Abstract: We introduce a notion of what it means for one graph to be a good spectral approximation of another, and prove that every graph can be well-approximated by a graph with few edges.
We ask how well a given graph can be approximated by a sparse graph. Expander graphs can be viewed as sparse approximations of complete graphs, with Ramanujan expanders providing the best possible approximations. We prove that every graph can be approximated by a sparse graph almost as well as the complete graphs are approximated by the Ramanujan expanders: our approximations employ at most twice as many edges to achieve the same approximation factor.
We also present an efficient randomized algorithm for constructing sparse approximations that only uses a logarithmic factor more edges than optimal.
Our algorithms follow from the solution of a problem in linear algebra. Given any expression of a rank-n symmetric matrix A 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.
Biosketch: Daniel Alan Spielman received his B.A. in Mathematics and Computer Science from Yale in 1992, and his Ph.D in Applied Mathematics from M.I.T. in 1995. He spent a year as a NSF Mathematical Sciences Postdoc in the Computer Science Department at U.C. Berkeley, and then taught in the Applied Mathematics Department at M.I.T. until 2005. Since 2006, he has been a Professor at Yale University. He is presently the Henry Ford II Professor of Computer Science, Mathematics, and Applied Mathematics.
He has received many awards, including the 1995 ACM Doctoral Dissertation Award, the 2002 IEEE Information Theory Paper Award, the 2008 Godel Prize, the 2009 Fulkerson Prize, the 2010 Nevanlinna Prize, an inaugural Simons Investigator Award, and a MacArthur Fellowship. He is a Fellow of the Association for Computing Machinery and a member of the Connecticut Academy of Science and Engineering. His main research interests include the design and analysis of algorithms, graph theory, machine learning, error-correcting codes and combinatorial scientific computing.