Timeline for An attempt to define expected value of a Riemannian manifold valued random variable - what'll go wrong?
Current License: CC BY-SA 4.0
6 events
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| Jul 18, 2023 at 21:51 | comment | added | Learning math | @ThomasKojar Not really, I keep track of (at east part of) the literature of manifold valued random variables. I guess I'm just unhappy that the Fréchet mean, defined in the OP and used in this literature, is not unique, and I'd have been happy of it was, hence that unsuccessful effort! Well somewhat generally, mean of manifold-valued random variables is a well-studied subject in medical imaging community and the definition of Fréchet mean that I wrote above is also common in the community, although it doesn't usually consist of probabilists. | |
| Jul 18, 2023 at 21:49 | comment | added | Learning math | @IosifPinelis Yes unfortunately, I see that, it's not even true for visualizable examples like $S^n\subset \mathbb{R}^{n+1}.$ | |
| Jul 18, 2023 at 21:25 | comment | added | R W | The first step to clarify the situation should consist in passing from "manifold-valued random variables" to their distributions, i.e., to probability measures on the manifold. | |
| Jul 18, 2023 at 20:08 | comment | added | Thomas Kojar | did you have some specific goal in mind that you wanted to try to formulate mean/variance for? Maybe we can help with that original goal instead. | |
| Jul 18, 2023 at 18:46 | comment | added | Iosif Pinelis | It will almost never be the case that the $\Phi$-image $\Phi(M)$ of $M$ is convex. So, generally, the mean of a distribution on $\Phi(M)$ will not be in $\Phi(M)$. So, I don't see how this could work. | |
| Jul 18, 2023 at 18:15 | history | asked | Learning math | CC BY-SA 4.0 |