metrics is exactly the weak convergence of probability measures.

The above example provides a motivation to study metric structures on spaces, even if they are also equipped with a norm. Another reason to shift attention from normed spaces to metric spaces is the fact that there has emerged a body of work to build a theory of analysis in metric spaces (see, e.g. this answer on mathoverflow or the book Gradient Flows: In Metric Spaces And In The Space Of Probability Measures by Ambrosio, Gigli and Savaré (which puts special emphasis on the space of probability measures)). Yet another motivation for the study of metrics in this way is the problem of comparing shapes (without being precisely defined yet): Which of these shapes look most alike?

(Note that shapes need not to be two dimensional figures, you may also think of more complex objects like surfaces in three dimensions or Riemannian manifolds.)

One may also ask the question how two compare different images defined on different shapes, i.e. different “distributions of colour” on two different shapes.

2. Comparing shapes: Metric spaces

Up to now we tried to compare different measures, defined on the same set. At least to me it seems that both the Prokhorov and the Wasserstein metrics are suited to measure the similarity of measures and in fact, they do so somehow finer than the usual norm does.

Let’s try to go one step further and ask ourselves, how we could compare two measures ${\mu_1}$ and ${\mu_2}$ which are defined on two different sets? While thinking about an answer one need to balance several things:

• The setup should be general enough to allow for the comparison of a wide range of objects.
• It should include enough structure to allow meaningful statements.
• It should lead to a measure which is easy enough to handle both analytically and computationally.

For the first and second bullet: We are going to work with measures not on arbitrary sets but on metric spaces. This will allow to measure distances between points in the sets and, as you probably know, does not pose a severe restriction. Although metric spaces are much more specific than topological spaces, we still aim at quantitative measures which are not provided by topologies. With respect to the last bullet: Note that both the Prokhorov and the Wasserstein metric are defined as infimums over fairly large and not too well structured sets (for the Prokhorov metric and need to consider all measurable sets and their ${\epsilon}$-neighborhoods, for the Wasserstein metric, one need to consider all measure couplings). While they can be handled quite well theoretically, their computational realization can be cumbersome.

In a similar spirit than Facundo Memoli’s paper we work our way up from comparing subsets of metric spaces up to comparing two different metric spaces with two measures defined on them.

2.1. Comparing compact subsets of a metric space: Hausdorff

Let ${(X,d)}$ be a compact metric space. Almost hundred years ago Hausdorff introduced a metric on the family of all non-empty compact subsets of a metric space as follows: The Hausdorff metric of two compact subsets ${A}$ and ${B}$ of ${X}$ is defined as

$\displaystyle d_H(A,B) = \inf\{\epsilon>0 \ :\ A\subset B_\epsilon,\ B \subset A_\epsilon\}$

(again, using the notion of ${\epsilon}$-neighborhood). This definition seems to be much in the spirit of the Prokhorov metric.

Proposition 2.1 in Facundo Memolis paper shows that the Hausdorff metric has an equivalent description as

$\displaystyle d_H(A,B) = \inf_R \sup_{(a,b) \in R} d(a,b)$

where the infimum is taken over all correspondences ${R}$ of ${A}$ and ${B}$, i.e., all subset ${R\subset A\times B}$ such that for all ${a\in A}$ there is ${b\in B}$ such that ${(a,b) \in R}$ and for all ${b\in B}$ there ${a\in A}$ such that ${(a,b)\in R}$. One may also say set coupling of ${A}$ and ${B}$ instead of correspondence.

Example 2 There is always the full coupling ${R = A\times B}$. Three different set couplings of two subsets ${A}$ and ${B}$ of the unit interval are shown here:

the “full one” ${A\times B}$ in green and two “slim” ones in red and orange. Other “slim” couplings can be obtained from surjective mappings ${f:A\rightarrow B}$ by ${R = \{(a,f(a))\ :\ a\in A\}}$ (or with the roles of ${A}$ and ${B}$ swapped): If you couple a set ${A}$ with itself, there is also the trivial coupling

$\displaystyle R = \{(a,a)\ : \ a\in A\}$

which is just the diagonal of ${A\times A}$

Note that the alternative definition of the Hausdorff metric is more in the spirit of the Wasserstein metric: It does not use enlarged objects (by ${\epsilon}$-neighborhoods) but couplings.

The Hausdorff metric is indeed a metric on the set ${\mathfrak{C}(X)}$ of all non-empty compact subsets of a metric space ${X}$ and if ${X}$ itself is compact it even holds that ${(\mathfrak{C}(X),d_H)}$ is a compact metric space (a result, known as Blaschke Selection Theorem).

One may say that we went up an abstraction ladder one step by moving from ${(X,d)}$ to ${(\mathfrak{C}(X),d_H)}$.

2.2. Comparing compact metric spaces: Gromov-Hausdorff

In the previous subsection we worked within one metric space ${X}$. In the book “Metric Structures for Riemannian and Non-Riemannian Spaces” Misha Gromov introduced a notion to compare two different metric spaces. For compact metric space ${X}$ and ${Y}$ the Gromov-Hausdorff metric is defined as

$\displaystyle d_{GH}(X,Y) = \inf_{Z,f,g} d_H(f(X),g(Y)) \ \ \ \ \ (2)$

where the infimum is taken over

• all metric spaces ${Z}$ and
• all isometric embeddings ${f}$ and ${g}$ which embed ${X}$ and ${Y}$ into ${Z}$ respectively.

In words: To compute the Gromov-Hausdorff metric, you try embed both ${X}$ and ${Y}$ into a common larger space isometrically such that they are as close as possible according to the Hausdorff metric in that space.

Strictly speaking, the above definition is not well stated as one can not form an infimum over all metric spaces since this collection does not form a set according to the rules of set theory. More precisely one should write that ${d_{GH}(X,Y)}$ is the infimum over all ${r>0}$ such that there exists a metric space ${Z}$ and isometric embeddings ${f}$ and ${g}$ of ${X}$ and ${Y}$, respectively, such that ${d_H(f(X),g(Y)).

As the Hausdorff metric could be reformulated with set couplings there is a reformulation of the Gromov-Hausdorff metric based on metric couplings: A metric coupling of two metric spaces ${(X,d_X)}$ and ${(Y,d_Y)}$ is a metric ${d}$ on the disjoint union ${X\sqcup Y}$ of ${X}$ and ${Y}$ such that for all ${x,x'\in X}$ and ${y,y'\in Y}$ it holds that ${d(x,x') = d_X(x,x')}$ and ${d(y,y') = d_Y(y,y')}$.

Example 3 We couple a metric space ${(X,d)}$ with itself. We denote with ${(X',d')}$ an identical copy of ${(X,d)}$ and look for a metric ${D}$ on ${X\times X'}$ that respects the metrics ${d}$ and ${d'}$ in the way a metric coupling has to.

To distinguish elements from ${X}$ and ${X'}$ we put a ${'}$ on all quantities from ${X'}$. Moreover, for ${x\in X}$ we denote by ${x'}$ its identical copy in ${X'}$ (and similarly for ${x'\in X'}$, ${x}$ is its identical twin). Then, for any ${\epsilon>0}$ we can define ${D_\epsilon(x,x') = D_\epsilon(x',x) = \epsilon}$ (i.e. the distance between any two identical twins is ${\epsilon}$. By the triangle inequality we get for ${x\in X}$ and ${y'\in X'}$ that ${D_\epsilon(x,y')}$ should fulfill

$\displaystyle D_\epsilon(x',y') - D_\epsilon(x',x) \leq D_\epsilon(x,y') \leq D_\epsilon(x,y) + D_\epsilon(y,y')$

and hence

$\displaystyle d(x,y) - \epsilon \leq D_\epsilon(x,y') \leq d(x,y) + \epsilon.$

Indeed we can choose ${D_\epsilon(x,y') = d(x,y)}$ if ${x\in X}$ and ${y'\in Y}$ leading to one specific metric coupling for any ${\epsilon}$. This couplings allow to distinguish identical twins and behave as a metric on the whole disjoint union. In the limiting case ${\epsilon\rightarrow 0}$ we do not obtain a metric but a semi-metric or pseudo-metric which is just the same as a metric but without the assumption that ${d(x,y) = 0}$ implies that ${x=y}$.

Example 4 The above example of a metric coupling of a metric space with itself was somehow “reproducing” the given metric as accurate as possible. There are also other couplings that put very different distances to points ${D(x,y')}$ and there is also a way to visualize metric couplings: When building the disjoint union of two metric spaces ${X}$ and ${Y}$, you can imagine this as isometrically embedding both in a larger metric space ${Z}$ in a non-overlapping way and obtain the metric coupling ${D}$ as the restriction of the metric on ${Z}$ to ${X\sqcup Y}$. For ${X=Y=[0,1]}$ you can embed both into ${Z = {\mathbb R}^2}$. A metric coupling which is similar (but not equal) to the coupling of the previous example is obtained by putting ${X}$ and ${Y}$ side by side at distance ${\epsilon}$ as here (one space in green, the other in blue).

A quite different coupling is obtained by putting ${X}$ and ${Y}$ side by side, but in a reversed way as here:

You may even embed them in a more weired way as here:

but remember that the embeddings has to be isometric, hence, distortions like here are not allowed.

This example illustrate that the idea of metric coupling is in similar spirit as of “embedding two spaces in a common larger one”.

With the notion of metric coupling, the Gromov-Hausdorff metric can be written as

$\displaystyle d_{GH}(X,Y) = \inf_{R,d} \sup_{(x,y)\in R} d(x,y) \ \ \ \ \ (3)$

where the infimum is taken over all set couplings ${R}$ of ${X}$ and ${Y}$ and all metric couplings ${d}$ of ${(X,d_X)}$ and ${(Y,d_Y)}$.

In words: To compute the Gromov-Hausdorff metric this way, you look for a set coupling of the base sets ${X}$ and ${Y}$ and a metric coupling ${d}$ of the metrics ${d_X}$ and ${d_Y}$ such that the maximal distance of two coupled points ${x}$ and ${y}$ is as small as possible. While this may look more complicated than the original definition from~(2), note that the original definition uses all metric spaces ${Z}$ in which you can embed ${X}$ and ${Y}$ isometrically, which seems barely impossible to realize. Granted, the new definition also considers a lot of quantities.

Also note that this definition is in spirit of the Wasserstein metric from~(1): If there were natural measures ${\mu_R}$ on the set couplings ${R}$ we could write \begin{equation*} d_{GH}(X,Y) = \inf_{R,d} \Big(\int d(x,y)^pd\mu_R\Big)^{1/p} \end{equation*} and in the limit ${p\rightarrow\infty}$ we would recover definition~(3).

Example 5 The Gromov-Hausdorff distance of a metric space ${(X,d_X)}$ to itself is easily seen to be zero: Consider the trivial coupling ${R = \{(x,x)\ :\ x\in X\}}$ from Example~2 and the family ${D_\epsilon}$ of metric couplings from Example~3. Then we have ${d_{GH}(X,X) \leq \epsilon}$ for any ${\epsilon >0}$ showing ${d_{GH}(X,X) = 0}$. Let’s take one of the next-complicated examples and compute the distance of ${X = [0,1]}$ and ${Y=[0,2]}$, both equipped with the euclidean metric. We couple the sets ${X}$ and ${Y}$ by ${R = \{(x,2x)\ : \ x\in X\}}$ and the respective metrics by embedding ${X}$ and ${Y}$ into ${{\mathbb R}^2}$ as follows: Put ${Y}$ at the line from ${(0,0)}$ to ${(2,0)}$ and ${X}$ at the line from ${(\tfrac12,\epsilon)}$ to ${(1\tfrac12,\epsilon)}$:

This shows that ${d_{GH}(X,Y) \leq \tfrac12}$ and actually, we have equality here.

There is another reformulation of the Gromov-Hausdorff metric, the equivalence of which is shown in Theorem 7.3.25 in the book “A Course in Metric Geometry” by Dmitri Burago, Yuri Burago and Sergei Ivanov:

$\displaystyle d_{GH}(X,Y) = \tfrac12\inf_R \sup_{\overset{\overset{x_{1/2}\in X}{y_{1/2}\in Y}}{(x_i,y_i)\in R}}\big| d_X(x_1,x_2) - d_Y(y_1,y_2)\big| \ \ \ \ \ (4)$

where the infimum is taken over all set couplings ${R}$ of ${X}$ and ${Y}$.

In words: Look for a set coupling such that any two coupled pairs ${(x_1,y_1)}$ and ${(x_2,y_2)}$ have the “most equal” distance.

This reformulation may have the advantage over the form (3) in that is only considers the set couplings and the given metrics ${d_X}$ and ${d_Y}$ and no metric coupling is needed.

Note that, as the previous reformulation~(3), it is also in the spirit of the Wasserstein metric: If there were natural measures ${\mu_R}$ in the set couplings ${R}$, we could write

$\displaystyle d_{GH}(X,Y) = \tfrac12\inf_R \Big(\int_{R\times R}\big| d_X(x_1,x_2) - d_Y(y_1,y_2)\big|^p d\mu_R(x_1,y_1)d\mu_R(x_2,y_2)\Big)^{1/p}.$

and recover the formulation~(4) in the limit ${p\rightarrow\infty}$.

One may say that we went up an abstraction ladder one step further by moving from ${(X,d)}$ to ${(\mathfrak{C}(X),d_H)}$ to ${(\text{All compact metric spaces},d_{GH})}$.

Since this post has been grown pretty long already, I decided to do the next step (which is the already announced metric on metric spaces which additionally carry some measure on them – so-called metric measure spaces) in a later post.