The Weizmann Institute of Science
Faculty of Mathematics and Computer Science
Walmart Lecture Series in Cryptography and Complexity
Lecture Hall, Room 1, Ziskind Building
on Sunday, May 15, 2011
at 11:00
Daniel Spielman
Yale University
will speak on
Spectral Sparsification of Graphs and Approximations of Matrices
Abstract:
Expander graphs can be viewed as sparse approximations of complete graphs, with
Ramanujan expanders providing the best possible approximation.
We formalize this notion of approximation and ask how well a given graph can be
approximated by a sparse graph. 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.