Cite this article. Anal. [2104.05802] Efficient Optimal Transport Algorithm by Accelerated Old TeXing habits die hard, I had not noticed the spurious spacing. Let us first define what it means to be a viscosity solution to the p-Poisson equation(1.1). Lecture 2: The Kantorovich Problem. Then either \(\vert \nabla \phi (x_{0}) \vert -1\leq 0\) or \(\vert \nabla \phi (x_{0}) \vert -1 > 0\), and in the latter case we obtain \(-\Delta _{\infty}\phi (x_{0})\leq 0\) by taking the limit \(i\to \infty \) and using the uniform boundedness of \(\mu _{p_{i}}\). Springer, Berlin (2012), Lellmann, J., Lorenz, D.A., Schnlieb, C., Valkonen, T.: Imaging with KantorovichRubinstein discrepancy. 27(1), 167 (1992), Medina, M., Ochoa, P.: On viscosity and weak solutions for non-homogeneous p-Laplace equations. $T_\#\mu=\nu$). Indeed, it is trivial to see that \(u_{t}\) is even a classical solution of (3.13) on \((-2,2)\setminus \{\pm t\}\). We prove a conjecture regarding the asymptotic behavior at infinity of the Kantorovich potential for the Multimarginal Optimal Transport with Coulomb and Riesz costs. 3.2 is devoted to the optimal transport characterization of cluster points, and Sect. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Provided by the Springer Nature SharedIt content-sharing initiative. This norm coincides with the geodesic Wasserstein distance of the positive and negative part of the measure and is defined as. Politec. Soc. Using the weak lower semicontinuity of the \(\mathrm{L}^{m}\)-norm, we obtain from (3.3) that, Taking the mth root and sending \(m\to \infty \) an application of Lemma3.1 shows, Using again lower semicontinuity and (3.6) yields, Taking the mth root and applying Lemma3.1 with \(p=m\) and \(k=0\) yields, Hence, we have established all inequalities in (3.2). J. Sci. In particular, a rigorous convergence proof of the finite-dimensional approximation of Poisson learning on weighted graphswhich is used in applicationswould be desirable. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Case2, \(x_{0}\in {\overline{\{\mu \neq 0\}}^{c}}\): We have to show that. Let us now define what precisely we mean by viscosity solutions to equation (3.13). Optimal Transport: how is this transport map Borel measurable. Ann. 3(3), 133140 (2014), MATH I've never seen them on MO. Soc. since the EulerLagrange equations of this problem precisely coincide with (2.3), cf. 273, 33273405 (2017), Kohn, W., Sham, L.J. Mat. Abstract We study the potential functions that determine the optimal density for \varepsilon -entropically regularized optimal transport, the so-called Schrdinger potentials, and their convergence to the counterparts in classical optimal transport, the Kantorovich potentials. Nutz, M., Wiesel, J. Entropic optimal transport: convergence of potentials. For the balanced case of two labelled classes with equal size, i.e., \(g:\mathcal{O}\to \{\pm 1\}\) and \(\overline{g}=0\), our main results can be interpreted as follows: The labelling function u arising as limit of solutions to Poisson learning as \(p\to \infty \) is directly connected to the solution of the optimal transport problem, which transports the empirical measure \(\sum_{i: g(x_{i})=+1}\delta _{x_{i}}\) of the points with label +1 to the empirical measure \(\sum_{i: g(x_{i})=-1}\delta _{x_{i}}\) of the points with label 1. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. For non-negative MathSciNet https://doi.org/10.1186/s13662-023-03754-8, DOI: https://doi.org/10.1186/s13662-023-03754-8. We can now introduce Kantorovich's formulation of the optimal transport problem. We now give the proof of this statement. LA - eng KW - Monge-Kantorovich problem; optimal transportation; mixed methods; finite elements; . Then the dual problem consists of whoever you are having this deal with optimizing their buy/sell-prices for the most profit, under the condition that you will never take an offer that is cheaper than (partially) transporting stuff. We remark that the full proof for the PDE on can be found in [21, Theorem1.8]. 653. rev2023.6.5.43477. What's the correct way to think about wood's integrity when driving screws? The method completely avoids solving an ODE during training. Soc., Providence (1999), Peral, I., Garca Azorero, J., Manfredi, J.J., Rossi, J.D. Probab. your institution. To obtain general uniqueness statements I suggest first determining the connected components of the support for your measures. This is a preview of subscription content, access via your institution. MathSciNet 2023 BioMed Central Ltd unless otherwise stated. 42(3), 576590 (1997), Villani, C.: Optimal Transport, Old and New, Volume 338 of Grundlehren der Mathematischen Wissenschaften. \end{aligned}$$, $$\begin{aligned} \limsup_{p\to \infty} \int _{\Omega} \vert \nabla u_{p} \vert ^{m}\,\mathrm{d}x \leq \vert \Omega \vert < \infty. Furthermore, since for points \(x,y\) that lie in a ball that is fully contained in it holds \(d_{\Omega}(x,y)= \vert x-y \vert \), it is easily seen (see [1, page 23]) that in fact. Letting \(u^{\pm }:= \max (\pm u,0)\) denote the positive/negative part of a function \(u:\Omega \to \mathbb{R}\) for all \(i\in \mathbb{N}\), it holds, Applying Lemma3.1 with \(k=1\) then yields, which by the upper semicontinuity of \(u_{\infty}\) is equivalent to \(\max_{\overline{\Omega }} u_{\infty }+ \operatorname{ess\,inf}_{\Omega }u_{\infty } \leq 0\). If \(\mu _{n}\overset{\ast }{\rightharpoonup }\mu \) in \(\mathcal {M}(\overline{\Omega })\) and \(u_{n}\to u\) uniformly in \(\mathrm{C}(\overline{\Omega })\), then it holds, With the abbreviation \(\langle \mu,u\rangle:=\int _{\Omega }u\,\mathrm{d}\mu \), we can compute, The BanachSteinhaus theorem (or the uniform boundedness principle) [16, Sect. nicolagigli@googlemail.com, Get access to the full version of this content by using one of the access options below. For a large class of transportation costs, the Kantorovich potential can give the optimal transportation map by a close-form formula. The p-Laplacian for \(p\in [1,\infty )\) is defined as, For \(\mathrm{C}^{2}\)-functions u, it admits the decomposition formula. Combining these two cases, we obtain (3.18). \(u_{\infty}\) is a viscosity solution of, Let \(x_{0}\in {\{\mu >0\}}\) and \(\phi \in \mathrm{C}^{2}(\Omega )\) such that \(u_{\infty}-\phi \) has a local minimum at \(x_{0}\). \end{aligned}$$, \(\sqrt[p]{\lambda _{p}}\to \lambda _{\infty}:= \frac{2}{\operatorname{diam}(\Omega )}\in (0,\infty )\), $$\begin{aligned} \limsup_{p\to \infty} \int _{\Omega} \vert u_{p} \vert ^{m} \, \mathrm{d}x \leq \frac{ \vert \Omega \vert }{\lambda _{\infty}^{m}}< \infty. Entropic optimal transport: convergence of potentials This could be helpful to extend the scope of your work. How to put white road markings on the asphalt of a highway in Geometry Nodes. Indeed a Kantorovich potential $\varphi$ is always Lipschitz (at least in bounded domains) hence differentiable Lebesgue-almost everywhere (Rademacher's theorem), and therefore also $\mu$-almost everywhere. Lincei Mat. \end{cases}\displaystyle \end{aligned}$$, \(\overline{\Omega }\setminus ({\{\mu >0\}}\cup{\{\mu <0\}}\cup{ \overline{\{\mu \neq 0\}}^{c}} )\), $$\begin{aligned} \textstyle\begin{cases} \min \lbrace \vert \nabla \phi (x_{0}) \vert -1,- \Delta _{\infty }\phi (x_{0}) \rbrace \leq 0& \text{if }x_{0} \in {\{\mu >0\}}, \\ -\Delta _{\infty }\phi (x_{0}) \leq 0& \text{if }x_{0}\in { \overline{\{\mu \neq 0\}}^{c}}, \\ \max \lbrace 1- \vert \nabla \phi (x_{0}) \vert ,- \Delta _{\infty }\phi (x_{0}) \rbrace \leq 0& \text{if }x_{0} \in {\{\mu < 0\}}, \end{cases}\displaystyle \end{aligned}$$, \(\max_{\overline{\Omega }} u + \operatorname{ess\,inf}_{\Omega }u \leq 0\), $$\begin{aligned} \textstyle\begin{cases} \min \lbrace \vert \nabla \phi (x_{0}) \vert -1,- \Delta _{\infty }\phi (x_{0}) \rbrace \geq 0& \text{if }x_{0} \in {\{\mu >0\}}, \\ -\Delta _{\infty }\phi (x_{0}) \geq 0& \text{if }x_{0}\in { \overline{\{\mu \neq 0\}}^{c}}, \\ \max \lbrace 1- \vert \nabla \phi (x_{0}) \vert ,- \Delta _{\infty }\phi (x_{0}) \rbrace \geq 0& \text{if }x_{0} \in {\{\mu < 0\}}, \end{cases}\displaystyle \end{aligned}$$, \(\operatorname{ess\,sup}_{\Omega }u + \min_{\overline{\Omega }}u \geq 0\), \(u_{\infty}\in \mathrm{C}(\overline{\Omega })\), \((p_{i})_{i\in \mathbb{N}}\subset (d,\infty )\), \((x_{i})_{i\in \mathbb{N}}\subset \Omega \), $$\begin{aligned} - \bigl( \bigl\vert \nabla \phi (x_{i}) \bigr\vert ^{p_{i}-2} \Delta \phi (x_{i}) + (p_{i}-2) \bigl\vert \nabla \phi (x_{i}) \bigr\vert ^{p_{i}-4}\Delta _{\infty}\phi (x_{i}) \bigr) = - \Delta _{p_{i}}\phi (x_{i}) \leq \mu _{p_{i}}(x_{i}). Then, for any supercritical speed t > T there exists a Kantorovich . Marcel Nutz. These results are proved for all continuous, integrable cost functions on Polish spaces. R. Soc. Feature Flags: { The Kantorovich Dual of the 1-Wasserstein distance $W_1(p,q)$ between two densities $p(x), q(x)$ is given by https://www.math.columbia.edu/~mnutz/docs/EOT_lecture_notes.pdf, Pal, S.: On the difference between entropic cost and the optimal transport cost. Once the potential functions have been trained, it is straightforward to perform density estimation and generative modeling through a CNF recovered from the learned potential functions. . Stat. .fp,B3t02?/o.;#ipT!VaW@Vb:CI('.qt:K> S\@ 1jQRk19J,#.zXRqPq7y0'g~zG.e oPvg,pS):Q,L(1-h5iIdi40v'fl%US: 8dN6)q6,a_*@zgC"*!_iLEk`7b0k. Google Scholar, Champion, T., De Pascale, L.: Asymptotic behaviour of nonlinear eigenvalue problems involving p-Laplacian-type operators. %PDF-1.4 Now we can prove the main theorem of this section. Let us now turn to the limiting PDE (1.3) satisfied by \(u_{\infty}\) for which we assume that the limiting data \(\mu \in \mathrm{C}(\overline{\Omega })\) are continuous.
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