Timeline for Uniqueness of Kantorovich potentials?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2023 at 20:09 | comment | added | leo monsaingeon | @Redeldio: up to some normalization yes, since they are only defined up to additive constants | |
| Sep 8, 2023 at 16:45 | comment | added | Redeldio | @leomonsaingeon Thank you! And obviously, they are also uniformly bounded with a bound which depends only on $c$, right? | |
| Sep 8, 2023 at 7:55 | comment | added | leo monsaingeon | For more comments I suggest you check F. Santambrogio's book [Optimal Transport for Applied Mathematicians], in partcular box 1.8 page 11 and subsequent discussions. | |
| Sep 8, 2023 at 7:54 | comment | added | leo monsaingeon | @Redeldio : this is because a pair of Kantorovich potentials can always be taken to be c-transforms of each other, i-e $(\phi,\psi)=(\phi,\phi^c)=(\phi^{cc},\phi^c)$ so that $\phi=(\phi^c)$ is itself a c-transform. And by definition a c-transform has the same modulus of continuity as the cost function $c$. So in bounded domain a smooth cost (such as the Euclidean one) is globally Lipschitz, and therefore so is the KP. An other way to think of this is that $T(x)-x=\nabla\phi(x)$, so in bounded domains $x,T(x)\in\Omega$ are bounded and therefore so is $\nabla\phi(x)$ | |
| Sep 7, 2023 at 15:40 | comment | added | Redeldio | @leomonsaingeon You said "Indeed a Kantorovich potential $\varphi$ is always Lipschitz (at least in bounded domains)". Why? Can you tell me a reference where to find such a result? Thank you | |
| Dec 20, 2022 at 8:54 | comment | added | Akira | @leomonsaingeon I have a question about Theorem 1.17 here. If you don't mind, please have a check in it. | |
| Jun 18, 2022 at 22:22 | answer | added | Shayan Hundrieser | timeline score: 4 | |
| Jun 13, 2022 at 11:29 | comment | added | JHM | @leomonsaingeon If the potentials are locally Lipschitz on $\Omega$, and their restrictions to $\partial \Omega$ are also locally Lipschitz, then Alexandrov theorem (applied to $\Omega$, $\partial \Omega$) would give a.e. uniqueneness with respect to $\mathcal{L}^d|_\Omega$ and $\mathcal{L}^{d-1}|_{\partial \Omega}$. I'm sure that's obvious to you, and therefore the gradients of the max Kantorovich potentials are a.e. uniquely defined by the OT. But pointwise uniqueness of $\phi, \psi$ is more difficult, and I don't know any references. | |
| Jun 12, 2022 at 13:12 | comment | added | leo monsaingeon | Thank you Loren. Old TeXing habits die hard, I had not noticed the spurious spacing. | |
| Jun 12, 2022 at 13:05 | comment | added | LSpice |
MathJax note: a $\newcommand\…{…}$ followed by a newline produces a spurious space in the rendered post. Distasteful as it is, one must run them together: $\newcommand\…{…}$Following text. I have edited accordingly.
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| Jun 12, 2022 at 13:04 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title; correct spurious space
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| Jun 12, 2022 at 13:02 | history | edited | Nawaf Bou-Rabee | CC BY-SA 4.0 |
added a link to the reference
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| Jun 12, 2022 at 12:56 | history | edited | leo monsaingeon | CC BY-SA 4.0 |
fixed typos, added the missing reference
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| Jun 12, 2022 at 5:04 | comment | added | leo monsaingeon | @JHM: I have my own application in mind, thank you very much. I just think it is not really worth explaining here (some nonstandard parabolic problem, formally a Wasserstein gradient flow). For the technical aspect: I need to take a first variation (Fréchet derivative) $\frac{\delta W^2(\mu,\nu)}{\delta\mu}$ with respect to $\mu$ ($\nu$ being fixed). IF the Kantorovich potential is unique then this first variation is indeed given by (the flat, $L^2$ action of) $\varphi_\mu$. Before bothering someone in person I though it was worth trying here on math.MO | |
| Jun 10, 2022 at 14:11 | comment | added | JHM | You have uniqueness of the Monge OT plan $\pi$, and in your setting the uniqueness of the subdifferentials of the dual Kantorovich potentials $\phi, \psi$. So why trouble yourself with uniqueness of the potentials themselves. Honestly why? It's very complicated question and IMO a waste of time. Are you just curious, or looking for research project, or do you have an actual application in mind? I would politely ask Jun Kitagawa or Robert McCann or Brendan Pass, if they know the answer. I've never seen them on MO. | |
| Jun 9, 2022 at 13:18 | history | asked | leo monsaingeon | CC BY-SA 4.0 |