Wasserstein Space Research Articles

Viscosity solutions of centralized control problems in measure spaces

This work focuses on a control problem in the Wasserstein space of probability measures over $\mathbb{R}^d$. Our aim is to link this control problem to a suitable Hamilton-Jacobi-Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This regularity allows to define a notion of viscosity and a Hamiltonian function relying on directional derivatives. Using a generalization of Ekeland's principle, we show that the corresponding HJB equation admits a comparison principle, and deduce that the value function is the unique solution in this viscosity sense. The PDE tools are developed in the general framework of Measure Differential Equations.

Convergence of Flow-Based Generative Models via Proximal Gradient Descent in Wasserstein Space

Convergence of Flow-Based Generative Models via Proximal Gradient Descent in Wasserstein Space

Efficient Convex PCA with Applications to Wasserstein GPCA and Ranked Data

Convex PCA, which was introduced by Bigot et al. modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this article is 3-fold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online.

Shadowing of the induced map for contracting homeomorphisms

Let f:X→X be a contracting homeomorphism of a metric space with positive diameter. We prove that the induced map f⁎ in the space of probability measures equipped with the Prokhorov metric does not have the shadowing property. However, if X is Polish, then the restriction of f⁎ to the Wasserstein space has the generalized shadowing property as per Boyarsky and Gora [4], concerning the Kantorovich-Rubinstein and Prokhorov metrics.

Wasserstein-Kaplan-Meier Survival Regression

Survival analysis plays a pivotal role in medical research, offering valuable insights into the timing of events such as survival time. One common challenge in survival analysis is the necessity to adjust the survival function to account for additional factors, such as age, gender, and ethnicity. We propose an innovative regression model for right-censored survival data across heterogeneous populations, leveraging the Wasserstein space of probability measures. Our approach models the probability measure of survival time and the corresponding non-parametric Kaplan-Meier estimator for each subgroup as elements of the Wasserstein space. The Wasserstein space provides a flexible framework for modeling heterogeneous populations, allowing us to capture complex relationships between covariates and survival times. We address an underexplored aspect by deriving the non-asymptotic convergence rate of the Kaplan-Meier estimator to the underlying probability measure in terms of the Wasserstein metric. The proposed model is supported with a solid theoretical foundation including pointwise and uniform convergence rates, along with an efficient algorithm for model fitting. The proposed model effectively accommodates random variation that may exist in the probability measures across different subgroups, demonstrating superior performance in both simulations and two case studies compared to the Cox proportional hazards model and other alternative models.

Stochastic gradient descent for barycenters in Wasserstein space

Abstract We present and study a novel algorithm for the computation of 2-Wasserstein population barycenters of absolutely continuous probability measures on Euclidean space. The proposed method can be seen as a stochastic gradient descent procedure in the 2-Wasserstein space, as well as a manifestation of a law of large numbers therein. The algorithm aims to find a Karcher mean or critical point in this setting, and can be implemented ‘online’, sequentially using independent and identically distributed random measures sampled from the population law. We provide natural sufficient conditions for this algorithm to almost surely converge in the Wasserstein space towards the population barycenter, and we introduce a novel, general condition which ensures uniqueness of Karcher means and, moreover, allows us to obtain explicit, parametric convergence rates for the expected optimality gap. We also study the mini-batch version of this algorithm, and discuss examples of families of population laws to which our method and results can be applied. This work expands and deepens ideas and results introduced in an early version of Backhoff-Veraguas et al. (2022), in which a statistical application (and numerical implementation) of this method is developed in the context of Bayesian learning.

Isometric rigidity of Wasserstein spaces over Euclidean spheres

We study the structure of isometries of the quadratic Wasserstein space W2(Sn,‖⋅‖) over the sphere endowed with the distance inherited from the norm of Rn+1. We prove that W2(Sn,‖⋅‖) is isometrically rigid, meaning that its isometry group is isomorphic to that of (Sn,‖⋅‖). This is in striking contrast to the non-rigidity of its ambient space W2(Rn+1,‖⋅‖) but in line with the rigidity of the geodesic space W2(Sn,∢). One of the key steps of the proof is the use of mean squared error functions to mimic displacement interpolation in W2(Sn,‖⋅‖). A major difficulty in proving rigidity for quadratic Wasserstein spaces is that one cannot use the Wasserstein potential technique. To illustrate its general power, we use it to prove isometric rigidity of Wp(S1,‖⋅‖) for 1≤p<2.

A finite-dimensional approximation for partial differential equations on Wasserstein space

This paper presents a finite-dimensional approximation for a class of partial differential equations on the space of probability measures. These equations are satisfied in the sense of viscosity solutions. The main result states the convergence of the viscosity solutions of the finite-dimensional PDE to the viscosity solutions of the PDE on Wasserstein space, provided that uniqueness holds for the latter, and heavily relies on an adaptation of the Barles & Souganidis monotone scheme (Barles and Souganidis, 1991) to our context, as well as on a key precompactness result for semimartingale measures. We illustrate our convergence result with the example of the Hamilton–Jacobi–Bellman and Bellman–Isaacs equations arising in stochastic control and differential games, and propose an extension to the case of path-dependent PDEs.

Well-posedness for Hamilton–Jacobi equations on the Wasserstein space on graphs

Well-posedness for Hamilton–Jacobi equations on the Wasserstein space on graphs

On adversarial robustness and the use of Wasserstein ascent-descent dynamics to enforce it

Abstract We propose iterative algorithms to solve adversarial training problems in a variety of supervised learning settings of interest. Our algorithms, which can be interpreted as suitable ascent-descent dynamics in Wasserstein spaces, take the form of a system of interacting particles. These interacting particle dynamics are shown to converge toward appropriate mean-field limit equations in certain large number of particles regimes. In turn, we prove that, under certain regularity assumptions, these mean-field equations converge, in the large time limit, toward approximate Nash equilibria of the original adversarial learning problems. We present results for non-convex non-concave settings, as well as for non-convex concave ones. Numerical experiments illustrate our results.

Coarse Embeddability of Wasserstein Space and the Space of Persistence Diagrams

AbstractWe prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e. Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the p-Wasserstein distance for $$1\le p\le 2$$ 1 ≤ p ≤ 2 remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams is equivalent to embeddability for Wasserstein space on $$\mathbb {R}^2$$ R 2 . When $$p &gt; 1$$ p &gt; 1 , Wasserstein space on $$\mathbb {R}^2$$ R 2 is snowflake universal (an obstruction to embeddability into any Banach space of non-trivial type) if and only if the space of persistence diagrams is snowflake universal.

Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

Viscosity Solutions of the Eikonal Equation on the Wasserstein Space

Distribution-on-distribution regression with Wasserstein metric: Multivariate Gaussian case

Distribution data refer to a data set in which each sample is represented as a probability distribution, a subject area that has received increasing interest in the field of statistics. Although several studies have developed distribution-to-distribution regression models for univariate variables, the multivariate scenario remains under-explored due to technical complexities. In this study, we introduce models for regression from one Gaussian distribution to another, using the Wasserstein metric. These models are constructed using the geometry of the Wasserstein space, which enables the transformation of Gaussian distributions into components of a linear matrix space. Owing to their linear regression frameworks, our models are intuitively understandable, and their implementation is simplified because of the optimal transport problem’s analytical solution between Gaussian distributions. We also explore a generalization of our models to encompass non-Gaussian scenarios. We establish the convergence rates of in-sample prediction errors for the empirical risk minimizations in our models. In comparative simulation experiments, our models demonstrate superior performance over a simpler alternative method that transforms Gaussian distributions into matrices. We present an application of our methodology using weather data for illustration purposes.

On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein Spaces

On the Viability and Invariance of Proper Sets under Continuity Inclusions in Wasserstein Spaces

A comparison principle for semilinear Hamilton–Jacobi–Bellman equations in the Wasserstein space

The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton–Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the 2-Wasserstein distance in the doubling of variables argument, which is done by introducing a further entropy penalization that ensures that the relevant optima are achieved at positive, Lipschitz continuous densities with finite Fischer information. This allows to prove uniqueness and stability of viscosity solutions in the class of bounded Lipschitz continuous (with respect to the 1-Wasserstein distance) functions. The result does not appeal to a mean field control formulation of the equation, and, as such, applies to equations with nonconvex Hamiltonians and measure-dependent volatility. For convex Hamiltonians that derive from a potential, we prove that the value function associated with a suitable mean-field optimal control problem with nondegenerate idiosyncratic noise is indeed the unique viscosity solution.

Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

Nonlinear sufficient dimension reduction for distribution-on-distribution regression

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space, while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

Convex functions defined on metric spaces are pulled back to subharmonic ones by harmonic maps

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Approximation of splines in Wasserstein spaces

This paper investigates a time discrete variational model for splines in Wasserstein spaces to interpolate probability measures. Cubic splines in Euclidean space are known to minimize the integrated squared acceleration subject to a set of interpolation constraints. As generalization on the space of probability measures the integral of the squared acceleration is considered as a spline energy and regularized by addition of the usual action functional. Both energies are then discretized in time using local Wasserstein-2 distances and the generalized Wasserstein barycenter. The existence of time discrete regularized splines for given interpolation conditions is established. On the subspace of Gaussian distributions, the spline interpolation problem is solved explicitly and consistency in the discrete to continuous limit is shown. The computation of time discrete splines is implemented numerically, based on entropy regularization and the Sinkhorn algorithm. A variant of Nesterov’s accelerated gradient descent algorithm is applied for the minimization of the fully discrete functional. A variety of numerical examples demonstrate the robustness of the approach and show striking characteristics of the method. As a particular application the spline interpolation for synthesized textures is presented.

W-entropy and Langevin deformation on Wasserstein space over Riemannian manifolds

W-entropy and Langevin deformation on Wasserstein space over Riemannian manifolds