Particle method for a nonlinear multimarginal optimal transport problem

Abstract

We study a nonlinear multimarginal optimal transport problem arising in risk management, where the objective is to maximize a spectral risk measure of the pushforward of a coupling by a cost function. Although this problem is inherently nonlinear, it is known to have an equivalent linear reformulation as a multimarginal transport problem with an additional marginal. We introduce a Lagrangian particle discretization of this problem, in which admissible couplings are approximated by uniformly weighted point clouds, and marginal constraints are enforced through Wasserstein penalization. We prove quantitative convergence results for this discretization as the number of particles tends to infinity. The convergence rate is shown to be governed by the uniform quantization error of an optimal solution, and can be bounded in terms of the geometric properties of its support, notably its box dimension. In the case of univariate marginals and supermodular cost functions, where optimal couplings are known to be comonotone, we obtain sharper convergence rates expressed in terms of the asymptotic quantization errors of the marginals themselves. We also discuss the particular case of conditional value at risk, for which the problem reduces to a multimarginal partial transport formulation. Finally, we illustrate our approach with numerical experiments in several application domains, including risk management and partial barycenters, as well as some artificial examples with a repulsive cost.

Figures (13)

• Figure 1: Conditional value at risk at level $m$, for an absolutely continuous probability measure $\mu$.
• Figure 2: We represent the decomposition of the unidimensional probability density induced by its optimal transport to the sum of Dirac masses, whose respective positions are indicated by the vertical lines. The left-hand side corresponds to standard optimal transport, with the Dirac weights summing to one, while the right-hand corresponds to partial optimal transport, with the Dirac weights summing to $m=\frac{1}{2}$.
• Figure 3: Numerical solution for the standard multimarginal problem with three marginals and surplus cost $c(x_1,x_2,x_3) = -|x_1+x_2+x_3|^2$. The three marginal densities are shown in the first row, overlapped with the respective histograms of the corresponding projections of our numerical solution. The different 3D views in the second row show that, as expected, the solution concentrates on the plane of equation $x_1+x_2+x_3 = 0$. We used $N = 3\ 000$ points and the sequence of penalty coefficients $\lambda \in 10^k : k = -2-,-2,\dots,4\}$.
• Figure 4: Numerical simulation for Kitagawa and Pass' example in the proof of kitagawa2015multi, which shows non-monotonicity of the partial barycenter (in green) in its mass $m$. The blue and orange densities correspond to the two probability marginals, with their active parts darkened. Both for these active parts and for the partial barycenter, we use a simple kernel density estimation to convert the corresponding point clouds obtained numerically into densities. The value $\varepsilon$ involved in the marginal densities defined by the aforementioned authors was set to $\frac{1}{3}$. We used $N = 1\ 000$ points, equal weights $\lambda_1 = \lambda_2 = \frac{1}{2}$, and the sequence of penalty coefficients $\lambda \in \{10^k : k = -1-,0,\dots,4\}$.
• Figure 5: Numerical solution for the multimarginal formulation of the partial barycenter problem between the triangle density $\mathrm{Tri}(0,1,2)$ and its translation by $0.7$. We used $N = 1\ 000$ points, mass $m = 0.4225$, equal weights $\lambda_1 = \lambda_2 = \frac{1}{2}$, and the sequence of penalty coefficients $\lambda \in \{ 10^k : k = -3-,-2,\dots,2\}$. The chosen mass is exactly equal to the amount of common mass of the two marginal probability measures. The orange point cloud in \ref{['fig:translated_pyramid-plan']} represents the partial transport plan we find whereas the green line is the standard optimal transport plan between the two marginals.

Theorems & Definitions (43)

• Theorem 1
• Theorem 2
• Remark 3
• Definition 4
• Remark 5: Domain of the spectral risk measure
• Remark 6
• Lemma 7
• Proposition 8: ennaji2024robust
• Remark 9
• Lemma 10: Hölder continuity of the objective function