Persistent Homology

A sample of the pictures we will look at this week:

cech_1 cech_2 cech_3 cech_4

Next week I’m going to the Young Topologists Meeting 2015, at EPFL in Lausanne, Switzerland. Over 180 young topologists are going and many of them will give short talks on their research. Alongside this, there are two invited speakers who will give mini lecture series:

  • Gunnar Carlsson of Stanford University, lecturing about Methods of applied topology
  • Emily Riehl of Harvard University, lecturing about Infinity category theory from scratch

I’ll try to write something about these courses, and this post will be a wee introduction to a tool introduced by Gunnar Carlsson which considers topology of data clouds: persistent homology. The pictures in this post were drawn by Paul Horrocks during our joint dissertation at undergrad: points to Paul!

The idea of persistent homology is to use a tool of topology – homology – to understand something about the structure or shape of a set of data points. But topology is to do with spaces, for example manifolds or surfaces. Therefore we want to make a space out of our data before we can work out the homology.

We do this by plotting our set of points, and around each point we draw a ball. This ball has a radius and we can vary the size of this radius:

Smaller radius

cech_1

Larger radius

cech_3

Once we have drawn these balls, we join two of the points by a line if their corresponding balls intersect, and colour in triangles formed by three lines if the balls corresponding to the three points of the triangle have a patch where they all intersect. For different radii we get different structures.

Smaller radius

cech_2

here only two of the balls intersected so there is only one line

Larger radius

cech_4

here many more balls intersected, and there is also one three way intersection.

We can also do this in higher dimensions: drawing a tetrahedron between 4 points if there is a four way intersection and so on.

Now we can work out the homology of the spaces we have created. Because we can do this for lots of different radii, we can vary the radius of the balls slowly, and see which features in the homology persist, hence the name! The persistent features are then considered to be the ‘interesting’ ones to think about.

If the radius is too small, we will have no structure and if it is too big then all the points will be joined together (think about this!) so it is only in a range that we have interesting homology. We can draw something called a barcode to tell us about this range, below is an example. Each bar is an interesting feature and the radius grows bigger from left to right. As the radius grows some of the features appear and dissappear.

barcode

For the radii indicated with dotted lines, the data set in question and its structure for balls of that radius are shown below:

complexes

Hopefully I will have more to say about this after my trip!

Advertisements

3 thoughts on “Persistent Homology

  1. Ole Andre Birkedal July 17, 2015 / 11:02 am

    Would love to see a follow up post on this, very good explanations

    Like

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s