A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
Sampling and integration of near log-concave functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
The Hadamard operator norm of a circulant and applications
SIAM Journal on Matrix Analysis and Applications
Geometric arguments yield better bounds for threshold circuits and distributed computing
Theoretical Computer Science
Scale-sensitive dimensions, uniform convergence, and learnability
Journal of the ACM (JACM)
A linear lower bound on the unbounded error probabilistic communication complexity
Journal of Computer and System Sciences - Complexity 2001
Playing large games using simple strategies
Proceedings of the 4th ACM conference on Electronic commerce
Communication Complexity Lower Bounds by Polynomials
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Limitations of learning via embeddings in euclidean half spaces
The Journal of Machine Learning Research
Quantum Search of Spatial Regions
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Random sampling and approximation of MAX-CSPs
Journal of Computer and System Sciences - STOC 2002
Tensor decomposition and approximation schemes for constraint satisfaction problems
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Additive Approximation for Edge-Deletion Problems
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Simulated annealing in convex bodies and an O*(n4) volume algorithm
Journal of Computer and System Sciences - Special issue on FOCS 2003
On Computation and Communication with Small Bias
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Games of fixed rank: a hierarchy of bimatrix games
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Nash equilibria in random games
Random Structures & Algorithms
Geometrical realization of set systems and probabilistic communication complexity
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Complexity measures of sign matrices
Combinatorica
Computational Complexity
Perturbed identity matrices have high rank: Proof and applications
Combinatorics, Probability and Computing
Lower bounds in communication complexity based on factorization norms
Random Structures & Algorithms
An Approximation Algorithm for Approximation Rank
CCC '09 Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity
Detecting high log-densities: an O(n¼) approximation for densest k-subgraph
Proceedings of the forty-second ACM symposium on Theory of computing
SIAM Journal on Computing
Proceedings of the forty-third annual ACM symposium on Theory of computing
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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.