Timeline for Upper bound Wasserstein distance by $\chi^2$ distance
Current License: CC BY-SA 4.0
14 events
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| Oct 27, 2022 at 14:31 | history | edited | Fei Cao | CC BY-SA 4.0 |
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| Oct 27, 2022 at 14:28 | comment | added | Fei Cao | @Efe I am sorry that you misunderstand my problem (and it's totally my fault), here the mean value of $X$ and $Y$ are fixed (hence it is equal to some fixed constant) and hence is not "scalable" | |
| Oct 27, 2022 at 1:47 | comment | added | Fei Cao | @GerryMyerson I am still a little sad about the fact that no such bound can be hoped for... | |
| Oct 27, 2022 at 1:31 | comment | added | Gerry Myerson | Wasserstein is almost an anagram of Weierstrass. | |
| Oct 27, 2022 at 0:39 | vote | accept | Fei Cao | ||
| Oct 26, 2022 at 23:57 | comment | added | Fei Cao | @Efe Actually in my problem I only care about the case when ${\bf q}$ is a Poisson distribution. I believe such estimate must have been done somewhere in the literature for a genetic ${\bf q}$ with full support on $\mathbb N$. | |
| Oct 26, 2022 at 23:56 | answer | added | Iosif Pinelis | timeline score: 1 | |
| Oct 26, 2022 at 23:50 | comment | added | Efe | In general it isn't? you need to impose a bunch of structure on q for that kind of control (think bakry-emery criterion and T2). I am not sure if mean is enough, as it seems I can move p_n -> p_5n and q_n -> q_5n which should scale the mean (and W1?), but not the chi squared? Now, if f is dependent on the mean, I dont have an example off the top of my head. | |
| Oct 26, 2022 at 23:36 | history | edited | Fei Cao | CC BY-SA 4.0 |
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| Oct 26, 2022 at 23:25 | comment | added | Fei Cao | @Efe Thank you for pointing out this very interesting reference! As I mentioned in the updated post, I forgot to mention that ${\bf p}$ and ${\bf q}$ have the same expected value. Also, the relative entropy does depend on the order of ${\bf p}$ and ${\bf q}$ but it is still a very "strong" measure of closeness between two distributions (stronger than Wasserstein), right? | |
| Oct 26, 2022 at 23:22 | history | edited | Fei Cao | CC BY-SA 4.0 |
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| Oct 26, 2022 at 22:43 | comment | added | Efe | Some basic thoughts: You can use chi squared to bound TV and then use that to bound W1 if your measures are defined on a finite subset (see arxiv.org/pdf/math/0209021.pdf). Otherwise, I fail to see how we are controlling W1 which depends on the order of the p_i s with chi squared which does not? | |
| Oct 26, 2022 at 21:30 | history | edited | Fei Cao | CC BY-SA 4.0 |
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| Oct 26, 2022 at 21:11 | history | asked | Fei Cao | CC BY-SA 4.0 |