Expected number of faces in Gaussian polytopes

Last week I was reading Active Learning via Perfect Selective Classification by El-Yaniv and Wiener, and came across a neat result due to Hug and Reitzner that they use in some of their bounds for active learning on Gaussian distributions.

The setup is the following : let X_1, X_2, \ldots, X_n be n jointly Gaussian vectors with distribution \mathcal{N}(0,I_d) in \mathbb{R}^d. The convex hull P_n of these points is called a Gaussian polytope. This is a random polytope of course, and we can ask various things about their shape : what is the distribution of the number of vertices, or the number of k-faces? Let f_k(P_n) be the number of k-faces Distributions are hard, but for general k the expected number of faces (as n \to infty) is given by

\mathbb{E}[ f_k(P_n)] = \frac{2^d}{\sqrt{d}} \binom{d}{k+1} \beta_{k,d-1}(\pi \ln n)^{(d-1)/2} (1 + o(1)),

where \beta_{k,d-1} is the internal angle of a regular (d-1)-simplex at one of its k-dimensional faces. What Hug and Reitzner show is a bound on the variance (which then El-Yaniv and Plan use in a Chebyshev bound) : there exists a constant c_d such that

\mathrm{Var}( F_k(P_n) ) \le c_d (\ln n)^{(d-1)/2}

So the variance of the number of k-faces can be upper bounded by something that does not depend at all on the actual value of k. In fact, they show that

f_k(P_n) (\ln n)^{-(d-1)/2} \to \frac{2^d}{\sqrt{d}} \binom{d}{k+1} \beta_{k,d-1} \pi^{(d-1)/2}

in probability as n \to \infty. That is, appropriately normalized, the number of faces converges to a constant.

To me this result was initially surprising, but after some more thought it makes a bit more sense. If you give me a cloud of Gaussian points, then I need k+1 points to define a k-face. The formula for the mean says that if I chose a random set of k+1 points, then approximately \frac{2^d}{\sqrt{d}} \beta_{k,d-1}(\pi \ln n)^{(d-1)/2} fraction of them will form a real k-face of the polytope. This also explains why the simplex-related quantity appears — points that are on “opposite sides” of the sphere (the level sets of the density) are not going to form a face together. As n \to \infty this fraction will change, but effectively the rate of growth in the number of faces with n is (\ln n)^{(d-1)/2}, regardless of k.

I’m not sure where this result will be useful for me (yet!) but it seemed like something that the technically-minded readers of the blog would find interesting as well.