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Introduction

This tutorial accompanies the paper "A fractal dimension for measures via persistent homology" by Henry Adams, Manuchehr Aminian, Elin Farnell, Michael Kirby, Chris Peterson, Joshua Mirth, Rachel Neville, and Clayton Shonkwiler, which is available at https://arxiv.org/abs/1808.01079. The tutorial relies on code from Ripser (https://github.com/Ripser/ripser).

Sampling random points from shapes

Uniformly random points from the square

Points from the Cantor set cross the interval

Persistent homology

We will be using Ripser (https://github.com/Ripser/ripser) in this tutorial. Some slides about how Ripser works are available at http://ulrich-bauer.org/ripser-talk.pdf.

Ripser in your browser - synthetic examples

The easiest way to run Ripser is in a live demo in your browser, for which no instalation (and in particular, no Python) is required. Try it out! The webpage is http://live.ripser.org/.

House example - distance matrix

For example, we can use Ripser to compute the persistent homology of the Vietoris-Rips complex of the following 5 points in the plane.

The file house_points.txt is in the folder PHdimension/persistent-homology/point_clouds. It is the 5x2 matrix corresponding to the above collection of 5 points in the plane; the entries of this file are

1, 0
1, 2
0, 3
-1, 2
-1, 0

At the Ripser live webpage, enter the following input Be sure to select the option point cloud!

Note that the 5 connected components merge into one, with merging events happening at scales (the square root of 2) and 2. There is a single 1-dimensional feature, beginning at scale parameter 2 and ending at scale parameter (the square root of 8).

Torus example

The following example computes the persistent homology barcodes of a 20 x 20 grid of 400 points on the unit torus S¹ x S¹ in R⁴, where a small amount of noise has been added to each point. Only a subset of the intervals are shown below. Note that the long barcodes (roughly between scale parameter 0.7 and 1.5) recover the homology of the torus, with a single connected component, with two 1-dimensional holes, and with a single 2-dimensional hole.

Ripser on your machine

A more advanced (but very useful) step is to now download Ripser to your machine and to run it locally. This allows you to perform larger computations. Ripser is written in C++. You may download the code for Ripser at https://github.com/Ripser/ripser, which also contains installation instructions. Minimal installation instructions are listed below

git clone https://github.com/Ripser/ripser.git
cd ripser
make
./ripser examples/sphere_3_192.lower_distance_matrix

For convenience, copy the Unix executable file ripser into the folder Top-ML/persistent-homology. You can use the flag --format distances to specify you are computing on a distance matrix, or --format point-cloud to specify you are computing on a point cloud. The flag --dim k specifies that homology is computed only up to dimension k, and the flag --threshold t specifies that persistent homology is computed only up to scale parameter t. For example, we can recreate all of the examples above with the following commands.

House example on the point cloud:

./ripser --format point-cloud point_clouds/house_points.txt

Torus example, up to 2-dimensional homology:

./ripser --format point-cloud --dim 2 point_clouds/torus_points.txt

Exercises on persistent homology

Exercise: Write a script that selects 500 points uniformly at random (or approximately uniformly at random) from the annulus {(x,y)|0.9²≤x²+y²≤1.1²} in R². Compute its persistent homology barcodes using ripser.

Exercise: Write a script (say in Python, or some other language) that selects 500 points uniformly at random (or approximately uniformly at random) from the "coconut shell" {(x,y,z)|0.9²≤x²+y²+z²≤1.1²} in R³. Compute its persistent homology barcodes using ripser.

Computing the minimal spanning tree

Computing the sum of persistent homology interval lengths

Computing a log-log plot estimating the persistent homology fractal dimension

Bibliography

M. A. Armstrong. Basic Topology. Springer, New York, Berlin, 1983.

H. Edelsbrunner and J. Harer. Computational Topology: An Introduction. American Mathematical Society, Providence, 2010.

H. Edelsbrunner, D. Letscher, and A. Zomorodian. Topological persistence and simplification. Discrete Comput. Geom., 28:511-533, 2002.

A. Hatcher. Algebraic Topology. Cambridge University Press, Cambridge, 2002.

A. Zomorodian. Advances in Applied and Computational Topology. American Mathematical Society, 2012.

A. Zomorodian and G. Carlsson. Computing persistent homology. Discrete Comput. Geom., 33:249-274, 2005.