Papers by Simone Di Marino
We study a generalization of the multi-marginal optimal transport problem, which has no fixed num... more We study a generalization of the multi-marginal optimal transport problem, which has no fixed number of marginals $N$ and is inspired of statistical mechanics. It consists in optimizing a linear combination of the costs for all the possible $N$'s, while fixing a certain linear combination of the corresponding marginals.
The Journal of Geometric Analysis
We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$ ( Y , d ) of curvature bounded f... more We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$ ( Y , d ) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$ μ on $$\mathrm{Y}$$ Y giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$ ( Y , d , μ ) is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$ W 1 , 2 ( Y , d , μ ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at $$x\in \mathrm{Y}$$ x ∈ Y is the tangent cone at x of $$\mathrm{Y}$$ Y . The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$ x ∈ Y such a cone is a $$\mathrm{CAT}(0)$$ CAT ( 0 ) space and, as such, has a Hilbert-like structure.
Abstract. We study a multimarginal optimal transportation problem in one dimension. For a sym-met... more Abstract. We study a multimarginal optimal transportation problem in one dimension. For a sym-metric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments. 1
Advanced Studies in Pure Mathematics
In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q ... more In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q (X, d, m), 1 < q < ∞, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.
arXiv: Optimization and Control, 2020
We introduce a new class of convex-regularized Optimal Transport losses, which generalizes the cl... more We introduce a new class of convex-regularized Optimal Transport losses, which generalizes the classical Entropy-regularization of Optimal Transport and Sinkhorn divergences, and propose a generalized Sinkhorn algorithm. Our framework unifies many regularizations and numerical methods previously appeared in the literature. We show the existence of the maximizer for the dual problem, complementary slackness conditions, providing a complete characterization of solutions for such class of variational problems. As a consequence, we study structural properties of these losses, including continuity, differentiability and provide explicit formulas for the its gradient. Finally, we provide theoretical guarantees of convergences and stability of the generalized Sinkhorn algorithm, even in the continuous setting. The techniques developed here are directly applicable also to study Wasserstein barycenters or, more generally, multi-marginal problems.
We show that, given a metric space (Y, d) of curvature bounded from above in the sense of Alexand... more We show that, given a metric space (Y, d) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure μ on Y giving finite mass to bounded sets, the resulting metric measure space (Y, d, μ) is infinitesimally Hilbertian, i.e. the Sobolev space W 1,2(Y, d, μ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at x ∈ Y is the tangent cone at x of Y. The conclusion then follows from the fact that for every x ∈ Y such a cone is a CAT(0) space and, as such, has a Hilbert-like structure.
Abstract. In this paper we study the structure theory of normed modules, which have been introduc... more Abstract. In this paper we study the structure theory of normed modules, which have been introduced by Gigli. The aim is twofold: to extend von Neumann’s theory of liftings to the framework of normed modules, thus providing a notion of precise representative of their elements; to prove that each separable normed module can be represented as the space of sections of a measurable Banach bundle. By combining our representation result with Gigli’s differential structure, we eventually show that every metric measure space (whose Sobolev space is separable) is associated with a cotangent bundle in a canonical way.
arXiv: Analysis of PDEs, 2021
We prove the conjectured first order expansion of the Levy-Lieb functional in the semiclassical l... more We prove the conjectured first order expansion of the Levy-Lieb functional in the semiclassical limit, arising from Density Functional Theory (DFT). This is accomplished by interpreting the problem as the singular perturbation of an Optimal Transport problem via a Dirichlet penalization.
Abstract. In this paper we study the structure theory of normed modules, which have been introduc... more Abstract. In this paper we study the structure theory of normed modules, which have been introduced by Gigli. The aim is twofold: to extend von Neumann’s theory of liftings to the framework of normed modules, thus providing a notion of precise representative of their elements; to prove that each separable normed module can be represented as the space of sections of a measurable Banach bundle. By combining our representation result with Gigli’s differential structure, we eventually show that every metric measure space (whose Sobolev space is separable) is associated with a cotangent bundle in a canonical way.
Topological Optimization and Optimal Transport, 2017
Journal of Scientific Computing, 2020
This paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Ana... more This paper exploit the equivalence between the Schrödinger Bridge problem (Léonard in J Funct Anal 262:1879–1920, 2012; Nelson in Phys Rev 150:1079, 1966; Schrödinger in Über die umkehrung der naturgesetze. Verlag Akademie der wissenschaften in kommission bei Walter de Gruyter u, Company, 1931) and the entropy penalized optimal transport (Cuturi in: Advances in neural information processing systems, pp 2292–2300, 2013; Galichon and Salanié in: Matching with trade-offs: revealed preferences over competing characteristics. CEPR discussion paper no. DP7858, 2010) in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schrödinger system. Our method extends also when we have more than two margina...
Rendiconti Lincei - Matematica e Applicazioni, 2021
The intent of this short note is to extend real valued Lipschitz functions on metric spaces, whil... more The intent of this short note is to extend real valued Lipschitz functions on metric spaces, while locally preserving the asymptotic Lipschitz constant. We then apply this results to give a simple and direct proof of the fact that Sobolev spaces on metric measure spaces defined with a relaxation approach à la Cheeger are invariant under isomorphism class of mm-structures.
Journal of Functional Analysis, 2021
We construct a regular random projection of a metric space onto a closed doubling subset and use ... more We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and C 1 functions. This way we prove more directly a result by Lee and Naor [LN05] and we generalize the C 1 extension theorem by Whitney [Whi34] to Banach spaces.
Comptes Rendus. Mathématique, 2020
We provide a quick proof of the following known result: the Sobolev space associated with the Euc... more We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure. Résumé. Nous fournissons une preuve courte du résultat connu suivant: l'espace de Sobolev associé à l'espace euclidien muni de sa distance euclidienne et d'une mesure arbitraire de Radon, est un espace d'Hilbert. Notre nouvelle approche repose sur des propriétés du fibré de décomposabilité introduit par Alberti et Marchese. En conséquence de nos arguments, nous prouvons aussi que si la norme de Sobolev est fermable dans les fonctions lisses à support compact, la mesure de référence est absolument continue par rapport à la mesure de Lebesgue.
SIAM Journal on Mathematical Analysis, 2019
We provide sharp conditions for the finiteness and the continuity of multimarginal optimal transp... more We provide sharp conditions for the finiteness and the continuity of multimarginal optimal transport with repulsive cost, expressed in terms of a suitable concentration property of the measure. To achieve this result, we analyze the Kantorovich potentials of the optimal plans, and we estimate the distance of any optimal plan from the regions where the cost is infinite.
Mathematical Models and Methods in Applied Sciences, 2019
We propose an entropy minimization viewpoint on variational mean-field games with diffusion and q... more We propose an entropy minimization viewpoint on variational mean-field games with diffusion and quadratic Hamiltonian. We carefully analyze the time discretization of such problems, establish [Formula: see text]-convergence results as the time step vanishes and propose an efficient algorithm relying on this entropic interpretation as well as on the Sinkhorn scaling algorithm.
Journal of the European Mathematical Society, 2015
Motivated by recent developments on calculus in metric measure spaces (X, d, m), we prove a gener... more Motivated by recent developments on calculus in metric measure spaces (X, d, m), we prove a general duality principle between Fuglede's notion [15] of p-modulus for families of finite Borel measures in (X, d) and probability measures with barycenter in L q (X, m), with q the dual exponent of p ∈ (1, ∞). We apply this general duality principle to study null sets for families of parametric and nonparametric curves in X. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on p-modulus ([21], [23]) and suitable probability measures in the space of curves ([6], [7]).
Journal of Functional Analysis, 2014
Canadian Journal of Mathematics, 2014
We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repuls... more We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that, given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.
Annali di Matematica Pura ed Applicata, 2013
A standard question arising in optimal transport theory is whether the Monge problem and the Kant... more A standard question arising in optimal transport theory is whether the Monge problem and the Kantorovich relaxation have the same infimum; the positive answer means that we can pass to the relaxed problem without loss of information. In the classical case with two marginals, this happens when the cost is positive, continuous, and possibly infinite and the first marginal has no atoms. We study a similar multimarginal symmetric problem, arising naturally in density functional theory, motivated by a recent paper by Buttazzo, De Pascale, and Gori Giorgi. The cost is the potential interaction between n charged particles (hence, it is symmetric, positive, continuous, and infinite whenever x i = x j), and the marginals are all equal with no atoms. We prove that also in this case, there is equality between the infimum in the cyclical Monge problem (the natural Monge problem in this context) and in the classical Kantorovich problem. This result is new even for 2 marginals, because we consider only transport maps which are involutions. The result is generalized to every symmetric continuous cost function on a Polish space.
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Papers by Simone Di Marino