2025Activity reportProject-Team‌​‌MOKAPLAN

3.1​​​‌ OT and related variational​ problems solvers

Participants: Flavien​‌ Léger, Jean-David Benamou​​, Guillaume Carlier,​​​‌ Thomas Gallouët, Guillaume​ Chazareix , Adrien Vacher​‌ , Paul Pegon.​​

• Asymptotic analysis of entropic​​​‌ OT
  for a small​ entropic parameter is well​‌ understood for regular data​​ on compact manifolds and​​​‌ standard quadratic ground cost​ 74, the team​‌ will extend this study​​ to more general settings​​​‌ and also establish rigorous​ asymptotic estimates for the​‌ transports maps. This is​​ important to provide a​​​‌ sound theoretical background to​ efficient and useful debiasing​‌ approaches like Sinkhorn Divergences​​ 84. Guillaume Carlier​​ , Paul Pegon and​​​‌ Luca Tamanini are investigating‌ speed of convergence and‌​‌ quantitative stability results under​​ general conditions on the​​​‌ cost (so that optimal‌ maps may not be‌​‌ continuous or even fail​​ to exist). Some sharp​​​‌ bounds have already been‌ obtained, the next challenging‌​‌ goal is to extend​​ the Laplace method to​​​‌ a nonsmooth setting and‌ understand what entropic OT‌​‌ really selects when there​​ are several optimal OT​​​‌ plans.
• High dimensional -‌ Curse of dimensionality
  We‌​‌ will continue to investigate​​ the computation or approximation​​​‌ of high-dimensional OT losses‌ and the associated transports‌​‌ 105 in particular in​​ relation with their use​​​‌ in ML. In particular‌ for Wasserstein 2 metric‌​‌ but also the repulsive​​ Density Functional theory cost​​​‌ 86.
• Back-and-forth
  The‌ back-and-forth method 91,‌​‌ 90 is a state-of-the-art​​ solver to compute optimal​​​‌ transport with convex costs‌ and 2-Wasserstein gradient flows‌​‌ on grids. Based on​​ simple but new ideas​​​‌ it has great potential‌ to be useful for‌​‌ related problems. We plan​​ to investigate: OT on​​​‌ point clouds in low‌ dimension, the principal-agent problem‌​‌ in economics and more​​ generally optimization under convex​​​‌ constraints 95, 98‌.
• Transport and diffusion‌​‌
  The diffusion induced by​​ the entropic regularization is​​​‌ fixed and now well‌ understood. For recent variations‌​‌ of the OT problem​​ (Martingale OT, Weak OT​​​‌ see 45) the‌ diffusion becomes an explicit‌​‌ constraint or the control​​ itself 88. The​​​‌ entropic regularisation of these‌ problems can then be‌​‌ understood as metric/ground cost​​ learning 64 (see also​​​‌ 101) and offers‌ a tractable numerical method.‌​‌
• Wasserstein Hamiltonian systems
  We​​ started to investigate the​​​‌ use of modern OT‌ solvers for the SG‌​‌ equation 76, 49​​ Semi-Discrete and entropic regularization.​​​‌ This is a special‌ instance Hamiltonian Systems in‌​‌ the sense of 42​​. with an OT​​​‌ component in the Energy.‌
• Nonlinear fourth-order diffusion equations‌​‌
  such as thin-films or​​ (the more involved) DLSS​​​‌ quantum drift equations are‌ WGF. Such WGF are‌​‌ challenging both in terms​​ of mathematical analysis (lack​​​‌ of maximum principle...) and‌ of numerics. They are‌​‌ currently investigated by Jean-David​​ Benamou , Guillaume Carlier​​​‌ in collaboration with Daniel‌ Matthes. Note also that‌​‌ Mokaplan already contributed to​​ a related topic through​​​‌ the TV-JKO scheme 66‌.
• Lagrangian approaches for‌​‌ fluid mechanics
  More generally​​ we want to extend​​​‌ the design and implementation‌ of Lagrangian numerical scheme‌​‌ for a large class​​ of problem coming from​​​‌ fluids mechanics (WHS or‌ WGF) using semi-discrete OT‌​‌ or entropic regularization. We​​ will also take a​​​‌ special attention to link‌ this approaches with problems‌​‌ in machine/statistical learning. To​​ achieve this part of​​​‌ the project we will‌ join forces with colleagues‌​‌ in Orsay University: Y.​​ Brenier, H. Leclerc, Q.​​​‌ Mérigot, L. Nenna.
• ${\mathrm{L‌}}^{\infty }$ optimal transport
  is‌​‌ a variant of OT​​ where we want to​​​‌ minimize the maximal displacement‌ of the transport plan,‌​‌ instead of the average​​ distance. Following the seminal​​​‌ work of 73,‌ and more recent developments‌​‌ 78, Guillaume Carlier​​​‌ , Paul Pegon and​ Luigi De Pascale are​‌ working on the description​​ of restrictable solutions (which​​​‌ are cyclically $\infty$-monotone)​ through some potential maps,​‌ in the spirit of​​ Mange-Kantorovich potentials provided by​​​‌ a duality theory. Some​ progress has been made​‌ to partially describe cyclically​​ quasi-motonone maps (related in​​​‌ some sense to cyclically​ $\infty$-monotone maps), through​‌ quasi-convex potentials.

3.2 Application​​ of OT numerics to​​​‌ non-variational and non convex​ problems

Participants: Flavien Léger​‌, Guillaume Carlier,​​ Jean-David Benamou.

• Market​​​‌ design
  Z-mappings form a​ theory of non-variational problems​‌ initiated in the '70s​​ but that has been​​​‌ for the most part​ overlooked by mathematicians. We​‌ are developing a new​​ theory of the algorithms​​​‌ associated with convergent regular​ splitting of Z-mappings. Various​‌ well-established algorithms for matching​​ models can be grouped​​​‌ under this point of​ view (Sinkhorn, Gale–Shapley, Bertsekas'​‌ auction) and this new​​ perspective has the potential​​​‌ to unlock new convergence​ results, rates and accelerated​‌ methods.
• Non Convex inverse​​ problems
  The PhD 109​​​‌ provided a first exploration​ of Unbalanced Sinkhorn Divergence​‌ in this context. Given​​ enough resources, a branch​​​‌ of PySit, a​ public domain software to​‌ test misfit functions in​​ the context of Seismic​​​‌ imaging will be created​ and will allow to​‌ test other signal processing​​ strategies in Full Waveform​​​‌ Inversion. Likewise the numerical​ method tested for 1D​‌ reflectors in 50 coule​​ be develloped further (in​​​‌ particular in 2D).
• Equilibrium​ and transport
  Equilibrium in​‌ labor markets can often​​ be expressed in terms​​​‌ of the Kantorovich duality.​ In the context of​‌ urban modelling or spatial​​ pricing, this observation can​​​‌ be fruitfully used to​ compute equilibrium prices or​‌ densities as fixed points​​ of operators involving OT,​​​‌ this was used in​ 47 and 46.​‌ Quentin Petit, Guillaume Carlier​​ and Yves Achdou are​​​‌ currently developing a (non-variational)​ new semi-discrete model for​‌ the structure of cities​​ with applications to tele-working.​​​‌
• Non-convex Principal-Agent problems
  Guillaume​ Carlier , Xavier Dupuis,​‌ Jean-Charles Rochet and John​​ Thanassoulis are developing a​​​‌ new saddle-point approach to​ non-convex multidimensional screening problems​‌ arising in regulation (Barron-Myerson)​​ and taxation (Mirrlees).

3.3​​​‌ Inverse problems with structured​ priors

Participants: Irène Waldspurger​‌, Antonin Chambolle,​​ Vincent Duval, Faniriana​​​‌ Rakoto Endor, Annette​ Dumas.

• Off-the-grid reconstruction​‌ of complex objects
  Whereas,​​ very recently, some methods​​​‌ were proposed for the​ reconstruction of curves and​‌ piecewise constant images on​​ a continuous domain (​​​‌57 and 67),​ those are mostly proofs​‌ of concept, and there​​ is still some work​​​‌ to make them competitive​ in real applications. As​‌ they are much more​​ complex than point source​​​‌ reconstruction methods, there is​ room for improvements (parametrization,​‌ introduction of several atoms...).​​ In particular, we are​​​‌ currently working on an​ improvement of the algorithm​‌ 57 for inverse problems​​ in imaging which involve​​​‌ Optimal Transport as a​ regularizer (see 82 for​‌ preliminary results). Moreover, we​​ need to better understand​​​‌ their convergence and the​ robustness of such methods,​‌ using sensitivity analysis.
• Correctness​​ guarantees for Burer-Monteiro methods​​
  The Burer-Monteiro strategy solves​​​‌ low-rank optimization problems by‌ writing the low-rank matrix‌​‌ to recover as a​​ product of two thinner​​​‌ matrices. It has an‌ integer parameter, namely the‌​‌ number of columns of​​ the product factors. It​​​‌ is important for numerical‌ efficiency that the parameter‌​‌ is chosen as small​​ as possible. Unfortunately, general​​​‌ correctness guarantees for the‌ method are only available‌​‌ for values of the​​ parameter above a certain​​​‌ threshold, much higher than‌ the values typically used‌​‌ by practitioners 54,​​ and we know that​​​‌ no such general guarantee‌ can hold below the‌​‌ threshold 107. Faniriana​​ Rakoto Endor and Irène​​​‌ Waldspurger have shown in‌ 83 that, below the‌​‌ threshold, it is still​​ possible to prove that​​​‌ the the Burer-Monteiro strategy‌ is correct (i.e. it‌​‌ does not have spurious​​ critical points which could​​​‌ trap the numerical solver)‌ for a large class‌​‌ of semidefinite problems. It​​ explains the favorable behavior​​​‌ of the Burer-Monteiro strategy‌ observed in a number‌​‌ of applications. Generalizations of​​ this result are currently​​​‌ under investigation.

3.4 Geometric‌ variational problems, and their‌​‌ interactions with transport

Participants:​​ Vincent Duval, Paul​​​‌ Pegon, Antonin Chambolle‌, Joao-Miguel Machado .‌​‌

• Approximation of measures with​​ geometric constraints
  Optimal Transport​​​‌ is a powerful tool‌ to compare and approximate‌​‌ densities, but its interaction​​ with geometric constraints is​​​‌ still not well understood.‌ In applications such as‌​‌ optimal design of structures,​​ one aims at approximating​​​‌ an optimal pattern while‌ taking into account fabrication‌​‌ constraints 52. In​​ Magnetic Resonance Imaging (MRI),​​​‌ one tries to sample‌ the Fourier transform of‌​‌ the unknown image according​​ to an optimal density​​​‌ but the acquisition device‌ can only proceed along‌​‌ curves with bounded speed​​ and bounded curvature 96​​​‌. Our goal is‌ to understand how OT‌​‌ interacts with energy terms​​ which involve, e.g. the​​​‌ length, the perimeter or‌ the curvature of the‌​‌ support... We want to​​ understand the regularity of​​​‌ the solutions and to‌ quantify the approximation error.‌​‌ Moreover, we want to​​ design numerical methods for​​​‌ the resolution of such‌ problems, with guaranteed performance.‌​‌
• Discretization of singular measures​​
  Beyond the (B)Lasso and​​​‌ the total variation (possibly‌ off-the-grid), numerically solving branched‌​‌ transportation problems requires the​​ ability to faithfully discretize​​​‌ and represent 1-dimensional structures‌ in the space. The‌​‌ research program of A.​​ Chambolle consists in part​​​‌ in developing the numerical‌ analysis of variational problems‌​‌ involving singular measures, such​​ as lower-dimensional currents or​​​‌ free surfaces. We will‌ explore both phase-field methods‌​‌ (with P. Pegon, V.​​ Duval) 70, 100​​​‌ which easily represent non-convex‌ problems, but lack precision,‌​‌ and (with V. Duval)​​ precise discretizations of convex​​​‌ problems, based either on‌ finite elements (and relying‌​‌ to the FEM discrete​​ exterior calculus 44,​​​‌ cf 71 for the‌ case of the total‌​‌ variation), or on finite​​ differences and possibly a​​​‌ clever design of dual‌ constraints as studied in‌​‌ 75, 72 again​​ for the total variation.​​​‌
• Transport problems with metric‌ optimization
  In urban planning‌​‌ models, one looks at​​​‌ building a network (of​ roads, metro or train​‌ lines, etc.) so as​​ to minimize a transport​​​‌ cost between two distributions,​ penalized by the cost​‌ for building the network,​​ usually its length. A​​​‌ typical transport cost is​ Monge cost $M{\mathrm{K‌}}_{\omega }$ with a metric​​ $\omega ={\omega }_{\mathrm{\Sigma ‌}}$ which is modified as​ a fraction of the​‌ euclidean metric on the​​ network $\Sigma$. We​​​‌ would like to consider​ general problems involving a​‌ construction cost to generate​​ a conductance field $\mathrm{\sigma ‌}$ (having in mind 1-dimensional​ integral of some function​‌ of $\sigma$), and​​ a transport cost depending​​​‌ on this conductance field.​ The afore-mentioned case studied​‌ in 62 falls into​​ this category, as well​​​‌ as classical branched transport.​ The biologically-inspired network evolution​‌ model of 89 seems​​ to provide such an​​​‌ energy in the vanishing​ diffusivity limit, with a​‌ cost for building a​​ 1-dimensional permeability tensor and​​​‌ an $L^2$ congested​ transport cost with associated​‌ resistivity metric ; such​​ a cost seems particularly​​​‌ relevant to model urban​ planning. Finally, we would​‌ like to design numerical​​ methods to solve such​​​‌ problems, taking advantage of​ the separable structure of​‌ the whole cost.

4.1 Natural​​​‌ Sciences

FreeForm Optics, Fluid​ Mechanics (Incompressible Euler, Semi-Geostrophic​‌ equations), Quantum Chemistry (Density​​ Functional Theory), Statistical Physics​​​‌ (Schroedinger problem), Porous Media.​

4.2 Signal Processing and​‌ inverse problems

Full Waveform​​ Inversion (Geophysics), Super-resolution microscopy​​​‌ (Biology), Satellite imaging (Meteorology)​

4.3 Social Sciences

Mean-field​‌ games, spatial economics, principal-agent​​ models, taxation, nonlinear pricing.​​​‌

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

Participants: Hugo Malamut,​ Maxime Sylvestre.

In​‌ 19, we study​​ the convergence of the​​​‌ transport plans ${\gamma }_{\mathrm{\epsilon }}$ towards ${\gamma }_0$ as​‌ well as the cost​​ of the entropy-regularized optimal​​​‌ transport $(c,{\gamma }_{\epsilon })$ towards​‌ $(c,{\mathrm{\gamma }}_0)$ 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 ${\mathrm{W}}^2({\gamma }_{\mathrm{\epsilon ‌}},{\gamma }_0)$ is asymptotically greater than​‌ $C\sqrt{\epsilon }$ and the​​ suboptimality $(c,‌{\gamma }_{\epsilon })-(c,{\mathrm{\gamma ‌}}_0)$ 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 $W^2({\gamma }_{\epsilon },{\mathrm{\gamma ‌}}_0)$ converges to​​ 0 at rate $\sqrt{\mathrm{\epsilon ‌}}$. Finally, if in​ addition the marginals have​‌ finite Fisher information, we​​ prove $(c,‌{\gamma }_{\epsilon })-(c,{\mathrm{\gamma ‌}}_0)\sim \mathrm{\epsilon }/2$ and we​​​‌ provide a companion expansion​ of $H\left({\mathrm{\gamma ‌}}_{\epsilon }\right)$. These​​ results are achieved by​​​‌ disentangling the role of​ the cost and the​‌ entropy in the regularized​​ problem.

5.2 Entropic approximations​​ of the semigeostrophic shallow​​​‌ water equations

Participants: Jean-David‌ Benamou, Hugo Malamut‌​‌.

In 31,​​ we develop a discretisation​​​‌ of the semigeostrophic rotating‌ shallow water equations, based‌​‌ upon their optimal transport​​ formulation, which takes the​​​‌ form of a Moreau-Yoshida‌ regularisation via the Wasserstein‌​‌ metric, whose solution provides​​ the shallow water layer​​​‌ depth represented as a‌ measure, which is the‌​‌ push forward of an​​ evolving measure under the​​​‌ semigeostrophic coordinate transformation. First,‌ we propose and study‌​‌ an entropic regularisation of​​ the rotating shallow water​​​‌ equations. Second, we discretise‌ the regularised problem by‌​‌ replacing both measures with​​ weighted sums of Dirac​​​‌ measures, and approximating the‌ (squared) L 2 norm‌​‌ of the layer depth​​ appearing as 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.‌​‌

5.3 Convergence rates for​​ regularized unbalanced optimal transport​​​‌

Participants: Luca Nenna,‌ Paul Pegon, Louis‌​‌ Tocquec.

Following our​​ study of convergence rates​​​‌ of entropy-regularized OT costs‌ 65, 99,‌​‌ we have are moving​​ on to the case​​​‌ of unbalanced OT. With‌ Luca Nennna and our‌​‌ PhD student Louis Tocquec​​ we have treated the​​​‌ case of unbalanced OT‌ in the discrete case,‌​‌ where the input measures​​ are finite sums of​​​‌ Dirac masses, and with‌ a general class of‌​‌ entropy functions 39.​​ We analyze the asymptotic​​​‌ behavior of both primal‌ transport plans and dual‌​‌ variables as the regularization​​ parameter tends to infinity.​​​‌ Under general convexity and‌ regularity assumptions on the‌​‌ marginal penalization, we establish​​ explicit convergence rates: at​​​‌ least $O\left(\mathrm{\epsilon ‌}\right)$ for the dual‌​‌ variables and $O(\sqrt{\epsilon })$ for the​​​‌ primal solutions. These theoretical‌ results are illustrated and‌​‌ validated through numerical experiments​​ for standard divergences, including​​​‌ the Kullback-Leibler and quadratic‌ marginal penalties.

5.4 Optimal‌​‌ quantization via branched optimal​​ transport distance

Participants: Paul​​​‌ Pegon, Mircea Petrache‌.

In 20 we‌​‌ study optimal quantization with​​ respect to branched optimal​​​‌ transport distances, addressing the‌ problem of approximating a‌​‌ target measure by an​​ atomic measure with NN​​​‌ atoms. Unlike the classical‌ Wasserstein setting, the associated‌​‌ optimal partitions lack an​​ explicit Voronoï structure and​​​‌ are expected to have‌ fractal interfaces. We analyze‌​‌ the asymptotic behavior of​​ optimal quantizers for absolutely​​​‌ continuous measures as $\mathrm{N‌}\to +\infty$,‌​‌ establishing a branched transport​​ analogue of Zador's theorem​​​‌ and identifying the limiting‌ distribution of point clouds.‌​‌ Our approach relies on​​ a $\Gamma$-convergence framework​​​‌ and on the uniform‌ Hölder regularity of the‌​‌ landscape function, a branched​​ counterpart of Kantorovich potentials.​​​‌ As an application, we‌ derive uniform separation and‌​‌ covering estimates for optimal​​ quantizers under Ahlfors regularity​​​‌ assumptions.

5.5 A Rockafellar‌ Theorem in the quasi-convex‌​‌ setting

Participants: Paul Pegon​​, Luigi De Pascale​​​‌.

In 40 we‌ investigate a quasi-convex analogue‌​‌ of Rockafellar's theorem, addressing​​​‌ the problem of integrating​ cyclically quasi-monotone maps through​‌ quasi-convex potentials. We establish​​ such an integration result​​​‌ for cyclically quasi-monotone maps​ that are $C^{\mathrm{1‌}}$ and non-vanishing, in arbitrary​​ dimension, as well as​​​‌ for general multi-valued maps​ in dimension one. The​‌ proof relies on the​​ construction of suitable preference​​​‌ relations and totally ordered​ families of convex sets,​‌ leading to lower semicontinuous​​ quasi-convex potentials whose normal​​​‌ cone operators contain the​ given maps. We also​‌ discuss connections with revealed​​ preference theory in economics​​​‌ and with $L^{\mathrm{\infty }}$ optimal transport, and provide​‌ examples illustrating the remaining​​ challenges in the general​​​‌ case.

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

Participants:​​​‌ Guillaume Carlier, Hugo​ Malamut, Maxime Sylvestre​‌.

In 33,​​ 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​ Beiglböck and Juillet.

5.7​‌ Graph Alignment via Birkhoff​​ Relaxation

Participants: Sushil Varma​​​‌, Irène Waldspurger,​ Laurent Massoulié.

In​‌ 25, we consider​​ the graph alignment problem,​​​‌ wherein the objective is​ to find a vertex​‌ correspondence between two graphs​​ that maximizes the edge​​​‌ overlap. The graph alignment​ problem is an instance​‌ of the quadratic assignment​​ problem (QAP), known to​​​‌ be NP-hard in the​ worst case even to​‌ approximately solve. In this​​ paper, we analyze Birkhoff​​​‌ relaxation, a tight convex​ relaxation of QAP, and​‌ present theoretical guarantees on​​ its performance when the​​​‌ inputs follow the Gaussian​ Wigner Model. More specifically,​‌ the weighted adjacency matrices​​ are correlated Gaussian Orthogonal​​​‌ Ensemble with correlation $\mathrm{1}/\sqrt{1+{\mathrm{\sigma ‌}}^2}$. Denote the​​ optimal solutions of the​​​‌ QAP and Birkhoff relaxation​ by ${\Pi }^{*}$ and​‌ $X^{*}$ respectively. We​​ show that $||‌X^{*}-{\mathrm{\Pi }}^{*}{\left|\right|}_{\mathrm{F‌}}^2=o(n)$ when $\mathrm{\sigma ‌}=o\left({\mathrm{n}}^{-1.\mathrm{25‌}}\right)$ and $||X^{*}-{\mathrm{\Pi ‌}}^{*}{\left|\right|}_{\mathrm{F}}^2=\Omega (‌n)$ when $\mathrm{\sigma }=\Omega \left({\mathrm{n‌}}^{-0.\mathrm{5}}\right)$. Thus, the​‌ optimal solution $X^{*}$ transitions from a small​​​‌ perturbation of ${\Pi }^{*}$ for small $\sigma$ to​‌ being well separated from​​ ${\Pi }^{*}$ as $\mathrm{\sigma }$ becomes larger than ${\mathrm{n‌}}^{-0.\mathrm{5‌}}$. This result allows‌​‌ us to guarantee that​​ simple rounding procedures on​​​‌ $X^{*}$ align $\mathrm{1‌}-o\left(\mathrm{1‌‌}\right)$ fraction of vertices​​ correctly whenever $\sigma =‌o(n^{-‌1.25})‌‌$. This condition on​​ $\sigma$ to ensure the​​​‌ success of the Birkhoff‌ relaxation is state-of-the-art.

5.8‌​‌ Benign landscape for Burer-Monteiro​​ factorizations of MaxCut-type semidefinite​​​‌ programs

Participants: Faniriana Rakoto‌ Endor, Irène Waldspurger‌​‌.

We study the​​ Burer-Monteiro factorization, which is​​​‌ a well-known heuristic to‌ reduce the computational cost‌​‌ of solving semidefinite programs​​ (SDP) in the case​​​‌ where the solution is‌ a priori known to‌​‌ be low rank. This​​ factorization reduces the dimension​​​‌ of the SDP at‌ the cost of its‌​‌ convexity, therefore possibly introducing​​ spurious second-order critical points​​​‌ which could trap the‌ optimization algorithm and prevent‌​‌ it from finding the​​ desired minimizer.

In 21​​​‌, for MaxCut-type SDP,‌ we give a sharp‌​‌ condition on the conditioning​​ of the associated Laplacian​​​‌ matrix under which any‌ second-order critical point of‌​‌ the non-convex problem is​​ a global minimizer. By​​​‌ applying our theorem, we‌ improve on recent results‌​‌ about the correctness of​​ the Burer-Monteiro factorization on​​​‌ ${\mathbb{Z}}^2$ -synchronization problems.‌

5.9 Nonnegative cross-curvature in‌​‌ infinite dimensions: synthetic definition​​ and spaces of measures​​​‌

Participants: Flavien Léger,‌ Gabriele Todeschi, François-Xavier‌​‌ Vialard.

In 17​​ we develop a synthetic​​​‌ notion of nonnegative cross-curvature.‌ Nonnegative cross-curvature (NNCC) is‌​‌ a geometric property of​​ a cost function defined​​​‌ on a product space‌ that originates in optimal‌​‌ transportation and the Ma-Trudinger-Wang​​ theory. Motivated by applications​​​‌ in optimization, gradient flows‌ and mechanism design, we‌​‌ propose a variational formulation​​ of nonnegative cross-curvature on​​​‌ c-convex domains applicable to‌ infinite dimensions and nonsmooth‌​‌ settings. The resulting class​​ of NNCC spaces is​​​‌ closed under Gromov-Hausdorff convergence‌ and for this class,‌​‌ we extend many properties​​ of classical nonnegative cross-curvature:​​​‌ stability under generalized Riemannian‌ submersions, characterization in terms‌​‌ of the convexity of​​ certain sets of c-concave​​​‌ functions, and in the‌ metric case, it is‌​‌ a subclass of positively​​ curved spaces in the​​​‌ sense of Alexandrov. One‌ of our main results‌​‌ is that Wasserstein spaces​​ of probability measures inherit​​​‌ the NNCC property from‌ their base space. Additional‌​‌ examples of NNCC costs​​ include the Bures-Wasserstein and​​​‌ Fisher-Rao squared distances, the‌ Hellinger-Kantorovich squared distance (in‌​‌ some cases), the relative​​ entropy on probability measures,​​​‌ and the 2-Gromov-Wasserstein squared‌ distance on metric measure‌​‌ spaces.

5.10 A Cahn–Hilliard–Willmore​​ phase field model for​​​‌ non-oriented interfaces

Participants: Antonin‌ Chambolle, Elie Bretin‌​‌, Simon Masnou.​​

In 13, we​​​‌ investigate a new phase‌ field model for representing‌​‌ non-oriented interfaces, approximating their​​ area and simulating their​​​‌ area-minimizing flow. Our contribution‌ is related to the‌​‌ approach proposed in arXiv:2105.09627​​ that involves ad hoc​​​‌ neural networks. We show‌ here that, instead of‌​‌ neural networks, similar results​​ can be obtained using​​​‌ a more standard variational‌ approach that combines a‌​‌ Cahn-Hilliard-type functional involving an​​​‌ appropriate non-smooth potential and​ a Willmore-type stabilization energy.​‌ We show some properties​​ of this phase field​​​‌ model in dimension 1​ and, for radially symmetric​‌ functions, in arbitrary dimension.​​ We propose a simple​​​‌ numerical scheme to approximate​ its L2-gradient flow. We​‌ illustrate numerically that the​​ new flow approximates fairly​​​‌ well the mean curvature​ flow of codimension 1​‌ or 2 interfaces in​​ dimensions 2 and 3.​​​‌

5.11 Convergence of discrete​ approximations of the mean​‌ curvature flow

Participants: Antonin​​ Chambolle, Daniele De​​​‌ Gennaro, Massimiliano Morini​.

In 35,​‌ we have pushed further​​ an idea developed last​​​‌ year for designing convergent​ time and space-discrete approximations​‌ of the crystalline mean​​ curvature flow, in any​​​‌ dimension. In the new​ paper, we define convolution-redistancing​‌ schemes in the fully​​ discrete setting which we​​​‌ show to converge to​ the curvature flow. A​‌ non-linear generalization allows to​​ explain why some machine-learning​​​‌ based approximations of the​ flow, developed by Bretin​‌ and collaborators in Lyon,​​ actually approximate such flows,​​​‌ since we show that​ any sufficiently symmetric discrete​‌ convolution kernel can be​​ used to approximate the​​​‌ curvature flow, provided the​ time and space discretization​‌ parameters are appropriately tuned.​​

5.12 Variational problems in​​​‌ linearized elasticity with cracks​

Participants: Antonin Chambolle,​‌ Vito Crismale.

Our​​ paper on a very​​​‌ general and complete compactness​ result for “$\mathrm{G‌}\left(S\right)\mathrm{B}D$” functions (which​​​‌ arise in variational models​ for fracture growth) was​‌ finally published this year​​ 14. Then, this​​​‌ year in 34,​ we gave a new​‌ characterization of these functions,​​ which is much simpler​​​‌ than the original definition​ of Dal Maso (2011).​‌ We also review the​​ theoretical results shown during​​​‌ the past 10 years​ on the existence and​‌ regularity for the Griffith​​ problem, a free discontinuity​​​‌ problem arising in this​ theory, and describe some​‌ proofs with a very​​ simple presentations, in the​​​‌ proceedings of the ICIAM​ 2023 conference 23.​‌

5.13 Analysis of Flow​​ Matching problems and their​​​‌ relationship to optimal transport​

Participants: Antonin Chambolle,​‌ Johannes Hertrich, Julie​​ Delon.

In 24​​​‌ we investigate the connections​ between rectified flows, flow​‌ matching, and optimal transport.​​ Flow matching is a​​​‌ recent approach to learning​ generative models by estimating​‌ velocity fields that guide​​ transformations from a source​​​‌ to a target distribution.​ Rectified flow matching aims​‌ to straighten the learned​​ transport paths, yielding more​​​‌ direct flows between distributions.​ Our first contribution is​‌ a set of invariance​​ properties of rectified flows​​​‌ and explicit velocity fields.​ In addition, we also​‌ provide explicit constructions and​​ analysis in the Gaussian​​​‌ (not necessarily independent) and​ Gaussian mixture settings and​‌ study the relation to​​ optimal transport. Our second​​​‌ contribution addresses recent claims​ suggesting that rectified flows,​‌ when constrained such that​​ the learned velocity field​​​‌ is a gradient, can​ yield (asymptotically) solutions to​‌ optimal transport problems. We​​ study the existence of​​​‌ solutions for this problem​ and demonstrate that they​‌ only relate to optimal​​ transport under assumptions that​​ are significantly stronger than​​​‌ those previously acknowledged. In‌ particular, we present several‌​‌ counter-examples that invalidate earlier​​ equivalence results in the​​​‌ literature, and we argue‌ that enforcing a gradient‌​‌ constraint on rectified flows​​ is, in general, not​​​‌ a reliable method for‌ computing optimal transport maps.‌​‌

5.14 One-dimensional approximation of​​ measures in Wasserstein distance​​​‌

Participants: Antonin Chambolle,‌ Joao-Miguel Machado, Vincent‌​‌ Duval.

In 15​​, we have proposed​​​‌ a variational approach to‌ approximate measures with measures‌​‌ uniformly distributed over a​​ 1-dimensional set. The problem​​​‌ consists in minimizing a‌ Wasserstein distance as a‌​‌ data term with a​​ regularization given by the​​​‌ length of the support.‌ As it is challenging‌​‌ to prove existence of​​ solutions to this problem,​​​‌ we propose a relaxed‌ formulation, which always admits‌​‌ a solution. In the​​ sequel we show that,​​​‌ under some assumption on‌ the original measure, any‌​‌ solution to the relaxed​​ problem is solution to​​​‌ the original one. Finally‌ we prove that, whenever‌​‌ the original measure has​​ a density in ${\mathrm{L‌}}^{d/(\mathrm{d‌}-1)}(‌‌{ℝ}^d)$,​​ any optimal solution is​​​‌ supported by an Ahlfors‌ regular set.

5.15 Phase-field‌​‌ approximation for 1-dimensional shape​​ optimization problems

Participants: Joao-Miguel​​​‌ Machado.

In 18‌, we propose an‌​‌ unified framework for the​​ phase field approximation of​​​‌ 1-dimensional shape optimization problems‌ with connectedness constraints in‌​‌ any dimension. In particular,​​ we focus on the​​​‌ average distance minimizers problem‌ and the Wasserstein-${\mathrm{H‌‌}}^1$ problem recently introduced​​ by Duval, Chambolle and​​​‌ Machado. The scheme relies‌ on the p-Ambrosio-Tortorelli energy‌​‌ and the diffuse connectedness​​ functional proposed by Dondl​​​‌ et al. that penalizes‌ how disconnected the level‌​‌ sets of phase fields​​ are. We argue that​​​‌ choosing p>d, not only‌ the optimal profiles coming‌​‌ from the Ambrosio Tortorelli​​ term present sharper transitions,​​​‌ but it also allows‌ us to control the‌​‌ level sets of phase​​ fields, enabling the analysis​​​‌ of the connectedness functional.‌ This leads to general‌​‌ $\Gamma$-liminf and limsup​​ inequalities that are easily​​​‌ adaptable to prove Γ-convergence‌ results for the average‌​‌ distance and Wasserstein-${\mathrm{H}}^1$ problems.

5.16 Memorization​​​‌ vs. Generalization in diffusion‌ models with the U-Net‌​‌ architecture

Participants: Sylvain Topeza​​, Antonin Chambolle,​​​‌ Vincent Duval.

During‌ the internship of Sylvain‌​‌ Topeza  41, we​​ have reproduced and extended​​​‌ the experiments of Kadkhodaie,‌ Guth, Simoncelli and Mallat‌​‌ 93 on generalization in​​ diffusion models. The original​​​‌ study reported that two‌ U-Net denoisers trained independently‌​‌ on non-overlapping subsets of​​ CelebA produce nearly identical​​​‌ samples when seeded with‌ the same noise, suggesting‌​‌ a form of deterministic​​ convergence. We confirm this​​​‌ qualitative behavior under constrained‌ settings (40×40 resolution) and‌​‌ develop a fully reproducible,​​ memory-efficient preprocessing pipeline. Beyond​​​‌ replication, we explore its‌ stability with respect to‌​‌ dataset correlation by introducing​​ attribute-controlled (eyeglasses/no-eyeglasses, male/female) and​​​‌ identity-disjoint splits, and by‌ assessing sample originality with‌​‌ a perceptual metric (LPIPS)​​ instead of pixel correlation.​​​‌ Our findings indicate that‌ convergence between independently trained‌​‌ denoisers remains strong when​​​‌ training subsets are correlated,​ but weakens as semantic​‌ heterogeneity increases. These observations​​ suggest that both architectural​​​‌ inductive biases and dataset​ redundancy contribute to the​‌ stability of the phenomenon​​ originally reported by Mallat​​​‌ and collaborators.

5.17 A​ variational method for curve​‌ extraction

Participants: Majid Arthaud​​, Antonin Chambolle,​​​‌ Vincent Duval.

In​ 22, we have​‌ addressed the problem of​​ extracting one-dimensional objects (curves)​​​‌ from images, which is​ of interest, e.g. when​‌ segmenting the blood vessels​​ in retina images. The​​​‌ originality of our approach​ is that it is​‌ variational and basd on​​ the discretization of an​​​‌ energy and Smirnov's decomposition​ theorem for vector fields.​‌ It is used to​​ design a bi-level minimization​​​‌ approach to automatically extract​ curves and 1D structures​‌ from an image, which​​ is mostly unsupervised.

In​​​‌ 30, we have​ extended then the method​‌ to curvature-dependent energies, using​​ a now classical lifting​​​‌ of the curves in​ the space of positions​‌ and orientations equipped with​​ an appropriate sub-Riemanian or​​​‌ Finslerian metric.

5.18 A​ synthetic approach to comparison​‌ principles for variational problems,​​ with applications to optimal​​​‌ transport

Participants: Flavien Léger​, Maxime Sylvestre.​‌

In 9, we​​ develop a synthetic, variational​​​‌ framework for deriving comparison​ principles in infinite-dimensional Banach​‌ spaces. Unlike traditional approaches​​ that rely on the​​​‌ regularity of minimizers and​ Euler–Lagrange equations, our method​‌ exploits the order-theoretic structure​​ of the energy. Central​​​‌ to our analysis is​ the notion of submodularity​‌ and its convex dual,​​ substitutability, which we extend​​​‌ here to the infinite-dimensional​ setting. We prove a​‌ duality theorem establishing that​​ a convex functional is​​​‌ submodular if and only​ if its conjugate is​‌ substitutable. We apply these​​ results to problems in​​​‌ optimal transport, and derive​ comparison principles for Kantorovich​‌ potentials in standard, entropic,​​ and unbalanced settings without​​​‌ requiring regularity assumptions on​ the cost or domain.​‌ Finally, we prove that​​ general transport costs are​​​‌ substitutable, yielding comparison principles​ for JKO schemes driven​‌ by internal energies.

6.1​​​‌ International initiatives

6.2 International​​ research visitors

6.3 National initiatives​​​‌

• PR[AI]RIE-PSAI (ANR-23-IACL-0008)
  (2024-2029) France‌ 2030 funded project (ANR-23-IACL-0008),‌​‌ chair of Antonin Chambolle​​ (2 PhD and 2​​​‌ post-docs during the 5‌ years, managed by the‌​‌ CNRS at CEREMADE/Paris-Dauphine)
• PDE​​ AI
  (2023-2027) Antonin Chambolle​​​‌ is the main coordinator‌ of the PDE-AI project,‌​‌ funded by the PEPR​​ IA (France 2030, ANR)​​​‌ and gathering 10 groups‌ throughout France working on‌​‌ PDEs and nonlinear analysis​​ for artificial intelligence.
• ANR​​​‌ GOTA
  (2023-2027) is a‌ JCJC grant (253k€) carried‌​‌ by Luca Nenna (PI),​​ Paul Pegon and Maxime​​​‌ Laborde , dealing with‌ some generalizations and applications‌​‌ of Optimal Transport theory​​ with a particular focus​​​‌ on three main topics:‌ multi-marginal optimal transport, urban‌​‌ planning and multi-population models,​​ and multi-marginal entropic optimal​​​‌ transport.
• ANR ESSTOS
  (2025-2029)‌ is a JCJC grant‌​‌ (215k€) carried by A​​​‌ . Monteil, managed by​ Université Paris Est-Créteil, on​‌ the relation between elliptic​​ systems and geometric objects​​​‌ such as minimal surfaces​ or minimal graphs. Paul​‌ Pegon is one of​​ the five members, involved​​​‌ in one of the​ two axes, which focuses​‌ on blending branched OT​​ with the different formulations​​​‌ of classical OT and​ congested OT.

7.1 Promoting scientific activities​​

7.2 Teaching​ - Supervision - Juries​‌ - Educational and pedagogical​​ outreach

7.3​​ Popularization

8.1 Major publications‌

• 1 inproceedingsP.-C.Pierre-Cyril‌​‌ Aubin-Frankowski, A.Anna​​ Korba and F.Flavien​​​‌ Léger. Mirror Descent‌ with Relative Smoothness in‌​‌ Measure Spaces, with application​​ to Sinkhorn and EM​​​‌.NeurIPS 2022 -‌ Thirty-sixth Conference on Neural‌​‌ Information Processing SystemsNew​​ Orleans, United States2022​​​‌HAL
• 2 articleJ.-D.‌Jean-David Benamou, G.‌​‌Guillaume Carlier, M.​​Marco Cuturi, L.​​​‌Luca Nenna and G.‌Gabriel Peyré. Iterative‌​‌ Bregman Projections for Regularized​​ Transportation Problems.SIAM​​​‌ Journal on Scientific Computing‌2372015,‌​‌ A1111-A1138HALDOI
• 3​​ articleJ.-D.Jean-David Benamou​​​‌, T.Thomas Gallouët‌ and F.-X.François-Xavier Vialard‌​‌. Second order models​​ for optimal transport and​​​‌ cubic splines on the‌ Wasserstein space.Foundations‌​‌ of Computational MathematicsOctober​​ 2019HALDOI
• 4​​​‌ articleC.Claire Boyer‌, A.Antonin Chambolle‌​‌, Y.Yohann de​​ Castro, V.Vincent​​​‌ Duval, F.Frédéric‌ de Gournay and P.‌​‌Pierre Weiss. On​​ Representer Theorems and Convex​​​‌ Regularization.SIAM Journal‌ on Optimization292‌​‌May 2019, 1260–1281​​HALDOI
• 5 article​​​‌G.Guillaume Carlier,‌ V.Vincent Duval,‌​‌ G.Gabriel Peyré and​​ B.Bernhard Schmitzer.​​​‌ Convergence of Entropic Schemes‌ for Optimal Transport and‌​‌ Gradient Flows.SIAM​​ Journal on Mathematical Analysis​​​‌492April 2017‌HALDOI
• 6 misc‌​‌G.Guillaume Carlier,​​ P.Paul Pegon and​​​‌ L.Luca Tamanini.‌ Convergence rate of general‌​‌ entropic optimal transport costs​​.June 2022HAL​​​‌
• 7 articleA.Antonin‌ Chambolle, V.Vincent‌​‌ Duval and J. M.​​João Miguel Machado.​​​‌ One-dimensional approximation of measures‌ in Wasserstein distance.‌​‌Journal de l'École polytechnique​​ — Mathématiques12January​​​‌ 2025, 101-145HAL‌DOI
• 8 miscF.‌​‌Flavien Léger and P.-C.​​Pierre-Cyril Aubin-Frankowski. Gradient​​​‌ descent with a general‌ cost.December 2023‌​‌HAL
• 9 miscF.​​Flavien Léger and M.​​​‌Maxime Sylvestre. A‌ synthetic approach to comparison‌​‌ principles for variational problems,​​ with applications to optimal​​​‌ transport.February 2026‌HALback to text‌​‌
• 10 miscF.Flavien​​ Léger and F.-X.François-Xavier​​​‌ Vialard. A geometric‌ Laplace method.December‌​‌ 2022HAL
• 11 article​​I.Irène Waldspurger.​​​‌ Phase retrieval with random‌ Gaussian sensing vectors by‌​‌ alternating projections.IEEE​​ Transactions on Information Theory​​​‌6452018,‌ 3301-3312HAL
• 12 article‌​‌I.Irène Waldspurger and​​ A.Alden Waters.​​​‌ Rank optimality for the‌ Burer-Monteiro factorization.SIAM‌​‌ Journal on Optimization30​​32020, 2577-2602​​​‌HALDOI

8.2 Publications‌ of the year

8.3 Cited​​​‌ publications

• 42 articleL.​Luigi Ambrosio and W.​‌Wilfrid Gangbo. Hamiltonian​​ ODEs in the Wasserstein​​​‌ space of probability measures​.Communications on Pure​‌ and Applied Mathematics: A​​ Journal Issued by the​​​‌ Courant Institute of Mathematical​ Sciences6112008​‌, 18--53back to​​ text
• 43 bookL.​​​‌Luigi Ambrosio, N.​Nicola Gigli and G.​‌Giuseppe Savaré. Gradient​​ flows in metric spaces​​​‌ and in the space​ of probability measures.​‌Lectures in Mathematics ETH​​ ZürichBirkhäuser Verlag, Basel​​​‌2008, x+334back​ to text
• 44 article​‌D. N.Douglas N.​​ Arnold, R. S.​​​‌Richard S. Falk and​ R.Ragnar Winther.​‌ Finite element exterior calculus,​​ homological techniques, and applications​​​‌.Acta Numerica15​2006, 1–155DOI​‌back to text
• 45​​ miscJ. D.Julio​​​‌ Daniel Backhoff-Veraguas and G.​Gudmund Pammer. Applications​‌ of weak transport theory​​.2020back to​​​‌ text
• 46 unpublishedX.​Xavier Bacon, G.​‌ G.Guillaume Guillaume Carlier​​ and B.Bruno Nazaret​​​‌. A spatial Pareto​ exchange economy problem.​‌December 2021, working​​ paper or preprintHAL​​​‌back to text
• 47​ articleC.César Barilla​‌, G.Guillaume Carlier​​ and J.-M.Jean-Michel Lasry​​​‌. A mean field​ game model for the​‌ evolution of cities.​​Journal of Dynamics and​​​‌ Games2021HALback​ to text
• 48 article​‌J.-D.Jean-David Benamou and​​ Y.Yann Brenier.​​​‌ A computational fluid mechanics​ solution to the Monge--Kantorovich​‌ mass transfer problem.​​Numerische Mathematik843​​​‌2000, 375--393DOI​back to text
• 49​‌ articleJ.-D.Jean-David Benamou​​ and Y.Y. Brenier​​​‌. Weak existence for​ the semigeostrophic equations formulated​‌ as a coupled Monge-Ampère/transport​​ problem.SIAM J.​​​‌ Appl. Math.585​1998, 1450--1461back​‌ to text
• 50 article​​J.-D.Jean-David Benamou,​​​‌ G.Guillaume Chazareix,​ G.Giorgi Rukhaia and​‌ W. L.Wilbert L​​ Ijzerman. Point Source​​​‌ Regularization of the Finite​ Source Reflector Problem.​‌Journal of Computational Physics​​May 2022HALback​​​‌ to text
• 51 book​M.Marc Bernot,​‌ V.Vicent Caselles and​​ J.-M.Jean-Michel Morel.​​​‌ Optimal Transportation Networks: Models​ and Theory.Lecture​‌ Notes in MathematicsBerlin​​ HeidelbergSpringer-Verlag2009,​​​‌ URL: https://www.springer.com/gp/book/9783540693147DOIback​ to text
• 52 article​‌M.M. Boissier,​​ G.G. Allaire and​​​‌ C.C. Tournier.​ Additive manufacturing scanning paths​‌ optimization using shape optimization​​ tools.Struct. Multidiscip.​​​‌ Optim.6162020​, 2437--2466URL: https://doi.org/10.1007/s00158-020-02614-3​‌DOIback to text​​
• 53 articleN.Nicolas​​​‌ Bonneel and J.Julie​ Digne. A survey​‌ of Optimal Transport for​​ Computer Graphics and Computer​​​‌ Vision.Computer Graphics​ Forum422_eprint:​‌ https://onlinelibrary.wiley.com/doi/pdf/10.1111/cgf.147782023, 439--460​​URL: https://onlinelibrary.wiley.com/doi/abs/10.1111/cgf.14778DOIback​​​‌ to text
• 54 article​N.N. Boumal,​‌ V.V. Voroninski and​​ A. S.A. S.​​ Bandeira. Deterministic guarantees​​​‌ for Burer-Monteiro factorizations of‌ smooth semidefinite programs.‌​‌preprinthttps://arxiv.org/abs/1804.020082018back​​ to text
• 55 article​​​‌N.Nicholas Boyd,‌ G.Geoffrey Schiebinger and‌​‌ B.Benjamin Recht.​​ The Alternating Descent Conditional​​​‌ Gradient Method for Sparse‌ Inverse Problems.SIAM‌​‌ Journal on Optimization27​​2Publisher: Society for​​​‌ Industrial and Applied Mathematics‌January 2017, 616--639‌​‌URL: https://epubs.siam.org/doi/10.1137/15M1035793DOIback​​ to text
• 56 article​​​‌A.Alessio Brancolini and‌ B.Benedikt Wirth.‌​‌ Equivalent Formulations for the​​ Branched Transport and Urban​​​‌ Planning Problems.106‌4October 2016,‌​‌ 695--724URL: https://www.sciencedirect.com/science/article/pii/S0021782416300083DOI​​back to text
• 57​​​‌ articleK.Kristian Bredies‌, M.Marcello Carioni‌​‌, S.Silvio Fanzon​​ and F.Francisco Romero​​​‌. A generalized conditional‌ gradient method for dynamic‌​‌ inverse problems with optimal​​ transport regularization.arXiv​​​‌ preprint arXiv:2012.117062020back‌ to textback to‌​‌ text
• 58 articleK.​​K. Bredies and H.​​​‌H.K. Pikkarainen. Inverse‌ problems in spaces of‌​‌ measures.ESAIM: Control,​​ Optimisation and Calculus of​​​‌ Variations1912013‌, 190--218back to‌​‌ text
• 59 articleY.​​Y. Brenier. Polar​​​‌ factorization and monotone rearrangement‌ of vector-valued functions.‌​‌Comm. Pure Appl. Math.​​4441991,​​​‌ 375--417URL: http://dx.doi.org/10.1002/cpa.3160440402DOI‌back to text
• 60‌​‌ articleY.Yann Brenier​​. The least action​​​‌ principle and the related‌ concept of generalized flows‌​‌ for incompressible perfect fluids​​.J.~Amer. Math. Soc.​​​‌21989, 225--255‌back to text
• 61‌​‌ articleS.S. Burer​​ and R. D.R.​​​‌ D. C. Monteiro.‌ A nonlinear programming algorithm‌​‌ for solving semidefinite programs​​ via low-rank factorization.​​​‌Mathematical Programming952‌2003, 329--357back‌​‌ to text
• 62 book​​G.Giuseppe Buttazzo,​​​‌ A.Aldo Pratelli,‌ E.Eugene Stepanov and‌​‌ S.Sergio Solimini.​​ Optimal Urban Networks via​​​‌ Mass Transportation.1961‌Lecture Notes in Mathematics‌​‌Berlin, HeidelbergSpringer Berlin​​ Heidelberg2009, URL:​​​‌ http://link.springer.com/10.1007/978-3-540-85799-0DOIback to‌ text
• 63 articleE.‌​‌ J.E. J. Candès​​ and C.C. Fernandez-Granda​​​‌. Towards a Mathematical‌ Theory of Super-Resolution.‌​‌Communications on Pure and​​ Applied Mathematics676​​​‌2014, 906--956back‌ to text
• 64 unpublished‌​‌G.Guillaume Carlier,​​ A.Arnaud Dupuy,​​​‌ A.Alfred Galichon and‌ Y.Yifei Sun.‌​‌ SISTA: Learning Optimal Transport​​ Costs under Sparsity Constraints​​​‌.October 2020,‌ working paper or preprint‌​‌HALback to text​​
• 65 articleG.Guillaume​​​‌ Carlier, P.Paul‌ Pegon and L.Luca‌​‌ Tamanini. Convergence rate​​ of general entropic optimal​​​‌ transport costs.Calculus‌ of Variations and Partial‌​‌ Differential Equations624​​May 2023, 116​​​‌HALDOIback to‌ text
• 66 articleG.‌​‌Guillaume Carlier and C.​​Clarice Poon. On​​​‌ the total variation Wasserstein‌ gradient flow and the‌​‌ TV-JKO scheme.ESAIM:​​ Control, Optimisation and Calculus​​​‌ of Variations2019HAL‌back to text
• 67‌​‌ articleY.Yohann de​​ Castro, V.Vincent​​​‌ Duval and R.Romain‌ Petit. Towards Off-the-grid‌​‌ Algorithms for Total Variation​​​‌ Regularized Inverse Problems.​Journal of Mathematical Imaging​‌ and VisionJuly 2022​​HALDOIback to​​​‌ text
• 68 articleY.​Yohann de Castro and​‌ F.Fabrice Gamboa.​​ Exact reconstruction using Beurling​​​‌ minimal extrapolation.Journal​ of Mathematical Analysis and​‌ Applications3951November​​ 2012, 336--354URL:​​​‌ https://linkinghub.elsevier.com/retrieve/pii/S0022247X12003952DOIback to​ text
• 69 articleP.​‌Paul Catala, V.​​Vincent Duval and G.​​​‌Gabriel Peyré. A​ Low-Rank Approach to Off-the-Grid​‌ Sparse Superresolution.SIAM​​ Journal on Imaging Sciences​​​‌123Publisher: Society​ for Industrial and Applied​‌ MathematicsJanuary 2019,​​ 1464--1500URL: https://epubs.siam.org/doi/abs/10.1137/19M124071XDOI​​​‌back to text
• 70​ articleA.Antonin Chambolle​‌, L. A.Luca​​ A. D. Ferrari and​​​‌ B.Benoit Merlet.​ Variational approximation of size-mass​‌ energies for $k$-dimensional​​ currents.ESAIM Control​​​‌ Optim. Calc. Var.25​2019, Paper No.​‌ 43, 39URL: https://doi.org/10.1051/cocv/2018027​​DOIback to text​​​‌
• 71 articleA.Antonin​ Chambolle and T.Thomas​‌ Pock. Crouzeix-Raviart approximation​​ of the total variation​​​‌ on simplicial meshes.​J. Math. Imaging Vision​‌626-72020,​​ 872--899URL: https://doi.org/10.1007/s10851-019-00939-3DOI​​​‌back to text
• 72​ articleA.Antonin Chambolle​‌ and T.Thomas Pock​​. Learning consistent discretizations​​​‌ of the total variation​.SIAM J. Imaging​‌ Sci.1422021​​, 778--813URL: https://doi.org/10.1137/20M1377199​​​‌DOIback to text​
• 73 articleT.Thierry​‌ Champion, L.Luigi​​ De Pascale and P.​​​‌Petri Juutinen. The​ $$-Wasserstein Distance: Local Solutions​‌ and Existence of Optimal​​ Transport Maps.40​​​‌1January 2008,​ 1--20URL: https://epubs.siam.org/doi/10.1137/07069938XDOI​‌back to text
• 74​​ inproceedingsL.Lenaic Chizat​​​‌, P.Pierre Roussillon​, F.Flavien Léger​‌, F.-X.François-Xavier Vialard​​ and G.Gabriel Peyré​​​‌. Faster Wasserstein Distance​ Estimation with the Sinkhorn​‌ Divergence.Neural Information​​ Processing SystemsAdvances in​​​‌ Neural Information Processing Systems​Vancouver, CanadaDecember 2020​‌HALback to text​​
• 75 articleL.L.​​​‌ Condat. Discrete total​ variation: new definition and​‌ minimization.SIAM J.​​ Imaging Sci.103​​​‌2017, 1258--1290URL:​ https://doi.org/10.1137/16M1075247back to text​‌
• 76 bookM. J.​​M. J. P. Cullen​​​‌. A Mathematical Theory​ of Large-Scale Atmosphere/Ocean Flow​‌.Imperial College Press​​2006, URL: https://books.google.fr/books?id=JxBqDQAAQBAJ​​​‌back to text
• 77​ inproceedingsM.Marco Cuturi​‌. Sinkhorn Distances: Lightspeed​​ Computation of Optimal Transport​​​‌.Proc. NIPS2013​, 2292--2300back to​‌ text
• 78 articleL.​​Luigi De Pascale and​​​‌ J.Jean Louet.​ A Study of the​‌ Dual Problem of the​​ One-Dimensional $L^{}$-Optimal Transport​​​‌ Problem with Applications.​27611June 2019​‌, 3304--3324URL: https://www.sciencedirect.com/science/article/pii/S0022123619300643​​DOIback to text​​​‌
• 79 articleS.Simone​ Di Marino, M.​‌Mathieu Lewin and L.​​Luca Nenna. Ground​​​‌ state energy is not​ always convex in the​‌ number of electrons.​​Journal of Physical Chemistry​​​‌ A12849This​ paper is dedicated to​‌ Trygve Helgaker on the​​ occasion of his 70th​​​‌ birthdayNovember 2024,​ 10697--10706HALDOIback​‌ to text
• 80 article​​V.Vincent Duval and​​ G.Gabriel Peyré.​​​‌ Exact Support Recovery for‌ Sparse Spikes Deconvolution.‌​‌Foundations of Computational Mathematics​​2014, 1--41URL:​​​‌ http://dx.doi.org/10.1007/s10208-014-9228-6DOIback to‌ text
• 81 articleV.‌​‌Vincent Duval and G.​​Gabriel Peyré. Sparse​​​‌ regularization on thin grids‌ I: the Lasso.‌​‌Inverse Problems335​​Publisher: IOP PublishingMarch​​​‌ 2017, 055008URL:‌ https://dx.doi.org/10.1088/1361-6420/aa5e12DOIback to‌​‌ text
• 82 articleV.​​Vincent Duval and R.​​​‌Robert Tovey. Dynamical‌ programming for off-the-grid dynamic‌​‌ inverse problems.ESAIM:​​ Control, Optimisation and Calculus​​​‌ of Variations302024‌, 7HALDOI‌​‌back to text
• 83​​ articleF. R.F.​​​‌ Rakoto Endor and I.‌I. Waldspurger. Benign‌​‌ landscape for Burer-Monteiro factorizations​​ of MaxCut-type semidefinite programs​​​‌.to appear in‌ SIAM Journal on Optimization‌​‌https://arxiv.org/abs/2411.031032024back to​​ text
• 84 unpublishedJ.​​​‌Jean Feydy, T.‌Thibault Séjourné, F.-X.‌​‌François-Xavier Vialard, S.-I.​​Shun-Ichi Amari, A.​​​‌Alain Trouvé and G.‌Gabriel Peyré. Interpolating‌​‌ between Optimal Transport and​​ MMD using Sinkhorn Divergences​​​‌.October 2018,‌ working paper or preprint‌​‌HALback to text​​
• 85 bookA.Alessio​​​‌ Figalli and F.Federico‌ Glaudo. An Invitation‌​‌ to Optimal Transport, Wasserstein​​ Distances, and Gradient Flows:​​​‌ Second Edition.25‌EMS Textbooks in Mathematics‌​‌EMS PressMay 2023​​, URL: https://ems.press/doi/10.4171/etb/25DOI​​​‌back to text
• 86‌ articleG.Gero Friesecke‌​‌ and D.Daniela Vögler​​. Breaking the Curse​​​‌ of Dimension in Multi-Marginal‌ Kantorovich Optimal Transport on‌​‌ Finite State Spaces.​​SIAM Journal on Mathematical​​​‌ Analysis5042018‌, 3996--4019URL: https://doi.org/10.1137/17M1150025‌​‌DOIback to text​​
• 87 bookA.Alfred​​​‌ Galichon. Optimal transport‌ methods in economics.‌​‌PrincetonPrinceton university press​​2016back to text​​​‌
• 88 articleI.Ivan‌ Guo and G.Grégoire‌​‌ Loeper. Path Dependent​​ Optimal Transport and Model​​​‌ Calibration on Exotic Derivatives‌.SSRN Electron.~J.Available‌​‌ at doi:10.2139/ssrn.330238401 2018​​DOIback to text​​​‌
• 89 articleJ.Jan‌ Haskovec, P.Peter‌​‌ Markowich, B.Benoît​​ Perthame and M.Matthias​​​‌ Schlottbom. Notes on‌ a PDE System for‌​‌ Biological Network Formation.​​138June 2016,​​​‌ 127--155URL: https://www.sciencedirect.com/science/article/pii/S0362546X15004344DOI‌back to text
• 90‌​‌ articleM.Matt Jacobs​​, W.Wonjun Lee​​​‌ and F.Flavien Léger‌. The back-and-forth method‌​‌ for Wasserstein gradient flows​​.ESAIM: Control, Optimisation​​​‌ and Calculus of Variations‌272021, 28‌​‌back to text
• 91​​ articleM.Matt Jacobs​​​‌ and F.Flavien Léger‌. A fast approach‌​‌ to optimal transport: The​​ back-and-forth method.Numer.​​​‌ Math.1462020,‌ 513--544URL: https://doi.org/10.1007/s00211-020-01154-8DOI‌​‌back to text
• 92​​ articleR.R. Jordan​​​‌, D.D. Kinderlehrer‌ and F.F. Otto‌​‌. The variational formulation​​ of the Fokker-Planck equation​​​‌.SIAM J. Math.‌ Anal.2911998‌​‌, 1--17back to​​ text
• 93 inproceedingsZ.​​​‌Zahra Kadkhodaie, F.‌Florentin Guth, E.‌​‌ P.Eero P Simoncelli​​ and S.Stéphane Mallat​​​‌. Generalization in diffusion‌ models arises from geometry-adaptive‌​‌ harmonic representations.International​​​‌ Conference on Learning Representations​122024back to​‌ text
• 94 articleS.​​Soheil Kolouri, S.​​​‌Serim Park, M.​Matthew Thorpe, D.​‌Dejan Slepčev and G.​​ K.Gustavo K. Rohde​​​‌. Optimal Mass Transport:​ Signal processing and machine-learning​‌ applications.IEEE signal​​ processing magazine344​​​‌July 2017, 43--59​DOIback to text​‌
• 95 articleT.Thomas​​ Lachand-Robert and E.Edouard​​​‌ Oudet. Minimizing within​ convex bodies using a​‌ convex hull method.​​SIAM Journal on Optimization​​​‌162January 2005​, 368--379HALback​‌ to text
• 96 article​​L.Léo Lebrat,​​​‌ F.Frédéric de Gournay​, J.Jonas Kahn​‌ and P.Pierre Weiss​​. Optimal Transport Approximation​​​‌ of 2-Dimensional Measures.​SIAM Journal on Imaging​‌ Sciences122January​​ 2019, 762--787URL:​​​‌ https://epubs.siam.org/doi/10.1137/18M1193736DOIback to​ text
• 97 miscF.​‌Flavien Léger and P.-C.​​Pierre-Cyril Aubin-Frankowski. Gradient​​​‌ descent with a general​ cost.2023,​‌ URL: https://arxiv.org/abs/2305.04917back to​​ text
• 98 articleJ.-M.​​​‌Jean-Marie Mirebeau. Adaptive,​ Anisotropic and Hierarchical cones​‌ of Discrete Convex functions​​.Numerische Mathematik132​​​‌435 pages, 11​ figures. (Second version fixes​‌ a small bug in​​ Lemma 3.2. Modifications are​​​‌ anecdotic.)2016, 807--853​HALback to text​‌
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