Narrow Results

In the setting of essentially non-branching metric measure spaces, we prove the equivalence between the curvature dimension condition $CD(K,N)$, in the sense of Lott–Sturm–Villani [Sturm, On the geometry of metric measure spaces. I, Acta Math.196(1) (2006) 65–131; On the geometry of metric measure spaces. II, Acta Math.196(1) (2006) 133–177; Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. (2) 169(3) (2009) 903–991], and a newly introduced notion that we call strong Brunn–Minkowski inequality $SBM(K,N)$. This condition is a reinforcement of the generalized Brunn–Minkowski inequality $BM(K,N)$, which is known to hold in $CD(K,N)$ spaces. Our result is a first step toward providing a full equivalence between the $CD(K,N)$ condition and the validity of $BM(K,N)$, which has been recently proved in [M. Magnabosco, L. Portinale and T. Rossi, The Brunn–Minkowski inequality implies the CD condition in weighted Riemannian manifolds, Nonlinear Anal.242 (2024) 113502] in the framework of weighted Riemannian manifolds.

We prove splitting theorems for mean convex open subsets in Riemannian curvature-dimension (RCD) spaces that extend results by Kasue et al. for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel’s theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable with respect to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.

Domain adaptation is a subfield of statistical learning theory that takes into account the shift between the distribution of training and test data, typically known as source and target domains, respectively. In this context, this paper presents an incremental approach to tackle the intricate challenge of unsupervised domain adaptation, where labeled data within the target domain is unavailable. The proposed approach, OTP-DA, endeavors to learn a sequence of joint subspaces from both the source and target domains using Linear Discriminant Analysis (LDA), such that the projected data into these subspaces are domain-invariant and well-separated. Nonetheless, the necessity of labeled data for LDA to derive the projection matrix presents a substantial impediment, given the absence of labels within the target domain in the setting of unsupervised domain adaptation. To circumvent this limitation, we introduce a selective label propagation technique grounded on optimal transport (OTP), to generate pseudo-labels for target data, which serve as surrogates for the unknown labels. We anticipate that the process of inferring labels for target data will be substantially streamlined within the acquired latent subspaces, thereby facilitating a self-training mechanism. Furthermore, our paper provides a rigorous theoretical analysis of OTP-DA, underpinned by the concept of weak domain adaptation learners, thereby elucidating the requisite conditions for the proposed approach to solve the problem of unsupervised domain adaptation efficiently. Experimentation across a spectrum of visual domain adaptation problems suggests that OTP-DA exhibits promising efficacy and robustness, positioning it favorably compared to several state-of-the-art methods.

The far-field reflector problem is a well-known inverse problem arising in geometric optics. It consists in creating a mirror that reflects a given point light source to a prescribed target light at infinity. In this article, we study this problem under the common design constraint that the mirror is convex and is the graph of a polynomial over a given plane. We propose a method that iteratively improves the optical properties of the mirror surface while strictly fulfilling the design constraints. At each iteration, we first create an initial reflector by solving an optimal transport problem on the sphere. We then parameterize this reflector by the graph of a function over the plane. We test our algorithm with classical target lights at infinity and also show that our approach allows to create reflectors with more complex target lights.

In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first-order systems modeling crowd motion with hard congestion effects, introduced recently by Maury et al. The monotonicity of the velocity field implies that the $2$-Wasserstein distance along two solutions is $\lambda$-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an $L^1$-contraction property. In this case, by the regularization effect of the nondegenerate diffusion, the result follows even if the given velocity field is only $L^{\infty }$ as in the standard Fokker–Planck equation.

Domain adaptation is an important area of research, as it aims to remedy the effects of the domain shift due to differences in the distribution between the source domain used for training and the target domain where prediction takes place. However, methods for characterizing the domain shift across datasets are lacking. In this work, we propose a domain shift metric called SpOT, which stands for spherical optimal transport, by operating on the spherical manifold. We realize our approach with a spherical network, used to obtain features, and an orthogonal projection loss, used to impose orthogonality in the feature space. The resulting spherical features have better inter-class separation and lower intra-class variation compared to features in Euclidean space. This type of feature clustering makes each domain representation more compact and more suitable for further analysis. The domain shift between the datasets is calculated using the optimal transport on the spherical features, which has a sound theoretical basis. Our results are further supported by experiments that show the correlation of SpOT with a new gain of transfer measure across domain adaptation datasets.

In many practical situations, we need to change the spatial distribution of some goods. In such situations, it is desirable to minimize the overall transportation costs. In the 1-D case, the smallest transportation cost of such a change is proportional to what is known as the Wasserstein metric. The same metric can be used to describe the distance between two probability distributions. In the last decades, it turned out that this metric can be successfully used in many economic applications beyond transportation. These successes are somewhat of a mystery: in principle, there are many different metrics on the set of all probability distributions, and it is not immediately clear why namely Wasserstein metric is so successful. In this paper, we show that the Wasserstein metric naturally appears in decision making situations. This fact explains the usefulness of the Wassterstein’s metric in economic applications.

Using neural networks, we compute bounds on the prices of multi-asset derivatives given information on prices of related payoffs. As a main example, we focus on European basket options and include information on the prices of other similar options, such as spread options and/or basket options on subindices. We show that, in most cases, adding further constraints gives rise to bounds that are considerably tighter. Our approach follows the literature on constrained optimal transport and, in particular, builds on the work of Eckstein & Kupper (2018) [Computation of optimal transport and related hedging problems via penalization and neural networks, Appl. Math. Optimiz. 1–29].

In this paper, we recover a class of displacement interpolations of probability measures, in the sense of the Optimal Transport theory, by means of semiclassical measures associated with solutions of Schrödinger equation defined on the flat torus. Moreover, we prove the completing viewpoint by proving that a family of displacement interpolations can always be viewed as a path of time-dependent semiclassical measures.

In this paper we prove that in a metric measure space $(X,d,𝔪)$ verifying the measure contraction property with parameters $K\in ℝ$ and $1<N<\infty$, any optimal transference plan between two marginal measures is induced by an optimal map, provided the first marginal is absolutely continuous with respect to $𝔪$ and the space itself is essentially non-branching. In particular this shows that there exists a unique transport plan and it is induced by a map.

We prove that H-type Carnot groups of rank $k$ and dimension $n$ satisfy the $\mathrm{\text{MCP}}(K,N)$ if and only if $K\le 0$ and $N\ge k+3(n-k)$. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.

The system of isentropic Euler equations in the potential flow regime can be considered formally as a second order ordinary differential equation on the Wasserstein space of probability measures. This interpretation can be used to derive a variational time discretization. We prove that the approximate solutions generated by this discretization converge to a measure-valued solution of the isentropic Euler equations. The key ingredient is a characterization of the polar cone to the cone of optimal transport maps.

We study an optimal transport problem where, at some intermediate time, the mass is either accelerated by an external force field or self-interacting. We obtain the regularity of the velocity potential, intermediate density, and optimal transport map, under the conditions on the interaction potential that are related to the so-called Ma–Trudinger–Wang condition from optimal transport [X.-N. Ma, N. S. Trudinger and X.-J. Wang, Regularity of potential functions of the optimal transportation problems, Arch. Ration. Mech. Anal.177 (2005) 151–183.].

A Wasserstein space is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be made precise.

In the first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a natural set of spaces generalizing the usual Hilbert cube. These invariants are very similar to concepts initiated by Rogers, but our variant is specifically suited to tackle Lipschitz comparison.

In the second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide uniform bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d. These arguments are very easily adapted to study the space of closed subsets of a compact metric space, partly generalizing results of Boardman, Goodey and McClure.

Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space .

In this paper we investigate the geometry of when X is a Hadamard space, by which we mean that X has globally non-positive sectional curvature and is locally compact. Although it is known that, except in the case of the line, is not non-positively curved, our results show that have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in .