Monge Problem Research Articles

Reducing Mode Collapse With Monge-Kantorovich Optimal Transport for Generative Adversarial Networks.

Mode collapse has been a persisting challenge in generative adversarial networks (GANs), and it directly affects the applications of GAN in many domains. Existing works that attempt to solve this problem have some serious limitations: models using optimal transport (OT) strategies (e.g., Wasserstein distance) lead to vanishing or exploding gradients; increasing the number of generators can cause several generators focusing on the same mode; and approaches that modify the loss also do not satisfactorily resolve mode collapse. In this article, we reduce mode collapse by formulating it as a Monge problem of OT map. We show that the Monge problem can be transformed to the distribution transformation problem in GAN, and a rectified affine neural network can be considered as a measurable function. In this way, we propose Monge GAN that uses this measurable function to transform the generated data distribution into the original data distribution. We utilize the Kantorovich formulation to obtain the OT cost, which is regarded as the OT distance between the two distributions. Finally, we conduct extensive experiments on both image and numerical datasets to validate our Monge GAN in reducing model collapse.

Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature R i c N \mathrm {Ric}_N is bounded from below by a real number K K in every timelike direction satisfies the timelike curvature-dimension condition T C D q ( K , N ) \mathrm {TCD}_q(K,N) for all q ∈ ( 0 , 1 ) q\in (0,1) . The converse and a nonpositive-dimensional version ( N ≤ 0 N \le 0 ) of this result are also shown. Our discussion is based on the solvability of the Monge problem with respect to the q q -Lorentz–Wasserstein distance as well as the characterization of q q -geodesics of probability measures. One consequence of our work is the sharp timelike Brunn–Minkowski inequality in the Lorentz–Finsler case.

Convergence of the vertical gradient flow for the Gaussian Monge problem

Convergence of the vertical gradient flow for the Gaussian Monge problem

Revisiting the Monge Problem in the Landauer Limit

We discuss the Monge problem of mass transportation in the framework of stochastic thermodynamics and revisit the problem of the Landauer limit for finite-time thermodynamics, a problem that got the interest of Krzysztof Gawedzki in the last years. We show that restricted to one dimension, optimal transportation is efficiently solved numerically by well-known methods from differential equations. We add a brief discussion about the relevance this has on optimising the processing in modern computers.

Gromov–Wasserstein distances between Gaussian distributions

AbstractGromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.

Dynamic Multiagent Assignment Via Discrete Optimal Transport

We propose an optimal solution to a dynamic assignment problem to assign a group of moving agents to another group of moving targets. Our approach leverages connections to the theory of discrete optimal transport to convert the problem into a tractable linear program. We simultaneously solve for the optimal assignment and the control of the individual agents. As a result, we can account for dynamics and capabilities of a heterogeneous set of agents and targets. In contrast to existing assignment schemes, this approach considers cost metrics informed by the underlying agent dynamics and capabilities rather than just distance. We show that the minimizer of the dynamic assignment problem is equivalent to the minimizer of the associated Monge problem from optimal transport. We prove that the resulting approach only requires a single assignment computation over the operating lifetime, rather than periodic reassignment. Furthermore, we demonstrate cost benefits that increase as the network size increases, achieving almost 50% cost reduction compared with distance-based metrics. We demonstrate our approach through simulation on several multiagent and multitarget tracking problems.

Approximate continuous optimal transport with copulas

Optimal Transport (OT) has become a powerful tool to compare probability distributions. However, it suffers from a severe computational burden for high dimensional and continuous distributions. To this end, we develop two novel methods for the Kantorovich and Monge formulations, which are the fundamental problems in OT. First, we learn the optimal joint distribution in the Kantorovich formulation and propose an algorithm, namely Cop-OT, which transforms the primal objective of the Kantorovich problem into a tractable objective with respect to the copula parameter. Second, based on the copula formulation of the joint distribution, we learn the optimal map in the Monge problem and propose an algorithm, namely Map-OT, which describes the optimal map using a parameterized function estimated by approximating the barycentric projection of the optimal joint distribution and then obtains a tractable objective with respect to parameters of interest. Both of them can be solved by stochastic optimization with a stable optimizing process. Empirical results demonstrate that Cop-OT and Map-OT can gain more accurate approximations of the Kantorovich and Monge problems compared with the baseline methods.

Generalized unnormalized optimal transport and its fast algorithms

We introduce fast algorithms for generalized unnormalized optimal transport. To handle densities with different total mass, we consider a dynamic model, which mixes the Lp optimal transport with Lp distance. For p=1, we derive the corresponding L1 generalized unnormalized Kantorovich formula. We further show that the problem becomes a simple L1 minimization which is solved efficiently by a primal-dual algorithm. For p=2, we derive the L2 generalized unnormalized Kantorovich formula, a new unnormalized Monge problem and the corresponding Monge-Ampère equation. Furthermore, we introduce a new unconstrained optimization formulation of the problem. The associated gradient flow is essentially related to an elliptic equation which can be solved efficiently. Here the proposed gradient descent procedure together with the Nesterov acceleration involves the Hamilton-Jacobi equation arising from the KKT conditions. Several numerical examples are presented to illustrate the effectiveness of the proposed algorithms.

Equivalence of ray monotonicity properties and classification of optimal transport maps for strictly convex norms

Abstract In this paper, we first define ray increasing and decreasing monotonicity of maps. If 𝑇 is an optimal transport map for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} in R n R^{n} or 𝑇 is an optimal transport map for the Monge problem with cost function d ⁢ ( x , y ) d(x,y) , the geodesic distance, in more general, non-branching geodesic spaces 𝑋, we show respectively equivalence of some previously introduced monotonicity properties and the property of ray increasing as well as ray decreasing monotonicity which we define in this paper. Then, by solving secondary variational problems associated with strictly convex and concave functions respectively, we show that there exist ray increasing and decreasing optimal transport maps for the Monge problem with cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} . Finally, we give the classification of optimal transport maps for the Monge problem such that the cost function ∥ y - x ∥ sc \lVert y-x\rVert_{\mathrm{sc}} further satisfies the uniform smoothness and convexity estimates. That is, all of the optimal transport maps for such Monge problem can be divided into three different classes: the ray increasing map, the ray decreasing map and others.

On the existence of dual solutions for Lorentzian cost functions

The dual problem of optimal transportation in Lorentz-Finsler geometry is studied. It is shown that in general no solution exists even in the presence of an optimal coupling. Under natural assumptions dual solutions are established. It is further shown that the existence of a dual solution implies that the optimal transport is timelike on a set of full measure. In the second part the persistence of absolute continuity along an optimal transportation under obvious assumptions is proven and a solution to the relativistic Monge problem is provided.

The Plateau problem from the perspective of optimal transport

Both optimal transport and minimal surfaces have received much attention in recent years. We show that the methodology introduced by Kantorovich on the Monge problem can, surprisingly, be adapted to questions involving least area, e.g., Plateau-type problems as investigated by Federer.

The Monge Problem of “Piles and Holes” on the Torus and the Problem of Small Denominators

We discuss the problem of existence of a smooth endomorphism of a closed n-dimensional manifold carrying a differential n-form into a prescribed volume form. Of course, we assume that the integrals of these forms over the whole manifold are equal. The solution of this problem for the n-dimensional torus reduces to the problem of small denominators well known in analysis.

The Entropic Regularization of the Monge Problem on the Real Line

We study the entropic regularization of the optimal transport problem in dimension 1 when the cost function is the distance c(x, y) = |y -- x|. The selected plan at the limit is, among those which are optimal for the non-penalized problem, the most 'diffuse' one on the zones where it may have a density.

On Sudakov’s type decomposition of transference plans with norm costs

We consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm costj j D min Z jT(x) xjD d (x); T : R d ! R d ; = T# ; with , probability measures in R d and absolutely continuous w.r.t. L d . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in Za R d , where fZaga2A R d are disjoint regions such that the construction of an optimal map Ta : Za! R d is simpler than in the original problem, and then to obtain T by piecing together the maps Ta. When the normj j D is strictly convex (25), the sets Za are a family of 1-dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map Ta is straightforward provided one can show that the disintegration ofL d (and thus of ) on such segments is absolutely continuous w.r.t. the 1-dimensional Hausdor measure (12). When the normjj D is not strictly convex, the main problems in this kind of approach are two: rst, to identify a suitable family of regionsfZaga2A on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper we show how these diculties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set Za and then in R d . The strategy is suciently powerful to be applied to other optimal transportation problems. The analysis requires

Lack of regularity of the transport density in the Monge problem

In this paper, we provide a family of counter-examples to the regularity of the transport density in the classical Monge–Kantorovich problem. We prove that the W1,p regularity of the source and target measures f± does not imply that the transport density σ is W1,p, that the BV regularity of f± does not imply that σ is BV and that f±∈C∞ does not imply that σ is W1,p, for large p.

Optimal transportation for generalized Lagrangian

This paper deals with the optimal transportation for generalized Lagrangian L = L(x, u, t), and considers the following cost function: $$c\left( {x,y} \right) = \mathop {\inf }\limits_{\begin{array}{*{20}{c}} {x\left( 0 \right) = x} {x\left( 1 \right) = y} {u \in U} \end{array}} \int_0^1 {L\left( {x\left( s \right),u\left( {x\left( s \right),s} \right),s} \right)ds} ,$$ where U is a control set, and x satisfies the ordinary equation $$\dot x\left( s \right) = f\left( {x\left( s \right),u\left( {x\left( s \right),s} \right)} \right).$$ It is proved that under the condition that the initial measure μ 0 is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation: $$\left\{ {_{V\left( {0,x} \right) = {\phi _0}\left( x \right).}^{{V_t}\left( {t,x} \right) + \mathop {\sup }\limits_{u \in U} \left\langle {{V_x}\left( {t,x} \right),f\left( {x,u\left( {x\left( t \right),t} \right),t} \right) - L\left( {x\left( t \right),u\left( {x\left( t \right),t} \right),t} \right)} \right\rangle = 0,}} \right.$$

The Monge problem with vanishing gradient penalization: Vortices and asymptotic profile

We investigate the approximation of the Monge problem (minimizing ∫Ω|T(x)−x|dμ(x) among the vector-valued maps T with prescribed image measure T#μ) by adding a vanishing Dirichlet energy, namely ε∫Ω|DT|2. We study the Γ-convergence as ε→0, proving a density result for Sobolev (or Lipschitz) transport maps in the class of transport plans. In a certain two-dimensional framework that we analyze in details, when no optimal plan is induced by an H1 map, we study the selected limit map, which is a new “special” Monge transport, possibly different from the monotone one, and we find the precise asymptotics of the optimal cost depending on ε, where the leading term is of order ε|log⁡ε|.

Invariance properties of the Monge-Kantorovich mass transport problem

We consider the multimarginal Monge-Kantorovich transport problem in an abstract setting. Our main results state that if a cost function and marginal measures are invariant by a family of transformations, then a solution of the Kantorovich relaxation problem and a solution of its dual can be chosen so that they are invariant under the same family of transformations. This provides a new tool to study and analyze the support of optimal transport plans and consequently to scrutinize the Monge problem. Birkhoff's Ergodic theorem is an essential tool in our analysis.

Monge problem in metric measure spaces with Riemannian curvature-dimension condition

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X,d,m) enjoying the Riemannian curvature-dimension condition RCD∗(K,N), with N<∞. For the first marginal measure, we assume that μ0≪m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge–Kantorovich problem, attain the same minimal value.Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.

Obstructions to regularity in the classical Monge problem

We provide counterexamples to regularity of optimal maps in the classical Monge problem under various assumptions on the initial data. Our construction is based on a variant of the counterexample in [10] to Lipschitz regularity of the monotone optimal map between smooth densities supported on convex domains.