Spectral Sparsification Spectral Sparsification: the Barrier Method and Its Applications

We survey recent literature focused on the following spectral sparsification question: Given an integer n and > 0, does there exist a function N(n, ) such that for every collection of C1, . . . ,Cm of n×n real symmetric positive semidefinite matrices whose sum is the identity, there exists a weighted subset of size N(n, ) whose sum has eigenvalues lying between 1− and 1 + ? We present the algorithms for solving this problem given in [4, 8, 10]. These algorithms obtain N(n, ) = O(n/ ), which is optimal up to constant factors, through use of the barrier method, a proof technique involving potential functions which control the locations of the eigenvalues of a matrix under certain matrix updates. We then survey the applications of this sparsification result and its proof techniques to graph sparsification [4, 10], low-rank matrix approximation [8], and estimating the covariance of certain distributions of random matrices [32, 26]. We end our survey by examining a multivariate generalization of the barrier method used in Marcus, Spielman, and Srivastava’s recent proof [19] of the Kadison-Singer conjecture. Acknowledgements I am indebted to my thesis adviser, Prof. Jelani Nelson, for introducing me to this wonderful research topic and for having countless captivating and patient conversations with me explaining these topics and many, many others. I would also like to give a heartfelt dedication to my mother and father for their lifelong support and encouragement in all aspects of my studies, as well as to the many friends who supported me during the process of writing this thesis.