Limit Theorems for Optimal Mass Transportation and Applications to Networks

It is shown that optimal network plans can be obtained as a limit of point allocations. These problems are obtained by minimizing the mass transportation on the set of atomic measures of prescribed number of atoms.


Introduction
Optimal mass transportation was introduced by Monge some 200 years ago and is, today, a source of a large number of results in analysis, geometry and convexity.
Optimal Network Theory was recently developed. It can be formulated in terms of Monge-transport corresponding to some non-standard metrics. For updated references on optimal networks via mass transportation see [BS, BCM].
In this paper we restrict ourselves to the transport of a finite number of points.
Consider N points {x 1 , . . . x N } (sources) in a state space (say, R k ), and another N points {y 1 , . . . y N } ⊂ R k (sinks). For each source x i we attribute a certain amount of mass m i ≥ 0. Similarly, m * i ≥ 0 is the capacity attributed to the sink y i , while We denote this system by an atomic measure λ := λ + − λ − where where δ (·) is the Dirac delta function.
The object is to transport the masses from the sources to sinks in an optimal way, such that the sinks are filled up according to their capacity. A natural cost was suggested by Xia [X]: For each q > 1, W (q) (λ) is defined below: Definition 1.1. Given λ as in (1.1), 1. An oriented, weighted graph (γ, m) associated with λ is a graphγ embedded in R k , composed of vertices V (γ) and edges E(γ). The orientation of an edge e ∈ E(γ) are the vertices composing the end points of e. The graphγ and the capacity function m : 2. The set of all weighted graphs associated with λ is denoted by Γ(λ). 3.
There are two special cases which should be noted. In the limit case q = 1 the op- for q ≥ 1 is the Wasserstein distance between λ + to λ − , , the minimum is taken in the set of N × N matrices satisfying In particular, W 1 depends only on the difference λ = λ + − λ − (which is not the case for q > 1).
The second case is the limit q = ∞. This is the celebrated Steiner Tree Problem [HRW]: where, this time, Γ(λ) is the set of all graphs satisfying {x i , y j ; m i , m * j > 0} ⊂ V (γ) and is, actually, independent of the masses m i and capacities m * i (assumed positive). In [W,Thm 2] it was shown that W 1 is obtained from W q by an asymptotic expression for the limit of infinite mass: If, in particular, λ is an atomic measure of the form (1.1), than it can be shown that for fixed M the minimizer of W q (µ + λ + , µ + λ − ) in B + M is an atomic measure of a finite number of atoms as well.
The main result of the current paper demonstrates that the network cost W (q) is obtained by similar expression, where the total mass M is replaced by the cardinality of the support of the atomic measure µ.
For each n ∈ N, let B +,n be the set of all atomic, positive measures of at most n atoms, that is: Theorem 2.1. For any q > 1 and λ as in (1.1) (2.1) The set B +,n is, evidently, not a compact one. Still we claim exists.
Remark 2.1. Note that W (n) q depends on each of the component λ ± while the limit Theorem 2.2. Let µ n be a regular 2 minimizer of W q (µ + λ + , µ + λ − ) in B +,n . Then the associated optimal plan spans a reduced weighted tree 3 (γ n , m n ) which converges (in Hausdorff metric) to an optimal graph (γ, m) ∈ Γ(λ) of (1.2) as n → ∞,
Definition 3.1. γ ∈ Γ(n, λ + , λ − ) is called a regular plan if it satisfies the following for any 1 ≤ i, j ≤ n + N: If γ is a regular plan, then µ ∈ B +,n is called a regular measure if for each i ∈ Lemma 3.1. For each Z ∈ (R k ) n and any plan γ ∈ Γ(n, λ + , λ − ) there exists a regular while γ r i,j = γ r 1 i,j otherwise. Then γ r verifies (3.1) while Lemma 3.2. The set of regular plans in B +,n associated with Γ(n, λ + , λ − ) (3.2) is compact.
Proof. Let z i be some point in the support of µ where µ({z i }) = Q. We show an apriori bound on Q (hence compactness). By (3.1) there exists a point z i 2 where γ i,i 2 ≥ Q/(N + n). We can define such a chain i = i 1 , i 2 , . . . where γ i l ,i l+1 > µ({z i l })/(n + N).
In particular it follows that µ({z i l }) ≥ Q/(n + N) l−1 . By part (a) of the definition of regular plans, this chain must be of length ar most n. By (3.1) it must end at some Corollary 3.1. For fixed Z ∈ (R k ) n , λ satisfying (1.1) and q > 1, the function F q admits a minimizer γ ∈ Γ(n, λ + , λ − ). Moreover, this minimizer is regular.
Proof. of lemma 2.1 For a fixed λ satisfying (1.1) and q > 1 it follows from Corollary 3.1 that It is also evident that F q is continuous and coercive on (R k ) n and that W (n) is a regular minimizer of (2.2).
It follows that any vertex v ∈ V (γ) must belong to a chain C i,j := ζ 1 , . . . ζ k where k ≤ n, ζ 1 = x i and ζ k = y j . By Definition 3.1-b there exists at most one such a chain for any pair (x i , y j ) ∈ {x 1 , . . . x N } × {y 1 , . . . y N }. In particular there exists at most N 2 such chains.
Let now ζ l ∈ C i,j . If the degree of ζ l is greater than 2, there exist deg(ζ l ) − 1 > 1 chains which contain ζ l . By Definition 3.1-b it follows that if two chains C i ′ ,j ′ , C i " ,j " intersect the chain C i,j then either C i ′ ,j ′ = C i " ,j " (and, in particular, they intersect C i,j at the same point), or i " = i ′ and j " = j ′ . Hence the number of chains crossing C i,j is bounded by 2N. As the number of chains {C i,j } is bounded by N 2 it follows that there exists at most 2N 3 chains which intersect other chains. Hence v∈V (γ) (deg(v) − 2) ≤ 2N 3 which implies the result.
Next, we elaborate some properties of an optimal regular plan.
Definition 3.3. A chain of a regular plan is a sequence of indices i 1 , . . . , i k such that γ i l ,i l+1 > 0 for k > l ≥ 1 while γ i l ,j = 0 for any j ∈ {1 . . . , n + 2N}. A maximal chain is a chain which is not contained in a larger chain.
Remark 3.1. By (3.1) we also get that γ i l ,i l+1 is a constant along any maximal chain i 1 , . . . , i k where 1 < l < k.
Lemma 3.4. If γ is a regular optimal plan then for any chain In particular, all points on a chain of the associated directed graph corresponding to an optimal plan are equally spaced on a line segment.
Proof. If γ R 0 is a regular optimal plan then Z = (z 1 , . . . z n ) is a minimizer of F q (Z, γ R 0 ) in (R k ) n . In particular ∂Fq ∂z j = 0 holds for any 1 ≤ j ≤ n. If j = i l is embedded in a chain then by definition and Remark 3.1 we obtain Let us now re-define the associated directed graph (γ, m) corresponding to an optimal regular plan (see Fig 2) Definition 3.4. The reduced weighted graph (γ R , m) associated with an optimal regular plan is obtained from (γ, m) (Definition 3.2) by identifying all edges corresponding to a maximal chain {i 1 , . . . i k } with a single edge [ζ i 1 , ζ i k ] and assigning the the common weight m e = γ i l ,i l+1 to this edge (recall Remark 3.1).
Corollary 3.2. A reduced weighted graph (γ n R , m) associated with an optimal regular plane in B +,n satisfies the following: (i) All the vertices ofγ n R are of degree at least 3.
(ii) The number of vertices ofγ n R is at most 2N 3 where N is the number of atoms of λ ± (in particular, independent of n). (iii) All the edges ofγ n R are line segments.
(iv) There exists C > 0, depending only on N, such that C > m e > 1/C for any e ∈ E(γ n R ).
(v) There is a compact set K ⊂ R k which containsγ n R for any n ∈ N.
Proof. Part (i) follows directly from Definition 3.4. Part (ii) from Lemma 3.3, part (iii) from Lemma 3.4. To prove part (iv) we repeat the proof of Lemma 3.2, with the additional information of (ii) (that is, the bound on the number of edges is independent of n). Part (v) is evident. If, moreover, (γ, m) is obtained from a regular plan γ ∈ Γ(n, λ + , λ − ) then where µ n ∈ B +,n associated with γ via (3.6). By Lemma 2.1 there exists an optimal measure µ n ∈ B +,n . Hence (4.2) holds with an equality for this choice of µ n . Moreover, µ n can be chosen to be a regular measure (Definition 3.1) hence, by (4.1,4.2) and by This implies the inequality lim inf To prove the reverse inequality in (2.1) we consider an optimal weighed graph (γ, m) of W q (λ) and construct µ n ∈ B +,n supported onγ which satisfy lim n→∞ n 1−1/q W q µ n + λ + , µ n + λ − = e∈E(γ) m 1/q e |e| = W (q) (λ) .
Assume n e is the number of points of µ n on the edge e, and any atom of µ n in e is of weight m e . The contribution to W q q (µ n + λ + , µ n + λ − ) from e is, then ≈ m e |e| n e q n e = m e |e| q n q−1 e n q−1 W q q (µ n + λ + , µ n + λ − ) ≈ n q−1 The constraint on n e is given by e∈E(γ) n e = n. Let us rescale w e := n e /n. Then we need to minimize m e |e| q w q−1 e subjected to e∈E(γ) w e = 1. Let α be the Lagrange multiplier with respect to the constraint e∈E(γ) w e . Since F is convex in w e we get that F is maximized at The minimizer is obtained at m 1/q e |e|α (q−1)/q . and the minimum is obtained at Proof. of Theorem 2.2: Let us consider the sequence of reduced weighted graphs (γ n R , m n ) (see Definition 3.4) associated with a regular minimizer γ n . By Corollary 3.2-(v) there exists a limitγ R (in the sense of Hausdorff metric) of a subsequence ofγ n R . By (ii-iv) of the Corollary, |E(γ R )| < 2N 3 and is E(γ R ) is composed of lines. Moreover, the weights m n : E(γ n R ) → R + converges also, along a subsequence, to m : E(γ R ) → R + so (m,γ R ) ∈ Γ(λ) (see q (λ + , λ − )n (q−1)/q = W q (λ) .
This and (4.5-4.7) yields e∈E(γ R ) m 1/q e |e| ≤ W q (λ) while the opposite inequality follows from the definition of W q .