Multiresolution Analysis Applied to the Monge-Kantorovich Problem

We give a scheme of approximation of the MK problem based on the symmetries of the underlying spaces. We take a Haar type MRA constructed according to the geometry of our spaces. Thus, applying the Haar type MRA based on symmetries to the MK problem, we obtain a sequence of transportation problem that approximates the original MK problem for each of MRA. Moreover, the optimal solutions of each level solution converge in the weak sense to the optimal solution of original problem.


Introduction
The optimal transport problem was first formulated by Monge in 1781 and concerned finding the optimal way in the sense of minimal transportation cost of moving a pile of soil from one site to another.This problem was given a modern formulation in the work of Kantorovich in 1942 and so is now known as the Monge-Kantorovich problem.
On the other hand, a big advantage over schemes of approximation was given in the seminal article [1]; it introduced approximation schemes for infinite linear program; in particular, it showed that under suitable assumptions the program's optimum value can be finite-dimensional linear programs and that, in addition, every accumulation point of a sequence of optimal solutions for the approximating programs is an optimal solution for the original problem.An example given in this article is the Monge-Kantorovich mass transfer (MK) problem on the space itself, where the underlying space is compact.
In [2], a general method of approximation for the MK problem is given, where  and  are Polish spaces; however, this method is noneasier implementation.Later, in [3], a scheme of approximation of MK problem is provided, which consists in giving a sequence of finite transportation problems underlying original MK problem (the space is compact); nevertheless, a general procedure is given, but the examples are in a two-dimensional cube and use the dyadic partition of the cube for approximation.Our objective is to give other families to approximate this kind of problems, based on Haar type multiresolution analysis (MRA).The advantage of using this technique is that we select a multiresolution analysis that is constructed according to the symmetries of the underlying space.Therefore, the new schemes of approximation take a lot of characteristics of the space into consideration.Note that the dyadic partition of the cube is a particular case of the MRA type Haar using translations and dilations of the underlying space.
The MRA is an important method to approximate functions in different context (signal processing, differential equations, etc.).In particular, we focus on Haar type MRA on R  ; the constructions of this kind of MRA are associated with the symmetries of the spaces; thus the approximations are related to the geometrical properties of the space.In this construction, the symmetries that we use are general dilations, rotations, reflections, translations, and so forth; for more details, see [4][5][6].
The main objective of this paper is giving a scheme of approximation of the MK problem based on the symmetries of the underlying spaces.We take a Haar type MRA constructed according to the geometry of our spaces.Thus,

Haar Type Multiresolution Analysis
We introduce the basic concepts of Haar type multiresolution analysis, following Gröchenig and Madych in [4] and Guo et al. in [5].Similar results have been obtained independently by Krishtal et al. to be published in [6].
Let Γ be a lattice such that Γ = Z  for any  ∈   (R).The classical multiresolution analysis (MRA) associated with a sequence of dilations {  } ∈Z = , where |det | ≤ 1, is a sequence {  } ∈Z of closed subspaces of L 2 (R  ), which satisfies the following conditions: (v) There exists  ∈  0 such that {  },  ∈ Γ, is an orthonormal basis for  0 .
Let  be a finite subgroup of   (R) such that |det | = 1 for all  ∈  and Γ = (Γ).The operator generator by the dilations   ,  ∈ , and the translations   ,  ∈ Γ form a group.The relation allows us to define the operation to  × Γ given by and we obtain a new group denoted by Γ.The Γ-invariant spaces are the closed subspaces  of L 2 (R  ) such that      ∈  whenever  ∈ ,  ∈ , and  ∈ Γ.This leads us to the following version of (v): (v  ) There exists  ∈  0 such that {    },  ∈  and  ∈ Γ, is an orthonormal basis for  0 .
In consequence, we have the following concept.
Definition 1.Let  = {  } ∈Z be dilatation set, let  be a finite subgroup of   (R) with |det | = 1, and let Γ be a complete lattice such that Γ = (Γ).The multiresolution analysis associated with the dilation set  and the group  or -MRA is a collection {  } ∈Z of closed subspaces of L 2 (R  ), which satisfies conditions (i), (ii), (iii), (iv), and (v  ).
The classical MRA is considered as an -MRA when  is the trivial group.Note that the space  0 is not generated by the Γ-translations of the single scaling function ; however, the relation      =      and the conditions (Γ) = Γ imply that the functions   , with  ∈ , are the generators of  0 .Also, we have the following set: Note that Γ  ⊂ Γ +1 and Γ  = Γ.
We consider the scaling function  =  Δ , where  Δ is the characteristic function of Δ ⊂ R  ,  ∈ R, and ‖‖ 2 = 1.The region Δ satisfies where int Δ  ∩ int Δ   = 0 for  ̸ =   and Δ  is the action of  = (,) on Δ.In addition, the symbol Δ   denotes the translation and scaling of the region Δ by  and   , respectively.Thus, the function         = |det | −/2  Δ   is denoted by    .And so we have the following relation: Finally, we define   as the orthogonal projection from L 2 (R  ) to   which is given by for all  ∈ L 2 (R  ).
Figure 2 can be found in [6].

Absolutely Continuous Measures with respect to Lebesgue
Measures.We consider the following conditions: (A1)  is a compact subset of R  .
(A2) The measure  is absolutely continuous with respect to , where  is the Lebesgue measure.
Condition (A2) guarantees the existence of functions  ∈ L 1 (), where  is the Radon-Nikodym derivative with respect to Lebesgue measure .
In this analysis, we also assume the following extra condition: (A3) The functions  also belong in L 2 (  ).
Remark 2. Note that the classical multiresolution analysis of Haar on L 2 (R 2 ) is a particular case of -MRA on R 2 , where the complete lattice is Γ = Z 2 , the group  is trivial, and the dilation associated is  = {2 − (1, 1)}.In this case, we have that scaling function is  Δ , where Δ = [0, 1] × [0, 1] is the fundamental domain associated with the action of Γ on R 2 .
We denote by   the projection from L 2 (R  ) into   , which is given by for all  ∈ L 2 (R  ).Moreover, if the function  has support in , then the above sum is finite, since  is compact.
From now on, we shall only functions with support in the compact set  and we have We know that the -MRA is dense in L 2 (); thus we have that         −      2 → 0, when  → ∞ (13) Using the fact that  is compact, we obtain that From the above equation, we define the approximation of the measure to the level  by which are measures in  for each .Now we want to prove that these measures are probability measures.
Definition 3. The expectation E with respect to Lebesgue measure  of function  is defined by where  is a Lebesgue measurable function.Also, the conditional expectation of  given , with  being a Lebesgue measurable set, is defined by In particular, the expectation on the Lebesgue measure  satisfies the following property.
, where   is the projection of the level  associated with -MRA {  }.
Proof.We take  ∈ L 2 (R  ); now we calculate E[  ]; thus We have that the functions    =    Δ   , where Δ   is the translation and dilation of the fundamental region Δ and   = (Δ   ) −1/2 ; this value does not depend on .From this, we have that Using the above equations, we have that Moreover, using the fact that As immediate consequence of the previous theorem, we get the following result.

Non-Absolutely-Continuous Measures with respect to
Lebesgue Measure.Now, we consider that  is a probability measure on the compact set , which is unnecessarily absolutely continuous measure with respect to Lebesgue measure .Note that each element of sequence (15) can be written by for all Borel measurable sets  on .Moreover, these approximations are well defined for every measure  on .Definition 6.Let {  ;  ≥ 0} be a sequence of probability measures on a metric space (, ).We say that   converges weakly to  and denote   →   if   () → () as  → ∞ for all bounded continuous functions  on , where () = ∫   .
Theorem 7. Given a measure  on the compact set , the sequence of measures {  } on  defined in (21) converge weakly to measure .
Proof.Let  :  → R be a continuous and bounded real function.From the fact that  is compact, we obtain that  is absolutely continuous function on ; thus, given  > 0, there exists  > 0 such that      −      <  implies We take  0 ∈ Z such that for all ,  ∈ Δ   implies | − | <  for all  ≥  0 and  ∈ Γ.Moreover, we know that there exist  1 , . . .,   ∈ Γ such that In consequence, we obtain the following relations:

Discretization of the Monge-Kantorovich Problem Using Multiresolution Analysis
Let ( × ) be the linear space of finite signed on  ×  and let  + ( × ) be the set of all nonnegative measures in ( × ).Given  ∈ ( × ), we denote by Π 1  and Π 2  the marginal of  on  and , respectively; that is, for all sets  and , such that ] 1 and ] 2 are measurable, respectively.The Monge-Kantorovich mass transfer problem is given as follows: MK: minimize ⟨, ⟩ fl ∫  d subject to: A measure  ∈ ( × ) is said to be a feasible solution for MK problem if it satisfies (28) and ⟨, ⟩ is finite.The MK problem is called consistent if the set of feasible solutions is nonempty, in which case its optimal value is defined as It is said that the MK problem is solvable if there is a feasible solution  * that attains the optimal value.In this case,  * is called an optimal solution for the MK problem and the value inf(MK) is written as min(MK) = ⟨ * , ⟩.
Note that since ] 1 and ] 2 are probability measures, a feasible solution for MK is also a probability measure.Moreover, if  is a continuous function on  ×  and  and  are compact subsets on R  , then the product measure  fl ] 1 × ] 2 is a feasible solution.Therefore the MK problem is consistent, and so the MK problem is solvable in this case.
We assume the following conditions:  and  are compact subsets of R  and  : × → R is a bounded continuous function.
Let {  } and {   } be -MRA on R  with scaling functions  1 =  , and   .Now we proceed to discretization of the MK-problem using the results of Section 2; then we have that   , ]  1 , and ]  2 are the projections to level  of the respective -MRA; thus, using these measures, we obtain the discretization of the MK-problem in the level , which is given by where   , ,   , , and  , are nonnegative real numbers.Now we calculate ⟨  , ⟩; thus and hence, using the above equation and the orthonormality of family { We consider that   is a fixed element in (Γ 1 ) 0 and we have that Δ  1,  is the region associated with   for  = 1, . . ., ; then we obtain On the other hand, we have that From the above equations, we have that the condition Analogously, for all the elements  1 , . . .,   of (Γ 2 )  0 , we have that Using Theorem 4, we obtain that In summary, we have that the   problem given by (30) is equivalent to the following problem: The problem MKD  , given the above system, has a feasible solution, so we know that  is bounded.Then we have that the problem MKD  has an optimal solution; for more details, see Chapter 10 in [7].
Let  * be the optimal solution of MK problem given in (28).We use the optimal solution  * to induce a sequence of measures   * such that   * =    * (see Section 3.2).Let   * be the optimal solution of MK  problem given in (30).
For  ∈ Z, we defined the following -algebras: where ⟨  ⟩ ∈ denotes the -algebra generated by the sets   with  ∈  and the lattices Γ Definition 8. Let F  be a sub--algebra of F, and let  ∈ L 1 be a random variable.We say that the random variable   is the conditional expectation of  with respect to F  and denote it by E[ | F  ] if and only if There are analogous results for  Proof.We suppose that  satisfies (28).Then we have the following relations: where [(Γ) where  and   are feasible solutions of the MK and   problems, respectively.
Proof.Let Δ   be a generator element of -algebra F  .Then In above equation, we use the fact that ] Definition 13.Let (, ) be a topological space, and let F be -algebra on  that contains the topology .Let M be a collection of measures defined on F. The collection M is called tight if, for any  > 0, there is a compact subset   of  such that, for all measures  ∈ M, ||( \   ) < , where || is the total variation measure of .√ 27; the lattice is Γ 2 =  1 Z 2 , where  1 = (1/4) ( 0 3√3 6 3 ); the finite group  2 is the rotation group of a regular hexagon with vertices in the unit circle; the scaling is given by  2 = {  }, where  = ( 1 −√3 √3 1 ).
We denote by    and    the squares and the triangles associated with level  of the multiresolution analysis presented above.Since  is a continuous function in each pair of sets (   ,    ), we have that |    ×   is approximated to (   ,    ), where    and    are the centroids of    and    , respectively.In particular, we present a discretization of MK problem given by (51) for the level  = 2; the graphical description of the process of discretization is given in Figure 3 (which was generated using Mathematica 11.
The optimal solution of this linear programming problem is 4.02996.In Table 1 we present the solution for some levels.

Conclusions
The application of the Haar type multiresolution analysis (MRA) to the problem of Monge-Kantorovich allows us to obtain a discretization scheme for this problem at each level of the MRA; moreover, this MRA is based on the symmetries of the underlying space.This is an advantage because it provided us a natural method of discretization based on the geometry.Each level of the MRA induces a soluble finite linear programming problem.So, we obtain a sequence of optimal solutions of these transport problems and this sequence converges weakly to the optimal solution of the original problem.

Figure 1
Figure 1 is reproduced from Krishtal et al. (2007) [under the Creative Commons Attribution License/public domain].
explicit expressions for the discretization of the measures and the cost function associated with MKproblem using the -MRA, which are given by   = ∑ (,)∈(Γ)
Let  be a feasible solution of the MK problem; then   is a feasible solution of the   -problem.