We introduce a notion of what it means for one graph to be a good spectral approximation of another. This induces the problem of spectral sparsification: finding a sparse graph that is a good spectral approximation of a given graph.
It turns out that Ramanujan expanders are the best sparse spectral approximations of complete graphs. We show that every graph may be approximated almost as well by a sparse graph as the complete graphs are by Ramanujan graphs. We also present an efficient randomized algorithm for construction sparse approximations which only uses a logarithmic factor more edges than optimal.
Our algorithms follow from the solution of a problem in linear algebra. Given a rank-n symmetric matrix A that is expressed as a sum of rank-1 symmetric matrices, we show that A can be well approximated by a weighted sum of only O(n) of those rank-1 matrices.
This is joint work with Joshua Batson, Nikhil Srivastava and Shang-Hua Teng.