What is the difference between Vietoris-Rips and Cech complexes?

[@robertmiotk3619]

Hi, I'm currently working on my thesis about TDA, and I meet the same inaccuracy over and over again. No offense, just wanted to let you know, that there is a mistake in your video. In 6:09 , you said that length of edges of empty triangle are at most r, but assuming r is a radius of shown balls and equals 0.312, you're wrong :( These two balls barely meet, so the edge has length 2r, so distance between two points (centers of balls) is not smaller than r. On the other hand, if r/2 = 0.312, then visualization for Vietori-Rips should be different. Balls still should have length 0.312 for Cech, but 0.624 for Vietoris-Rips. It is affecting on complexes, because Vietoris-Rips in 6:09 shouldn't even consist edges of this triangle. Sorry if I'm wrong, correct me then, but this is something I would love to make clear ;)

[@aatrn1]

Vietoris-Rips complexes have all edges of length at most r. There are two different conventions for Cech complexes --- either the nerve of all balls of radius r, or the nerve of all balls of radius r/2. Both conventions are commonly used, but they differ by a factor of two. This video is using the opposite convention for Cech complexes than the one you are used to.

[@robertmiotk3619]

​@@aatrn1 Excatly: "Vietoris-Rips complexes have all edges of length at most r." When 2 balls with radii r are touched in one point, then distance between these two centers is 2r. Not r ;) Then edge of this triangle is 2r. So it shouldnt be simplex of Vietoris-Rips of parameter r.

[@aatrn1]

In this video, I'm using Cech complexes that are defined to have radius r/2. So, when two balls with radius r/2 touch in one point, then the distance between these two centers is at most r, and hence the edge is a simplex of the Vietoris-Rips complex at parameter r. So, there are two different common definitions of Cech complexes; one with balls of radius r, and the other with balls of radius r/2. This video is using the latter definition, not the former definition.

[@nksupreme1]

This is explanatory and really helpful. Thanks!

[@aatrn1]

Glad to hear it!

[@stsui96]

Thank you so much for this nice explanation. This is really helpful!

[@aatrn1]

Thanks!

[@haneenballan7337]

Thanks a bunch, Dr. Henry for this great video, it is really useful! I have tried opening the Mathematica demo but to no avail, however, the other two demos worked properly. Can you, please, help with this. I really need this demo for my project. All appreciation for your great efforts!

[@jacobcleveland1594]

This is awesome! Very concise and clear difference between the two, I’ve been confused about this for a long time. Couple of questions: are there upper bounds on the scale parameter, maybe in terms of the diameter of the vertex set? Also is there a distinction between using closed balls and open balls in the definition of the Čech complex?

[@aatrn1]

Thanks! Once the scale parameter r reaches the diameter of the dataset, the Vietoris-Rips complex doesn't change anymore. But typically (due to size constraints) you have to stop computing much sooner before you reach the diameter of the dataset! For finite datasets, there's not much difference between using closed balls and open balls in the definition of the Čech complex. Indeed, if r is a scale parameter where the Čech complex with closed balls disagrees with that with open balls, then there's an arbitrarily small epsilon perturbation you can make to r so that the complexes agree again. Probably a better way of saying this is that when changing from closed balls to open balls or visa versa, all that can change in the persistent homology for finite datasets is the endpoints of the persistent homology bars --- i.e. which endpoints are included or not. Same for Vietoris-Rips complexes, when using the

[@lordnaive]

Hi Henry, what should be the perfect choice of filtration technique for point clouds in 3D or higher dim domains?

[@aatrn1]

For 3D or lower (not what you ask about), you often want to use an alpha complex. For higher dimensions, Vietoris-Rips is probably what most folks try first. You can move to witness complexes for smaller approximating complexes. I don’t know of too many software packages implementing Cech; probably GUDHI does?

[@aaaab384]

It took me ages to understand that what you write as a big ALPHA is actually THE NUMBER 2.

[@aatrn1]

Whoops!

[@leboeapolinyane6410]

I would like to cite those demos for my project on tda. How do I do that please? I am on LaTex. Also, how can I do those on Python?

[@aatrn1]

Hi, you could cite the mathematica demonstrations using a format something like the following: @software{vr-cech-mathematica, author = {Adams, Henry and Segert, Jan}, title = {Mathematica demo on \v{C}ech and Vietoris--Rips complexes}, url = {www.math.colostate.edu/~adams/research/}, date = {2020}, } For doing these on python, you may have to code them up yourself, unless perhaps somebody else has already done so! The algorithm we use for Cech complexes is called the "miniball" algorithm.

[@mehdinategh8876]

awesome

[@aatrn1]

Thanks!

[@adwaithvijayakumar9034]

Really awesome explanation. Thanks professor Henry🙂

[@aatrn1]

Great, thanks!