Topological Data Analysis for Machine Learning I: Algebraic Topology
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- čas přidán 13. 09. 2020
- In which we discuss an introduction to computational topology, the utility of Betti numbers, simplicial homology (with examples) and simplicial complexes, as well as how to compute all of this manually.
(this video is part of a lecture at ECML PKDD 2020, the European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases) - Věda a technologie
Thank you! I'm currently taking a seminar on GCNs and am fascinated by topology.
The explanation is very nice. Thank you. When I go to college, I am going to take electives in topology. :)
one of the greatest explanation that i had ever seen..
At 22:50 it should say the set of R^{nxn} INVERTIBLE matrices with matrix multiplication is a group, namely GL(n,R).
The statement before it about quadratic matrices can also be misunderstood, since formally "the set of quadratic matrices" consists of all quadratic matrices, meaning M in R^{nxn} for any n, i.e. R^{1x1} union R^{2x2} union R^{3x3} union ...
While what is meant here is supposedly (R^{nxn}, +) for some fixed n.
Slide 4, from about 7:00 is perhaps one of the most clear explanations of what Betti numbers are. I also loved the examples with the cube-sphere-torus that added value to the definition as a nice example of why Betti numbers are a topological invariance and in essence can show the qualitative difference of two topological spaces. Thank you very much !
Thanks for your kind words 🙏
This was such a good lecture. Well-explained and digestible enough to enjoy. 🙌🏼
Thanks a lot!
Nice. Thank you for this!
Thank you for posting this lecture! I learned the basics of Betti Numbers. Thank you too for discussing the needed algebraic concepts.
Cool! Thanks!!
This is a really cool field of study. Does anyone have a list of open problems? I'm interested in trying to do some research in this area.
Computational Topology: An Introduction by Herbert Edelsbrunner contains such a list
@@hansenmarcwhat a nice book. Thanks for the recommendation!
Hi, Bastian, Thanks a lot for giving such invaluable tutorial. It'll be much helpful for TDA novice if some related practicals are also accompanied. Could you suggest some repositories on practical codes which we could look for? Many thanks~
Sure thing! One of the best resources currently available is `giotto-tda`: giotto-ai.github.io/gtda-docs/latest/#
They have an absolutely comprehensive documentation and a very cool '30 seconds guide'.
Another good one is GUDHI: gudhi.inria.fr/
I have only read about it before. Let me know if you have any other questions!
@@Pseudomanifold Thanks a lot for your recommendation :)
@@ernstroell Thanks a lot for your recommendation :)
Amazing lecture! Do you have any other resources for understanding mathematical topology?
Thanks! I'd start with Computational Topology by Edelsbrunner, and Algebraic Topology by Hatcher.
Dear Bastian, what do you understand for "quadratic matrices"? square matrices? I am confused because square matrices doest' form a group for matrix multiplication, not all of them are invertible (they mus have non zero determinant)
Yes, it should be square matrices with non-zero determinant.
Greetings. What I learned is, for both diagrams 15:53 to 17:30, that both of them are simplicial complexes but neither is properly joined. Or neither of them is a simplicial complex, depends on your definition. But the point in the middle of the line facing nearly to the right in the first diagram makes it also improper in any case.
No, the one on the left does not contain a point within a single line. It's a vertex with two edges outgoing. Hope that helps :-)
@@Pseudomanifold Ah, yes, right. My bad! For the box/sphere at the beginning, I don't get however why the sphere is "higher" - basically, given matching dimensions, there even is a homeomorphism between both and thus neither is "higher". They are topologically equivalent.
@@andreasbeschorner1215 The box and sphere are supposed to be equivalent...hope I did not state anything else?
@@Pseudomanifold Only that the sphere is "higher" :-) Otherwise very good and condensed overview. A little issue with the definition of the quotient group is that the "group" parts is more like a proposition using the fact that N is a normal subgroup. Otherwise it is not a group necessarily. However, looking very much forward to seeing the other parts of the lecture!
@@andreasbeschorner1215 Sure; admittedly, this is hard to get right in such a lecture setting.
This was an amazing lecture. Could you recommend some books for someone who has decent knowledge in machine learning but has never explored topological aspect of it? Or just some book for starting out in topology? Thanks in advance
Thanks, happy you liked it. For starting out, I would recommend Edelsbrunner's book "Computational Topology".
@@Pseudomanifold which one? He has two
@@mohammadrizvi3326 Not to my knowledge, but there's different variants and PDFs around.
What is the intuition of the set of elements mapped to 0 in case of the kernel
This is the definition of a kernel in linear algebra. Here, these are all elements that don't have a boundary.
Hey Bastian, great lecture! I'm considering writing a Medium article inspired by these lecture notes, would that be okay with you? I would give full credit to you. Thanks again for the consideration and informative introduction to TDA!
That would be lovely! You can also ping me via email or something in case you want to discuss more
@@Pseudomanifold Thanks Bastian, Will do!
link us to that article though!
For quotient group how is it that Z/2Z consists if only 0 and 1
I means removing all 2Z elements from Z leads to all odd numbers.
Technically, you 'condense' the group based on the subgroup of all even integers. In the quotient group, you then look at the cosets that you get, and these are either odd or even numbers, hence, there's only two cosets, leading to the representation I described in the tutorial.
Got it. Thanks for the quick reply.
Is beta_3 for the cube, sphere and torus equal to 1?
No, it's zero, since the objects don't have higher-order holes.
@@Pseudomanifold Oh OK, thanks !
22:18 Did you know that the matrix [1 1; 1 1] have no inverse?
Yes, the statement only holds for invertible matrices.
An algebraic topologist is someone who can tell his a$$ from two holes in the ground.
Guess I am not an algebraic topologist...
the definition in 14:17 does not fit with Wiki and also your later explaining...
What are you missing? There are different, equivalent definitions
@@Pseudomanifold is it? but you use the "other" definition in the non-example to try to show that's a non-example, and that's where I got confused...
@@tongzhu6714 Sorry about that, I'll clarify that in a revision.