Friday, June 22, 2012

CGWeek II

Some highlights from day 2 of CGWeek:

On Laplacians of Random Complexes, by Anna Gundert and Uli Wagner.

The Cheeger inequality is a well known inequality in spectral graph theory that connects the "combinatorial expansion" of a graph with "spectral expansion". Among other things, it's useful for clustering, because you can split a graph using Cheeger's inequality to find a "thin cut" and then repeat. There's been work recently on a "higher-order" Cheeger inequality, that allows you to cut a graph into $k$ pieces instead of two by connecting this to observations about the first $k$ eigenvectors of the Laplacian.

The above paper generalizes the notion of combinatorial expansion versus spectral expansion in a very different way. Think of a graph as the 1-skeleton of a simplicial complex (i.e the set of faces of dimension at most 1). Is there a way to define the Laplacian of the complex itself ? And is there then an equivalent of Cheeger's inequality ?

It turns out that this can indeed be done. Recent work by Gromov and others have shown how to define the notion of "edge expansion" for a simplicial complex. Roughly speaking, you compute the edge expansion of a cochain (a function of a chain) by endowing it with a norm and then looking at the norm of the edge operator with respect to the norm (sort of) of the cochain itself. What is interesting is that if you choose the underlying coefficient field as $\mathbb{R}$ and the norm as $\ell_2$, you get the spectral equivalent, and if you choose instead $\mathbb{Z}_2$ and the Hamming distance, you get the combinatorial equivalent.

It's known that for 1-skeletons, and even for things like persistence, the underlying field used to define homology doesn't really matter. However, for this problem, it matters a lot. The authors show that there is no equivalent Cheeger's inequality for simplicial complexes ! They also look at random complexes and analyze their properties (just like we can do for random graphs).

Suppose you have three Gaussians in the plane, and you look at the resulting normalized distribution. You expect to see three bumps (modes), and if the Gaussians merge together, you'd expect to see the modes come together in a supermode.

Can you ever get more than three modes ?

This is the question the above paper asks. It was conjectured that this cannot happen, and in fact in 2003 it was shown that it was possible to get 4 modes from three Gaussians (you can get a little bump in the middle as the three Gaussians pull apart). In this paper, they show that in fact you can get a super-linear number of "bumps" or critical points for $n$ Gaussians, and these modes are not transient - they "persist" in a certain sense.

This is quite surprising in and of itself. But it's also important. A plausible approach to clustering a mixture of Gaussians might look for density modes and assign cluster centers there. What this paper says is that you can't really do that, because of the "ghost" centers that can appear.

Other quick hits:
• Chris Bishop ran a very interesting workshop on Analysis and Geometry. While I couldn't attend all the talks, the first one was by Ranaan Schul  gave an overview of the analytic TSP, in which you're given a continuous set of  points in the plane, and want to know if there's a finite length curve that passes through all the points. The techniques used here relate to multiscale analysis of data and things like local PCA.
• One of the cool innovations in CGWeek was the fast-forward session: each afternoon before the workshops started, speakers were allowed to give a 1-slide overview of what would happen in their events. The slides were all placed in a single presentation ahead of time, and it was clocked so that people couldn't run on. It was great - I didn't have to do careful planning, and got a much better idea of what was coming up. We should do this for all talks at the conference !