arXiv subjects

Hugo Malamut

Publications and source records attributed to Hugo Malamut.

5 recordsLinked to original sources

arXiv2026

A Brenier-Strassen Theorem on CAT(kappa) Spaces

We extend the Brenier-Strassen theorem about projections in convex order to non-flat spaces with curvature bounded from above. Precisely, for probability measures $\mu$, $\nu$ of finite second moment on a complete separable CAT(0) space, we prove that $\mu$ admits a unique W 2 -projection \bar{\mu} to the set of probability measures dominated by $\nu$ in convex order. Moreover, the unique optimal coupling from $\mu$ to \bar{\mu} is induced by a 1-Lipschitz map, without any absolute-continuity assumption on $\mu$. Our proof identifies the projection problem with a weak optimal transport problem whose cost is the squared distance to the set of convex means. We also establish a localized version on CAT(kappa) spaces with kappa \geq 0, where the optimal map is 1/2-H{\"o}lder continuous. Finally, we give a Strassen-type characterization of one-step barycentric martingales on proper CAT(0) spaces.

math.FA↗

arXiv2025

Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization

We consider weak optimal problems (possibly entropically penalized) incorporating both soft and hard (including the case of the martingale condition) moment constraints. Even in the special case of the martingale optimal transport problem, existence of Lagrange multipliers corresponding to the martingale constraint is notoriously hard (and may fail unless some specific additional assumptions are made). We identify a condition of qualification of the hard moment constraints (which in the martingale case is implied by well-known conditions in the literature) under which general dual attainment results are established. We also analyze the convergence of entropically regularized schemes combined with penalization of the moment constraint and illustrate our theoretical findings by numerically solving in dimension one, the Brenier-Strassen problem of Gozlan and Juillet and a family of problems which interpolates between monotone transport and left-curtain martingale coupling of Beiglb\"{o}ck and Juillet.

math.OC↗

arXiv2025

Entropic approximations of the semigeostrophic shallow water equations

We develop a discretisation of the semigeostrophic rotating shallow water equations, based upon their optimal transport formulation. This takes the form of a Moreau-Yoshida regularisation of the Wasserstein metric. Solutions of the optimal transport formulation provide the shallow water layer depth represented as a measure, which is itself the push forward of an evolving measure under the semigeostrophic coordinate transformation. First, we propose and study an entropy regularised version of the rotating shallow water equations. Second, we discretise the regularised problem by replacing both measures with weighted sums of Dirac measures, and approximate the (squared) L2 norm of the layer depth, which defines the potential energy. We propose an iterative method to solve the discrete optimisation problem relating the two measures, and analyse its convergence. The iterative method is demonstrated numerically and applied to the solution of the time-dependent shallow water problem in numerical examples.

math.NA↗

arXiv2024

Well-posedness and convergence of entropic approximation of semi-geostrophic equations

We prove existence and uniqueness of solutions for an entropic version of the semi-geostrophic equations. We also establish convergence to a weak solution of the semi-geostrophic equations as the entropic parameter vanishes. Convergence is also proved for discretizations that can be computed numerically in practice as shown recently in [6].

math.AP↗

arXiv2023

Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy

We study the convergence of the transport plans $\gamma_\epsilon$ towards $\gamma_0$ as well as the cost of the entropy-regularized optimal transport $(c,\gamma_\epsilon)$ towards $(c,\gamma_0)$ as the regularization parameter $\epsilon$ vanishes in the setting of finite entropy marginals. We show that under the assumption of infinitesimally twisted cost and compactly supported marginals the distance $W_2(\gamma_\epsilon,\gamma_0)$ is asymptotically greater than $C\sqrt{\epsilon}$ and the suboptimality $(c,\gamma_\epsilon)-(c,\gamma_0)$ is of order $\epsilon$. In the quadratic cost case the compactness assumption is relaxed into a moment of order $2+\delta$ assumption. Moreover, in the case of a Lipschitz transport map for the non-regularized problem, the distance $W_2(\gamma_\epsilon,\gamma_0)$ converges to $0$ at rate $\sqrt{\epsilon}$. Finally, if in addition the marginals have finite Fisher information, we prove $(c,\gamma_\epsilon)-(c,\gamma_0) \sim d\epsilon/2$ and we provide a companion expansion of $H(\gamma_\epsilon)$. These results are achieved by disentangling the role of the cost and the entropy in the regularized problem.

math.OC↗