
A gentle introduction to persistent homology - cbock90
https://christian.bock.ml/posts/persistent_homology/
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duality
Can anyone working in this field comment on how it's applied? For example,
what is computed? Homology groups, just the ranks of these groups, or
something else? Given these, what does one learn about the data set under
analysis?
[https://en.wikipedia.org/wiki/Topological_data_analysis#Appl...](https://en.wikipedia.org/wiki/Topological_data_analysis#Applications)
is pretty scant on detail.

~~~
kaitai
There are a number of approaches; since it's still relatively new, there's a
lot of "playing around" with techniques.

1) The first idea is to just see which topological features persist as the
filtration/threshold parameter varies. I personally find the barcode
illustration, rather than the birth-death diagram, to be more intuitive in
this respect. For instance, say you had LIDAR data about a vehicle. Looking at
the persistent homology of this point cloud could allow you to ascertain the
size and number of windows (smaller windows would have shorter persistence
than larger windows, and the number of persistent bars would correspond to the
number of windows or "openings" on the vehicle). This might allow you to
figure out if the vehicle is a van or an SUV.

2) For applications like those in neuroscience (I like, for instance, the talk
you can find on Youtube titled "Kathryn Hess (6/27/17) Bedlewo: Topology meets
neuroscience") there is instead a look at how these ranks behave over time. A
rough sketch is this: look at the topology of the pattern of activation in
neurons as a mouse (or an AI) learns something. As the learning process
happens, what happens to the Betti numbers?

3) Sometimes one might want explicit generators. A thought-provoking small
case of this can be found in "Mind the Gap:A Study in Global Development
through Persistent Homology"
[https://arxiv.org/abs/1702.08593](https://arxiv.org/abs/1702.08593) This
paper looks at statistics like GDP and infant mortality of countries around
the world and find explicit oddities in the data. It's a proof of concept, to
me; I'm interested to see where it'll go.

There is a lot more of course but that brings you to three interesting yet
different directions.

