Timeline for Convexity of a set of probability densities
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 28, 2021 at 22:52 | comment | added | leo monsaingeon | just for the record: the notion of displacement convexity is due to Robert McCann in his beautiful paper "A convexity principle for interacting gases" | |
| Oct 28, 2021 at 22:51 | comment | added | leo monsaingeon | yes, it is specific, and displacement convexity is somehow tricky. I can also suggest giving a look at Villani's "books, as well as chapter 9 in the book by Ambrosio-Gigli-Savaré (where an alternative notion of generalized geodesics is provided, very useful at least for PDE purposes) | |
| Oct 28, 2021 at 22:47 | comment | added | 900edges | @leomonsaingeon looking these terms up, thank you for sharing! Is displacement convexity a concept specific to Wasserstein space? And does it mean "$F$ is displacement convex if it is convex along optimal transport plans"? | |
| Oct 28, 2021 at 22:43 | comment | added | leo monsaingeon | Just a comment: as pointed out by pseudocydonia, a key word is "displacement convexity". Essentially you can think of a (displacement) convex set as the sublevelset of a (displacement) convex function. So your question somehow amounts to: how to determin whether a functional is displacement convex or not? this turns out to be a difficult question, it is an active field of research, and geometry really plays a role (for example convexity of the entropy is equivalent to Ricci lower bound in the sense of Lott-Sturm-Villani, almost by definition). | |
| Jun 19, 2021 at 5:57 | comment | added | pseudocydonia | @900edges, there is a notion called either "displacement convexity" or "geodesic convexity" which I believe is what you are looking for. | |
| Jun 18, 2021 at 13:19 | comment | added | 900edges | @Steve interpolation wrt $W_2$ is closer to what I was thinking of! I'm interested in convexity of a set in the ambient space $(P,W_2)$ which is why this definition makes more sense. So far, I'm not convinced that {product densities} is not convex wrt $W_2$... | |
| Jun 18, 2021 at 6:56 | comment | added | Steve | On the one hand, you can take the simple convex combination of two measures $\mu, \nu$ as $\lambda \mu + (1-\lambda) \nu$. On the other hand you can take the displacement interpolation motivated by the Wasserstein distance: Take $\pi \in \Pi(\mu, \nu)$ to be an optimizer of $W_2(\mu, \nu)$, then if $(X, Y) \sim \pi$, the interpolation is the distribution of $\lambda X + (1-\lambda) Y$. Notice that the two concepts are very different even for Dirac measures (they result in either $\lambda \delta_x + (1-\lambda) \delta_y$ or $\delta_{\lambda x + (1-\lambda) y}$). What do you mean? | |
| Jun 17, 2021 at 17:50 | comment | added | 900edges | @DieterKadelka I figured that the distance metric would affect the definition of the path between two points, but I guess that doesn't have to be the case. Thanks for pointing that out | |
| Jun 17, 2021 at 17:08 | comment | added | Dieter Kadelka | Not quite a counterexample to assertion 2. (no densities, but this can be solved): $\mu :=\frac{1}{2} \delta_{(0,0)} + \frac{1}{2} \delta_{(1,1)}$ is a convex combination of two probability measures with independent marginals, but $\mu$ is not of this form. | |
| Jun 17, 2021 at 16:53 | comment | added | Robert Israel | The line segment between two mixtures of $N$ Gaussians consists of mixtures of $2N$ Gaussians. | |
| Jun 17, 2021 at 16:46 | comment | added | Dieter Kadelka | I don't understand. Convexity is a purely algebraic property. Where comes the 2-Wasserstein distance in? | |
| Jun 17, 2021 at 15:31 | history | asked | 900edges | CC BY-SA 4.0 |