# Spectral Sparsification of Hypergraphs

In this post we will construct a “spectral sparsifier” of a given hypergraph in a way that is similar to how Spielman and Srivastava construct spectral graph sparsifiers. We will assign a probability ${p_e}$ to each hyperedge, we will sample each hyperedge ${e}$ with probability ${p_e}$, and we will weigh it by ${1/p_e}$ if selected. We will then bound the “spectral error” of this construction in terms of the supremum of a Gaussian process using Talagrand’s comparison inequality and finally bound the supremum of the Gaussian process (which will involve matrices) using matrix Chernoff bounds. This is joint work with Nikhil Bansal and Ola Svensson.

# Spielman-Srivastava Sparsification à la Talagrand

This is the second in a series of posts explaining a result on hypergraph sparsification that uses Talagrand’s work on Gaussian processes.

In the previous post we talked about Gaussian and sub-Gaussian processes and generic chaining.

In this post we talk about the Spielman-Srivastava probabilistic construction of graph sparsifiers. Their analysis requires a bound on the largest eigenvalue of a certain random matrix, that can be derived from matrix Chernoff bounds.

We will then make our life harder and we will also derive an analysis of the Spielman-Srivastava construction by casting the largest eigenvalue of that random matrix as the sup of a sub-Gaussian process, and then we will apply the machinery from the previous post.

This will be more complicated than it needs to be, but the payoff will be that, as will be shown in the next post, this more complicated proof will also apply, with some changes, to the setting of hypergraphs.