Timeline for Fréchet subdifferentiation on riemannian manifolds
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 17, 2020 at 10:03 | history | edited | dohmatob | CC BY-SA 4.0 |
deleted 22 characters in body
|
| Nov 17, 2020 at 8:42 | comment | added | dohmatob | On second thought, if $M$ is closed and sufficiently smooth, then it can be embedded into some euclidean space of dimension $m=O(n^3)$. Then one can lift $F$ from $\mathbb R^n$ to $\mathbb R^m$ in an obvious way. Then the tentative definition in my comment (based on inverse retractions) should reduce to the attempt in my question above, modulo an $\epsilon$ which we later limit to $0^+$. What do you think ? | |
| Nov 17, 2020 at 8:41 | comment | added | user35593 | Yes its always possible but it might be difficult to find a concrete embedding for a specific manifold. Also it might be favorable to have a method that is independent of the embedding. Therefore the exponential map. | |
| Nov 17, 2020 at 8:36 | comment | added | dohmatob | BTW, I don't understand you remark about "isometrically embedded". Do you mean embedded in euclidean space of any dimension (this is always possible if $M$ is closed, thanks to the Nash embedding theorem) or embedded in $R^n$ (i.e a constraint on the embedding dimension) ? | |
| Nov 17, 2020 at 8:22 | comment | added | dohmatob | Thanks for the suggestion. Definnition 2.1 is referring to differentiability. I guess you where referring to something like $$ \partial f(w) := \left\{w^\star \in T_wM \mid \liminf_{x,y \to w,\; x \ne y}\frac{F(x) - F(y) -\langle w^\star, \exp^{-1}_w(y)-\exp^{-1}_w(x)\rangle}{\|\exp_w^{-1}(y) - \exp_w^{-1}(x)\|} \ge 0\right\} $$ | |
| Nov 17, 2020 at 8:08 | comment | added | user35593 | That works if the manifold is isometrically embedded in Euclidean space. If not you might use Definition 2.1. of researchgate.net/publication/… | |
| Nov 17, 2020 at 7:07 | history | edited | dohmatob | CC BY-SA 4.0 |
added 120 characters in body
|
| Nov 17, 2020 at 2:21 | history | edited | LSpice | CC BY-SA 4.0 |
Redundant accent: \'é -> é, and minor proofreading
|
| Nov 17, 2020 at 2:11 | history | asked | dohmatob | CC BY-SA 4.0 |