Timeline for Persistent homology of Gaussian fields in Euclidean space
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| May 15, 2020 at 13:52 | history | edited | YCor | CC BY-SA 4.0 |
minor formatting, added tag, replaced tag
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| Oct 12, 2017 at 19:23 | history | edited | j.c. | CC BY-SA 3.0 |
replace dead link (I'm guessing by the previous larry.pdf filename that this article in the festschrift for Lawrence D. Brown is the correct one)
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| Aug 23, 2015 at 6:04 | vote | accept | Ryan Budney | ||
| Aug 23, 2015 at 6:04 | answer | added | Ryan Budney | timeline score: 10 | |
| S Aug 24, 2013 at 9:29 | history | suggested | Kelly Davis |
Added tag persistent-homology
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| Aug 24, 2013 at 9:10 | review | Suggested edits | |||
| S Aug 24, 2013 at 9:29 | |||||
| Dec 8, 2012 at 8:19 | comment | added | Ryan Budney | @Vel: it's nothing sophisticated, or rigorous. The idea is to assume the points look like they're on a cubical lattice. You scale the lattice appropriately to achieve the required density of points. The nice thing is thati its easy to describe the persistent homology of $\mathbb Z^n \subset \mathbb R^n$ as a module over the group of translations $\mathbb Z^n$, so this computation allows you to guestimate the `density' homology classes in the collection of points given by a Gaussian distribution. I doubt this result is correct but I'm just trying to get a ballpark estimate of what to expect | |
| Dec 8, 2012 at 8:07 | history | edited | Ryan Budney | CC BY-SA 3.0 |
H_0 does not go to zero as N goes to infinity in this formula
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| Dec 8, 2012 at 7:30 | comment | added | Vidit Nanda | Ryan, two further questions about your recent edit: 1. how did you triangularize the point cloud, and 2. did you try another probability distribution function whose sub-level sets are homologically similar to the Gaussian one, eg a Poisson distribution? That is, can you tell the Gaussian and Poisson barcodes apart even in 1D? | |
| Dec 8, 2012 at 0:43 | history | edited | Ryan Budney | CC BY-SA 3.0 |
added a rough guestimate of barcode shapes; added 10 characters in body
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| Dec 7, 2012 at 1:29 | comment | added | Ryan Budney | If you think of a Gaussian distribution of points in $\mathbb R^n$ as something of a probabilistic version of an $n$-dimensional $0$-handle, you could think of this question as something like "persistent homology is the first instance of a probabilistic homology theory, so what are its coefficients?" | |
| Dec 7, 2012 at 1:27 | comment | added | Ryan Budney | Vel, I'm not asking for a precise prediction of homology classes, but just a general prediction of the shape of the barcodes. Interpret that broadly -- it can be a request for the expected proportion of various Betti numbers, for example. Also, it doesn't have to be an asymptotic prediction -- the prediction could depend on the sample size and it's fine to say "with a sampling of N points one would expect barcodes in this range, X times out of Y", etc. | |
| Dec 7, 2012 at 0:26 | comment | added | Vidit Nanda | I'm surprised no one has asked this yet, but: how many sample points? If you have only one sample point, the barcode is not goint to be terribly hard to describe. Perhaps by asymptotic behavior you mean "let the sample size go to infinity" at which point generically nothing survives for too long. In short, I don't see a sample size invariant answer to your question that is also interesting. What do you have in mind? | |
| Dec 5, 2012 at 16:27 | answer | added | JSE | timeline score: 5 | |
| Dec 5, 2012 at 11:50 | answer | added | Mikael Vejdemo-Johansson | timeline score: 8 | |
| Dec 4, 2012 at 23:06 | comment | added | Ryan Budney | Presumably the cycles in the homology will predominantly sit in thin spherical layers, fairly far out on the bell-curve, so to speak. The $H_0$ classes will be on the most distant spherical shell, the $H_1$ classes next closest, then the $H_n$ classes will be in the central most spherical shell before all the homology vanishes in the core of the distribution. The main issue is the relative thickness of the shells I would imagine. | |
| Dec 4, 2012 at 22:39 | comment | added | Steve Huntsman | I suspect Adler's work is the state of the art on marrying random fields or their ilk with persistent homology...You should bear in mind that a high-dimensional Gaussian contains almost all its mass in a thin spherical shell far from the origin. The concomitant sparsity of the samples away from the shell will manifest in the Vietoris-Rips complex. Does your wing look like what you'd get with points on a big sphere? | |
| Dec 4, 2012 at 20:40 | history | asked | Ryan Budney | CC BY-SA 3.0 |