@inproceedings{1cdaad9cac274f74b6c0fa69c8007aeb,

title = "The approximate rank of a matrix and its algorithmic applications",

abstract = "We study the ε-rank of a real matrix A, defined for any ε > 0 as the minimum rank over matrices that approximate every entry of A to within an additive ε. This parameter is connected to other notions of approximate rank and is motivated by problems from various topics including communication complexity, combinatorial optimization, game theory, computational geometry and learning theory. Here we give bounds on the ε-rank and use them for algorithmic applications. Our main algorithmic results are (a) polynomial-time additive approximation schemes for Nash equilibria for 2-player games when the payoff matrices are positive semidefinite or have logarithmic rank and (b) an additive PTAS for the densest subgraph problem for similar classes of weighted graphs. We use combinatorial, geometric and spectral techniques; our main new tool is an algorithm for efficiently covering a convex body with translates of another convex body.",

keywords = "Approximate rank, Convex body, Covering number, Nash equilibria",

author = "Noga Alon and Troy Lee and Adi Shraibman and Santosh Vempala",

year = "2013",

doi = "10.1145/2488608.2488694",

language = "English (US)",

isbn = "9781450320290",

series = "Proceedings of the Annual ACM Symposium on Theory of Computing",

pages = "675--684",

booktitle = "STOC 2013 - Proceedings of the 2013 ACM Symposium on Theory of Computing",

note = "45th Annual ACM Symposium on Theory of Computing, STOC 2013 ; Conference date: 01-06-2013 Through 04-06-2013",

}