Timeline for Sampling from a Manifold
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 4, 2015 at 18:59 | comment | added | Joseph Zambrano | Thank you everyone, this has been incredible helpful. Vidit Nanda - yes, my question was actually inspired by a very brief look at their work, I really must take the time to thoroughly go through it. Joseph O'Rourke - wonderful links. | |
| Jan 4, 2015 at 13:36 | comment | added | Joseph O'Rourke | Related questions: "Probing a manifold with geodesics," and "Can one recover a metric from geodesics?." The answer to the latter question is No. But see Robert Bryant's Yes answer to the former question, Yes under certain assumptions. | |
| Jan 4, 2015 at 9:06 | comment | added | Vidit Nanda | I hope you are familiar with the work of niyogi, smale and weinberger (people.cs.uchicago.edu/~niyogi/papersps/NiySmaWeiHom.pdf) which assumes that the manifold in question has been embedded in euclidean space and provides bounds on the sample size in terms of the injectivity radius to extract homotopy type with high confidence. | |
| Jan 4, 2015 at 6:53 | comment | added | Manfred Weis | examples of canonical subsets of the geodesics could be those that constitute to: a minimum-weight matching, a minimum-weight apanning tree, a delaunay triangulation, or the shortest tour through all points of the sample. | |
| Jan 4, 2015 at 0:56 | comment | added | Joseph Zambrano | Ryan Budney - if possible, could you elaborate on your definition of "short geodesic" and what obstructions we might face in recovering a PL structure? | |
| Jan 3, 2015 at 23:06 | comment | added | Joseph Zambrano | I will first apologize for stating this question very loosely. To be quite honest I do not know what assumptions would be particularly useful. My intuition tells me that it would be beneficial to have, for example, some information about geodesic loops - hopefully telling us something about the homology/homotopy. But I should have been clear in stating that I don't have anything particular in mind. | |
| Jan 3, 2015 at 23:04 | comment | added | Ryan Budney | If you chose "short" geodesics in an appropriate way you should be able to derive some kind of simplicial approximation to the manifold. You'd have to put strong restrictions to get the PL type, but you should be able to recover the homotopy type if you choose your criterion carefully. | |
| Jan 3, 2015 at 22:57 | comment | added | Marco Golla | What sort of assumptions would you make on your subset of geodesics? Do you have any specific example (say, all shortest length geodesics)? | |
| Jan 3, 2015 at 22:12 | history | asked | Joseph Zambrano | CC BY-SA 3.0 |