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1) Code up an algorithm to compute the 0th and 1st persistent homology PH0 (X)r and PH1(X)r for a data
set (X,d) given in the form of a finite metric space. Run it on an actual data set of around 10 – 15
points in the plane or in 3-space for a set of maybe three “r” values. The “r” value should be an
input to your program.
2) Code up a program to compute the Vietoris-Rips complex for relevant values of “r” as an abstract
simplicial complex. Start with a data set X in 3-space having 8 -10 points viewed as a finite metric
space (X,d). Note: You will have as many as 45 distinct distances, so you could have up to 45
different abstract simplicial complexes. Just output the abstract complex for 3 or 4 select r’s in
the range between the minimum non-zero distance and about half the maximum distance (diameter
of the finite metric space.) The output will be the simplices of various sizes, i.e. subsets of X.
3) Analyze either an actual data set you find online or create an artificial data set in 3 or 4 dimensions
with interesting topological features. (I can suggest how to do this.)
4) Code up an algorithm to create a finitely generated persistence vector space from a filtered set
(X, rho) where X has up to 10 points and compute its barcode representation.
5) Create an abstract simplicial complex with 8-10 vertices and simplices no bigger that 3-simplices.
Compute it’s homology. Write down all the matrices in chain complex and compute the cycles
and boundaries, etc.
6) Find the barcode for a PH1 for a suitable data set using Ripser. See me on how to create the data
set. Bauer, Ulrich, “Ripser: a lean C++ code for computation of Vietoris-Rips barcodes.
https://github.com/Ripser/ripser
Writing Project
Write a 3-4 page paper related to persistent homology in data analysis. Possible topics:
A detailed analysis of a real world application of persistent homology. Explain the data used, the
algorithms employed, what the results were and what they are used for, and various positives or
negatives that you see in the methods and results.
Explain an algorithm (or algorithms) to compute persistent homology as though you were giving a
quick tutorial. Discuss issues with the algorithm(s) you selected – pros and cons – problems, etc.
Investigate what is believed to be one of the best (fastest) algorithm for computing persistent
homology and/or barcodes
Sample Solution
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