arXiv subjects
Publications and source records attributed to Jun Kitagawa.
In this paper, we establish a higher order Morrey's inequality in the framework of %non-collapsed $\mathsf{RCD}(K,N)$-spaces for $K\in\mathbb{R}$ and $N\in\mathbb{N}$. We do so by first introducing an alternate version of the second order Sobolev space $W^{2, p}(X)$, which contains amply many functions even when $p>N$.
We establish several quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under H\"older regularity assumptions, we prove Lipschitz $L^2$ (respectively $C^{1,\alpha}$) stability estimates for optimal transport maps in terms of the 2-Monge-Kantorovich distance (respectively $L^{p}$ distances) between pairs of source and target densities. When the continuity assumption is removed, we obtain a Lipschitz $L^2$ stability estimate for the Brenier potentials in terms of the $L^2$ distance between the source and target densities. The proofs rely on a precise characterization of the linear response of the Brenier potential along smooth interpolations of the data, obtained by linearizing the Monge-Amp\`ere equation in divergence form. As a further application of this approach, we derive an explicit formula for the second variation of the quadratic Monge-Kantorovich distance.
We prove existence and duality on a wide class of metric spaces, and uniqueness results on any connected, complete Riemannian manifold, with or without boundary, for classical Monge--Kantorovich barycenters. In particular, this is the first and only uniqueness result with no restriction on the geometry of the manifold aside from connectedness and completeness. We obtain these via the corresponding results for barycenter problems associated to a new two-parameter family of metrics on probability measures on a general metric fiber bundle, called the $\textit{disintegrated Monge--Kantorovich metrics}$ (previously introduced by the authors).
We develop a new framework for branched transport between probability measures which are allowed to vary in time. This framework can be used to model problems where the underlying transportation network displays a branched structure, but the source and target mass distributions can change cyclically over time, such as road networks or circulatory systems. We introduce the notion of time-dependent transport paths along with associated energies and distances, and prove existence of transport paths whose energy achieves the distance. We also show the time-dependent transport yields a metric structure on subsets of appropriately defined measure-valued Sobolev spaces.
Here, we study Hamilton-Jacobi-Bellman equations on graphs. These are meant to be the analog of any of the following types of equations in the continuum setting of partial differential and nonlocal integro-differential equations: Hamilton-Jacobi (typically first order and local), Hamilton-Jacobi-Bellmann-Isaacs (first, second, or fractional order), and elliptic integro-differential equations (nonlocal equations). We give conditions for the existence and uniqueness of solutions of these equations, and work through a long list of examples in which these assumptions are satisfied. This work is meant to accomplish three goals: complement and unite earlier assumptions and arguments focused more on the Hamilton-Jacobi type structure; import ideas from nonlocal elliptic integro-differential equations; and argue that nearly all of the operators in this family enjoy a common structure of being a monotone function of the differences of the unknown, plus ``lower order'' terms. This last goal is tied to the fact that most of the examples in this family can be proven to have a Bellman-Isaacs representation as a min-max of linear operators with a graph Laplacian structure.
We prove quantitative bounds on the stability of optimal transport maps and Kantorovich potentials from a fixed source measure $\rho$ under variations of the target measure $\mu$, when the cost function is the squared Riemannian distance on a Riemannian manifold. Previous works were restricted to subsets of Euclidean spaces, or made specific assumptions either on the manifold, or on the regularity of the transport maps. Our proof techniques combine entropy-regularized optimal transport with spectral and integral-geometric techniques. As some of the arguments do not rely on the Riemannian structure, our work also paves the way towards understanding stability of optimal transport in more general geometric spaces.
We define a new two-parameter family of metrics on subsets of Borel probability measures on general metric fiber bundles, called the $ \textit{disintegrated Monge--Kantorovich metrics}$. This family contains the classical Monge-Kantorovich metrics, linearized optimal transport distance, and fibered Wasserstein distances, and certain cases admit isometric embeddings of the sliced and max-sliced Wasserstein spaces. We prove these metrics are complete, separable (except an endpoint case), and geodesic, with a dual representation. Our results cannot be obtained by applying the theory of $L^q$ maps valued in spaces of probability measures, in fact the $L^q$ map case can be recovered from our results by taking the underlying bundle as a trivial product bundle, and the geodesicness and duality results are new even in the fibered Wasserstein case.
We introduce a one-parameter family of metrics on the space of Borel probability measures on Euclidean space with finite $p$th moment for $1\leq p <\infty$, called the $\textit{sliced Monge--Kantorovich metrics}$, which include the sliced Wasserstein and max-sliced Wasserstein metrics. We then show that these are complete, separable metric spaces that are topologically equivalent to the classical Monge--Kantorovich metrics and these metrics have a dual representation. However, we also prove these sliced metrics are $\textit{not}$ bi-Lipschitz equivalent to the classical ones in most cases, and also the spaces are (except for an endpoint case) $\textit{not}$ geodesic. The completeness, duality, and non-geodesicness are new even in the sliced and max-sliced Wasserstein cases, and non bi-Lipschitz equivalence is only known for a few specific cases. In particular this indicates that sliced and max-sliced Wasserstein metrics are not suitable direct replacements for the classical Monge--Kantorovich metrics in problems where the specific metric or geodesic structure are critical.
We prove that if $\Omega\subset \mathbb{R}^{n+1}$ is a (not necessarily strictly) convex, $C^1$ domain, and $\mu$ and $\bar{\mu}$ are probability measures absolutely continuous with respect to surface measure on $\partial \Omega$, with densities bounded away from zero and infinity, whose $2$-Monge-Kantorovich distance is sufficiently small, then there exists a continuous Monge solution to the optimal transport problem with cost function given by the quadratic distance on the ambient space $\mathbb{R}^{n+1}$. This result is also shown to be sharp, via a counterexample when $\Omega$ is uniformly convex but not $C^1$. Additionally, if $\Omega$ is $C^{1, \alpha}$ regular for some $\alpha$, then the Monge solution is shown to be H\"older continuous.
We prove existence of equal area partitions of the unit sphere via optimal transport methods, accompanied by diameter bounds written in terms of Monge--Kantorovich distances. This can be used to obtain bounds on the expectation of the maximum diameter of partition sets, when points are uniformly sampled from the sphere. An application to the computation of sliced Monge--Kantorovich distances is also presented.
Fix a pair of smooth source and target densities $\rho$ and $\rho^*$ of equal mass, supported on bounded domains $\Omega, \Omega^* \subset \mathbb{R}^n$. Also fix a cost function $c_0 \in C^{4,\alpha}(\overline{\Omega} \times \overline{\Omega^*})$ satisfying the weak regularity criterion of Ma, Trudinger, and Wang, and assume $\Omega$ and $\Omega^*$ are uniformly $c_0$- and $c_0^*$-convex with respect to each other. We consider a parabolic version of the optimal transport problem between $(\Omega,\rho)$ and $(\Omega^*,\rho^*)$ when the cost function $c$ is a sufficiently small $C^4$ perturbation of $c_0$, and where the size of the perturbation depends on the given data. Our main result establishes global-in-time existence of a solution $u \in C^2_xC^1_t(\overline\Omega \times [0, \infty))$ of this parabolic problem, and convergence of $u(\cdot,t)$ as $t \to \infty$ to a Kantorovich potential for the optimal transport map between $(\Omega,\rho)$ and $(\Omega^*,\rho^*)$ with cost function $c$. A noteworthy aspect of our work is that $c$ does \emph{not} necessarily satisfy the weak Ma-Trudinger-Wang condition.
In this short note, we show that given a cost function $c$, any coupling $\pi$ of two probability measures where the second is a discrete measure can be associated to a certain bipartite graph containing a perfect matching, based on the value of the infinity transport cost $\norm{c}_{L^\infty(\pi)}$. This correspondence between couplings and bipartite graphs is explicitly constructed. We give two applications of this result to the $\mathcal{W}_\infty$ optimal transport problem when the target measure is discrete, the first is a condition to ensure existence of an optimal plan induced by a mapping, and the second is a numerical approach to approximating optimal plans.
In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second boundary value problem. Our main observation is that this problem can be posed as an optimal transport problem where the target is a subset of the lower hemisphere of $\mathbb{S}^n$. As a result we obtain existence and regularity of solutions under mild assumptions on the curvature, as well as a quantitative version of a gradient blowup result due to Urbas, which turns out to fall within the optimal transport framework.
We show quantitative stability results for the geometric "cells" arising in semi-discrete optimal transport problems. Our results show two types of stability, the first is stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. The second is stability in Hausdorff measure, under a Poincar{\`e}-Wirtinger inequality and a regularity assumption equivalent to the Ma-Trudinger-Wang conditions of regularity in Monge-Amp{\`e}re. This last result also yields stability in the uniform norm of the dual potential functions, all three stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Amp{\`e}re regularity theory.
We present an iterative method based on repeatedly inverting the Monge-Amp\`ere operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain $\Omega \subset \mathbb{R}^n$. We prove that the iterates $u_k$ generated by this method converge as $k \to \infty$ to a solution of the Monge-Amp\`ere eigenvalue problem $$\begin{cases} \text{det} D^2u = \lambda_{MA} (-u)^n & \quad \text{in } \Omega,\\ u = 0 & \quad \text{on } \partial \Omega. \end{cases}$$ Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates $\hat{u}_k := \frac{u_k}{||u_k||_{L^{\infty}(\Omega)}}$ converge to the eigenfunction of unit height. In addition, we show that $\lim\limits_{k \to \infty} R(u_k) = \lim\limits_{k \to \infty} R(\hat{u}_k) = \lambda_{MA}$, where the Rayleigh quotient $R(u)$ is defined as $$R(u) := \frac{\int_{\Omega} (-u) \ \text{det} D^2u}{\int_{\Omega} (-u)^{n+1}}.$$ Our method converges for a wide class of initial choices $u_0$ that can be constructed explicitly, and does not rely on prior knowledge of the Monge-Amp\`ere eigenvalue $\lambda_{MA}$.
We introduce and prove convergence of a damped Newton algorithm to approximate solutions of the semi-discrete optimal transport problem with storage fees, corresponding to a problem with hard capacity constraints. This is a variant of the optimal transport problem arising in queue penalization problems, and has applications to data clustering. Our result is novel as it is the first numerical method with proven convergence for this variant problem; additionally the algorithm applies to the classical semi-discrete optimal transport problem but does not require any connectedness assumptions on the support of the source measure, in contrast with existing results. Furthermore we find some stability results of the associated Laguerre cells. All of our results come with quantitative rates. We also present some numerical examples.
We introduce and investigate properties of a variant of the semi-discrete optimal transport problem. In this problem, one is given an absolutely continuous source measure and cost function, along with a finite set which will be the support of the target measure, and a "storage fee" function. The goal is then to find a map for which the total transport cost plus the storage fee evaluated on the masses of the pushforward of the source measure is minimized. We prove existence and uniqueness for the problem, derive a dual problem for which strong duality holds, and give a characterization of dual maximizers and primal minimizers. Additionally, we find some stability results for minimizers.
We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Amp\`ere type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We derive a differential Harnack inequality for a special class of functions that solve the linearized problem. Using this Harnack inequality and certain techniques specific to mass transport, we control the oscillation in time of solutions to the parabolic equation, and obtain exponential convergence. Additionally, in the course of the proof, we present a connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport, which is interesting in its own right.