Timeline for Relation between optimal transport cost and difference between topological invariants?
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27 events
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| Oct 9, 2022 at 15:07 | comment | added | JHM | The problem with this question is that $\chi(M)=0$ for many (most?) manifolds $M$, and the Euler characteristic $\chi$ is not a strong topological invariant. So the right hand side function $h$ will not be uniquely defined, especially on $h(0,0)$. | |
| Jan 5, 2021 at 12:56 | answer | added | JHM | timeline score: 2 | |
| Sep 21, 2020 at 14:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| May 24, 2020 at 12:31 | comment | added | Ben McKay | Is the function $d$ the distance as measured in an arbitrary metric, or one which induces the manifold topology, or some special type of metric, or just some function? | |
| May 24, 2020 at 12:29 | comment | added | Ben McKay | Why do you call the continuous mappings $C^{\infty}$? That symbol is used, in the parts of mathematics I am familiar with, for mappings with derivatives of all orders. | |
| May 24, 2020 at 12:28 | comment | added | Ben McKay | What is a regular manifold? | |
| May 24, 2020 at 12:07 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jan 25, 2020 at 11:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 26, 2019 at 10:29 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title (the question was bumped anyway)
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| Dec 26, 2019 at 5:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 28, 2019 at 3:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 30, 2019 at 2:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 31, 2018 at 2:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Dec 1, 2018 at 1:45 | answer | added | Morino_Hikari | timeline score: 0 | |
| Dec 1, 2018 at 1:33 | comment | added | Morino_Hikari | Thanks a lot :) As for me, two trivial cases currently can be safely stated. 1. If $\mathcal{M}$ is $C^{\infty}$-diffeomorphic to $\mathcal{N}$, then $(*) = 0$. 2. In the toy case so far considered, $ (*) = p\inf\int{d^{2}(y_{0}, y)}d\nu(y)$ exactly, since one can always construct a $C^{\infty}$-diffeomorphism from $D^{1}\backslash{\{(0,0)\}}$ to $S^{1}$. Since $\beta_{1}(D^{1}) = 0$, $\beta_{1}(S^{1}) = 1$, $\beta_{1}(D^{1}\backslash{\{(0,0)\}}) = 0$, is there some 'supernatural' relation between the different betti numbers and the non-vanishing $(*)$ in the toy case? | |
| Dec 1, 2018 at 1:22 | history | edited | Morino_Hikari | CC BY-SA 4.0 |
improved formatting
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| Nov 30, 2018 at 11:19 | comment | added | Martin Kell | In my opinion adding a delta might only lead to a value that is equal to $p \cdot b(\mathcal{N},\nu)$ where $b(\mathcal{N},\nu) = \inf \int d^2(y_0,y)d\nu(y)$. | |
| Nov 30, 2018 at 11:19 | comment | added | Martin Kell | Here some idea of a general construction: There is no relationship of $T$ and $\nu$ meaning continuous map would do the job. Via approximation it's enough to look an open domain $V$ of full $\nu$-measure that is starshaped from a point fixed point. Now construct a smooth map $T_n$ that maps an (starshaped) closed domain $K_n$ of $\mathcal{M}$ of measure $\mu(\mathcal{M})-\frac{1}{n}$ into the chosen domain and is constant outside of $K_n$. Altering the map inside of $K_n$ a bit one can assume the push-forward of the smooth part of $\mu$ is equal to (the smooth part) of $\nu$ on $T(K_n)$. | |
| Nov 30, 2018 at 5:35 | history | edited | Morino_Hikari | CC BY-SA 4.0 |
fix some confusions in optimization objective (*)
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| Nov 29, 2018 at 14:20 | comment | added | Morino_Hikari | And I thought the simplification you made in the first reply is based on the existence of T that push $\mu$ forward to $\nu$, which is however not obvious to me... | |
| Nov 29, 2018 at 14:02 | comment | added | Morino_Hikari | Oh, I see. Thanks for your swift feedbacks!And I have also considered a non-uniform case on two-dimension where $\mu = p\delta_{0} + (1-p)\text{Unif}(D^{1})$ and $\nu = \text{Unif}(S^{1})$ as I thought the negligible measure at point $(0,0)$ would make the conjectured 'topological obstruction' covered up by integral. And with experiments, I found out that even a small $p$ would make the transportation cost un-vanishing. By the way, have you ever read about some similar issues in recent literature? Since as far as I known, mathematician would not include the $T$ term in their setting. | |
| Nov 29, 2018 at 13:52 | comment | added | Martin Kell | Your problem does not demand continuous transport of $\mu$ to $\nu$. Any coupling and any continuous/smooth $T$ would do. Hence let $T(r,s)=s$ where $D^1$ is the unit disk and $S^1$ is parametrized by $[-1,1)$. The push-forward is a absolutely continuous measure on the circle with density having a bump at $0$ and being zero at $\pm 1$. It's even possible to find a continuous map such that the push forward is exactly the uniform measure on $S^1$. Choose a coupling concentrated on $\{ (r,s,s) | r,s \in [-1,1]\}$. Problems arise if $\dim \mathcal{M} < \dim \mathcal{N}$. | |
| Nov 29, 2018 at 13:40 | history | edited | Morino_Hikari | CC BY-SA 4.0 |
deleted 12 characters in body
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| Nov 29, 2018 at 13:29 | comment | added | Morino_Hikari | For (1), em.. I am thinking about a general form of $h$ since I am not sure if anyone has seen a similar formulation before. Concretely, we can view $c$ as the geodesic distance on $\mathcal{N}$ if it is Riemannian. | |
| Nov 29, 2018 at 13:24 | comment | added | Martin Kell | (1) Which cost function are you thinking off? (2) As there is lots of choice for $T$ the answer seems to be $(*) = \inf_{S:\mathcal{N}\to\mathcal{N} \int c(S(y),y)d\nu(y)$ if the dimensions of the manifolds agree. | |
| Nov 29, 2018 at 13:15 | history | edited | Morino_Hikari | CC BY-SA 4.0 |
Relation of Optimal Transport Cost and Difference between Topological Invariants
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| Nov 29, 2018 at 13:07 | history | asked | Morino_Hikari | CC BY-SA 4.0 |