Timeline for Is the Wasserstein distance to the empirical measure minimized by the underlying distribution?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 31, 2023 at 9:53 | comment | added | joemrt | I see, thanks a lot for this detailed counterexample. | |
| Jan 30, 2023 at 20:00 | comment | added | Iosif Pinelis | @joemrt : As is now shown, the only exception to the strict inequality is when the support of $\mu$ consists of at most two points: mathoverflow.net/a/439727/36721 | |
| Jan 30, 2023 at 16:50 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 30, 2023 at 16:31 | comment | added | Iosif Pinelis | @joemrt : Condition on $X$ and then apply Jensen's inequality to the zero-mean random variable $Y$. At least for $p>1$, the inequality will be strict, because then the function $|\cdot|^p$ is strictly convex. I am sure the inequality will be strict even for $p=1$ (and nondegenerate $\mu$), but cannot prove this at the moment. | |
| Jan 30, 2023 at 16:26 | comment | added | joemrt | Thanks Iosif for you answer. I admit I don't see entirely why it's Jensen's inequality but I see that it's true for a normal distribution which is enough to give me a counterexample. Thanks! | |
| Jan 30, 2023 at 16:25 | vote | accept | joemrt | ||
| Jan 30, 2023 at 16:06 | vote | accept | joemrt | ||
| Jan 30, 2023 at 16:22 | |||||
| Jan 30, 2023 at 15:46 | history | edited | Iosif Pinelis | CC BY-SA 4.0 |
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| Jan 30, 2023 at 15:40 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |