Persistent homology of Gaussian Fields in Euclidean space - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T16:22:08Z http://mathoverflow.net/feeds/question/115442 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/115442/persistent-homology-of-gaussian-fields-in-euclidean-space Persistent homology of Gaussian Fields in Euclidean space Ryan Budney 2012-12-04T20:40:12Z 2012-12-08T08:07:16Z <p>If you generate points in $\mathbb R^n$ via a process that respects a Gaussian normal distribution, then compute the persistent homology / barcodes, to my eye something fairly regular seems to be happening, with the barcodes tending towards something like a "wing" shape, fat in lower dimensions, thinning out towards dimension $n$. </p> <p>Has anyone proven any theorems that describe the asymptotic "shape" of the barcodes? </p> <p>Ideally I'd like a test so that I can look at some barcodes and say "that's typical of a Gaussian normal distribution". </p> <p>The closest thing I've been able to find is experiments and results on the expected Euler characteristic of the persistent homology, in the references below. </p> <p><a href="http://webee.technion.ac.il/people/adler/larry.pdf" rel="nofollow">http://webee.technion.ac.il/people/adler/larry.pdf</a></p> <p><a href="http://arxiv.org/abs/1003.5175" rel="nofollow">http://arxiv.org/abs/1003.5175</a></p> <p>edit: </p> <p>I did a very rough computation to try and get some kind of guess as to what the distribution of barcodes should look like. So I made a very coarse estimate based on a distribution of points that is roughly locally cubical' and approximately respecting a normal distribution.</p> <p>The density is given by:</p> <p>$$\mu = N e^{-r^2}$$</p> <p>where $r$ is the distance from the origin. Then if $\epsilon$ is the parameter for persistent homology, it appears that $H_0$ is rank approximately</p> <p>$$N \int_{\sqrt{\ln(N\epsilon^{1/n})}}^\infty r^{n-1}e^{-r^2} dr$$</p> <p>and $H_k$ for $k \in \{1,2,\cdots,n-1\}$ has rank approximately</p> <p>$${n \choose k+1}\frac{(\sqrt{\ln(N\epsilon^{1/n}/\sqrt{k}))}^{n-2}}{4\sqrt{k}\epsilon^{1/n}}$$</p> <p>These are fairly coarse estimates, and in no way rigorous. But if something like this is actually true it seems to be saying that for $N$ large and $n \geq 3$, the $H_0$ betti number tends to some asymptote (dependent on $\epsilon$), and $H_1, \cdots, H_{n-1}$ are non-trivial but small. So most of the points in the distribution are in a giant homology black hole' at the centre and persitent homology sees the thin crust around the outside. </p> <p>I'd be curious if people have done other similar guestimates (or better) and if they had similar-looking results. </p> http://mathoverflow.net/questions/115442/persistent-homology-of-gaussian-fields-in-euclidean-space/115493#115493 Answer by Mikael Vejdemo-Johansson for Persistent homology of Gaussian Fields in Euclidean space Mikael Vejdemo-Johansson 2012-12-05T11:50:37Z 2012-12-05T11:50:37Z <p>The closest I can find spontaneously would be Matthew Kahle's work on random topology; <a href="http://arxiv.org/abs/0910.1649" rel="nofollow">http://arxiv.org/abs/0910.1649</a> looks like it would be directly related to your question, and <a href="http://arxiv.org/abs/1009.4130" rel="nofollow">http://arxiv.org/abs/1009.4130</a> seems related too.</p> http://mathoverflow.net/questions/115442/persistent-homology-of-gaussian-fields-in-euclidean-space/115512#115512 Answer by JSE for Persistent homology of Gaussian Fields in Euclidean space JSE 2012-12-05T16:27:48Z 2012-12-05T16:27:48Z <p>For what it's worth, Laura Balzano and I ran some experiments on this precise question for Gaussian clouds in R^2, trying to understand what kind of barcodes are rare under this "null hypothesis" of data without topological structure. We focused on the question "how long a bar in R^1 constitutes a surprisingly long bar?" and ran some tests on this. No theorems, though. I agree with your implicit assertion that this is an important question for the theory!</p> <p><a href="http://www.math.wisc.edu/~ellenber/topoData_icassp-3.pdf" rel="nofollow">http://www.math.wisc.edu/~ellenber/topoData_icassp-3.pdf</a></p>