Ramified optimal transportation in geodesic metric spaces
An optimal transport path may be viewed as a geodesic in the space of probability measures under a suitable family of metrics. This geodesic may exhibit a tree-shaped branching structure in many applications such as trees, blood vessels, draining and irrigation systems. Here, we extend the study of ramified optimal transportation between probability measures from Euclidean spaces to a geodesic metric space. We investigate the existence as well as the behavior of optimal transport paths under various properties of the metric such as completeness, doubling, or curvature upper boundedness. We also introduce the transport dimension of a probability measure on a complete geodesic metric space, and show that the transport dimension of a probability measure is bounded above by the Minkowski dimension and below by the Hausdorff dimension of the measure. Moreover, we introduce a metric, called “the dimensional distance”, on the space of probability measures. This metric gives a geometric meaning to the transport dimension: with respect to this metric, the transport dimension of a probability measure equals to the distance from it to any finite atomic probability measure.
Key words and phrases:optimal transport path, branching structure, dimension of measures, doubling space, curvature
2000 Mathematics Subject Classification:Primary 49Q20, 51Kxx; Secondary 28E05, 90B06
The optimal transportation problem aims at finding an optimal way to transport a given measure into another with the same mass. In contrast to the well-known Monge-Kantorovich problem (e.g. , , , , , , , ), the ramified optimal transportation problem aims at modeling a branching transport network by an optimal transport path between two given probability measures. An essential feature of such a transport path is to favor transportation in groups via a nonlinear (typically concave) cost function on mass. Transport networks with branching structures are observable not only in nature as in trees, blood vessels, river channel networks, lightning, etc. but also in efficiently designed transport systems such as used in railway configurations and postage delivery networks. Several different approaches have been done on the ramified optimal transportation problem in Euclidean spaces, see for instance , , , , , , , , , , , and . Related works on flat chains may be found in , ,  and .
This article aims at extending the study of ramified optimal transportation from Euclidean spaces to metric spaces. Such generalization is not only mathematically nature but also may be useful for considering specific examples of metric spaces later. By exploring various properties of the metric, we show that many results about ramified optimal transportation is not limited to Euclidean spaces, but can be extended to metric spaces with suitable properties on the metric. Some results that we prove in this article are summarized here:
When is a geodesic metric space, we define a family of metrics on the space of atomic probability measures on for a (possibly negative) parameter . The space is still a geodesic metric space when . A geodesic, also called an optimal transport path, in this space is a weighted directed graph whose edges are geodesic segments.
Moreover, when is a geodesic metric space of curvature bounded above, we find in §2, a universal lower bound depending only on the parameter for each comparison angle between edges of any optimal transport path. If in addition is a doubling metric space, we show that the degree of any vertex of an optimal transport path in is bounded above by a constant depending only on and the doubling constant of . On the other hand, we also provide a lower bound of the curvature of by a quantity related to the degree of vertices.
Furthermore, when is a complete geodesic metric space, we consider optimal transportation between any two probability measures on by considering the completion of the metric space . A geodesic, if it exists, in the completed metric space is viewed as an optimal transport path between measures. The existence of an optimal transport path is closely related to the dimensional information of the measures. As a result, we consider the dimension of measures on by introducing a new concept called the transport dimension of measures, which is analogous to the irrigational dimension of measures in Euclidean spaces studied by . We show in §4.2.3 and 4.3.4 that the transport dimension of a measure is bounded below by its Hausdorff dimension and above by its Minkowski dimension. Furthermore, we show that the transport dimension has an interesting geometric meaning: under a metric (called the dimensional distance), the transport dimension of a probability measure equals to the distance from it to any atomic probability measure.
In §5, when is a compact geodesic doubling metric space with Assouad dimension and the parameter , then we show that the space of probability measures on with respect to is a geodesic metric space. In other words, there exists an -optimal transport path between any two probability measures on .
1. The metrics on atomic probability measures on a metric space
1.1. Transport paths between atomic measures
We first extend some basic concepts about transport paths between measures of equal mass as studied in , with some necessary modifications, from Euclidean spaces to a metric space.
Let be a geodesic metric space. Recall that a (finite, positive) atomic measure on is in the form of
with distinct points and positive numbers where denotes the Dirac mass located at the point . The measure is a probability measure if the mass . Let be the space of all atomic probability measures on .
Given two atomic measures
on of the same mass, a transport path from to is a weighted directed acyclic graph consisting of a vertex set , a directed edge set and a weight function such that and for any vertex , there is a balance equation
where each edge is a geodesic segment in from the starting endpoint to the ending endpoint .
Note that the balance equation (1.1.3) simply means the conservation of mass at each vertex. In terms of polyhedral chains, we simply have .
Here, a directed graph is called acyclic if it contains no directed cycles in the sense that for any vertex , there does not exist a list of vertices such that and is a directed edge in for each . Reasons for introducing this constraint were given in [29, Remark 2.1.5].
For any two atomic measures and on of equal mass, let Path be the space of all transport paths from to . Now, we define the transport cost for each transport path as follows.
For any real number and any transport path , we define
We now consider the following optimal transport problem:
Given two atomic measures and of equal mass on a geodesic metric space , find a minimizer of
among all transport paths .
An minimizer in is called an optimal transport path from to .
1.2. The metrics
For any , we define
for any .
Let and be two atomic measures of equal mass , and let and be the normalization of and . Then, for any transport path , we have is a transport path from to with . Thus, we also set
It is easy to see that is a metric on when . But to show that is still a metric when , we need some estimates on the lower bound of when .
We denote (and , respectively) to be the sphere (and the closed ball, respectively) centered at of radius . Note that for any transport path , the restriction of on any closed ball gives a transport path between the restriction of measures.
Suppose and are two atomic measures on a geodesic metric space of equal total mass, and is a transport path from to . For each , if the intersection of as sets is nonempty for almost all for some , then
where for each , the set
is the family of all edges of that intersects with the sphere .
For every edge of , let and be the points on such that
Then, since is a geodesic segment in ,
where is the characteristic function of the interval . By assumption, is nonempty for almost all . Also, observe that if and only if . Therefore,
The following corollary implies a positive lower bound on when .
Let the assumptions be as in Lemma 1.2.3 and . Then
where is an upper bound of the weight for every edge in . In particular, for any atomic measure
on with mass , we have the following estimate
Suppose . For any in the form of (1.1.1), and , we have
Suppose , and is a geodesic metric space. Then defined in definition 1.2.1 is a metric on the space of atomic probability measures on .
1.3. The metric viewed as a metric induced by a quasimetric
When , another approach of the metric was introduced in , which says that the metric is the intrinsic metric on induced by a quasimetric 111A function is a quasimetric on if satisfies all the conditions of a metric except that satisfies a relaxed triangle inequality for some , rather than the usual triangle inequality. . Let us briefly recall the definition of the quasimetric here.
Let and be two fixed atomic probability measures in the form of (1.1.2) on a metric space , a transport plan from to is an atomic probability measure
in the product space such that
for each and . Let be the space of all transport plans from to .
For any atomic probability measure in of the form (1.3.1) and any , we define
where is the given metric on .
Using , we define
For any given natural number , let be the space of all atomic probability measures
on with , and then is the space of all atomic probability measures on .
In [28, Proposition 4.2], we showed that defines a quasimetric on . Moreover, is a complete quasimetric on if is a complete metric space. The quasimetric has a very nice property in the sense that this quasimetric is able to induce an intrinsic metric on .
Moreover, in [28, remark 4.16] we have a simple formula for the cost. Suppose for some . If each edge of is a geodesic curve between its endpoints in the geodesic metric space , then there exists an associated piecewise metric Lipschitz curve such that
where the quasimetric derivative
exists almost everywhere.
Suppose is a complete geodesic metric space. Then, is a complete geodesic metric space for each .
Since , and is a geodesic space for each , we have the following existence result of optimal transport path:
[28, proposition 4.02] Suppose is a complete geodesic metric space. Then, is a geodesic metric space for each . Moreover, for any , every optimal transport path from to is a geodesic from to in the geodesic space . Vice versa, every geodesic from to in is an optimal transport path from to .
2. Transportation in metric spaces with curvature bounded above
In this section, we will show that when is a geodesic metric space with curvature bounded above, then there exists a universal upper bound for the degree of every vertex of every optimal transport path on .
We now recall the definition of a space of bounded curvature . For a real number , the model space is the simply connected surface with constant curvature . That is, if , then is the Euclidean plane. If , then is obtained from the sphere by multiplying the distance function by the constant . If , then is obtained from the hyperbolic space by multiplying the distance function by the constant . The diameter of is denoted by for and for .
Let be a geodesic metric space, and let be a geodesic triangle in with geodesic segments as its sides. A comparison triangle is a triangle in the model space such that and , where denotes the distance function in the model space . Such a triangle is unique up to isometry. Also, the interior angle of at is called the comparison angle between and at .
A geodesic metric space is a space of curvature bounded above by a real number if for every geodesic triangle in and every point in the geodesic segment , one has
where is the point on the side of a comparison triangle in such that .
Now, let be a geodesic metric space with curvature bounded above by a real number . Suppose and is an optimal transport path between two atomic probability measures . We will show that the comparison angle of any two edges from a common vertex of is bounded below by a universal constant depending only on . Moveover, when is in addition a doubling space, then the degree of any vertex of is bounded above by a constant depending only on and the doubling constant of .
More precisely, let be any vertex of and be any two distinct directed edges with (or simultaneously) and weight for . Also, for , let be the point on the edge with for some satisfying and .
Now, we want to estimate the distance . To do it, we first denote
and have the following estimates for :
For each , the infimum of is given by
For each , the supremum of is given by
Also, when , then . When , then . When , then .
When , we will show later in lemma 2.0.3.
We first denote
as in [24, Example 2.1]. Note that and
By considering the function
for and , we have . Using Calculus, one may check that for each ,
when , the function is strictly concave up and
when , the function is strictly concave down and
when , the function is strictly concave down and
has constant values when .
Using these facts, we get the estimates for for each . ∎
Now, we have the following key estimates for the distance :
Assume that and . Then, we have the following estimates for :
If , then
If , then
If , then
Let be the point on the geodesic from to with
for and some to be chosen later in (2.0.5) with . For any , let be the point on the geodesic from to such that
where . For , let be a comparison triangle of in the model space .
Let be the point on the side of such that and let . Since has curvature bounded above by , we have . Let
since is a vertex of an optimal transport path . This implies that if exists. Now, we may calculate the derivative as follows.
When , by applying the spherical law of cosines to triangles and we have
where is the angle . Thus,
Taking derivative with respect to at and using the fact , we have
Therefore, for ,
Applying these expressions to , we have
as a complex number, we have
On the other hand, as , we have
as a complex number for some . Thus, inequality (2.0.4) becomes
Since , we have . Then it is easy to see that . Let
we have the inequality . That is,
By simplifying this inequality, we get
The proof for the cases and are similar when using the ordinary (or the hyperbolic) law of cosines in the model space . ∎
Let be defined as in (2.0.1). For any and , we have
By the triangle inequality, we have . We now use the estimates in lemma 2.0.2.
When , then
This yields .
When , then
When , then
as . Therefore, we still have . ∎
The following proposition says that when is negative, the weights on any two directed edges from a common vertex of an optimal transport path are comparable to each other.
If , then for each ,
where is defined as in (2.0.3).
Without losing generality, we may assume that . By proposition 2.0.3, we have . That is,
Simplify it, we have
Since and , we have
Simplify it again using