Communications of the ACM
Probabilistic communication complexity
Journal of Computer and System Sciences
The probabilistic communication complexity of set intersection
SIAM Journal on Discrete Mathematics
On the distributional complexity of disjointness
Theoretical Computer Science
Surveys in combinatorics, 1993
Efficient noise-tolerant learning from statistical queries
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
On the Size of Weights for Threshold Gates
SIAM Journal on Discrete Mathematics
Weakly learning DNF and characterizing statistical query learning using Fourier analysis
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
An introduction to computational learning theory
An introduction to computational learning theory
The nature of statistical learning theory
The nature of statistical learning theory
Communication complexity
On randomized one-round communication complexity
Computational Complexity
The BNS-chung criterion for multi-party communication complexity
Computational Complexity
A Tutorial on Support Vector Machines for Pattern Recognition
Data Mining and Knowledge Discovery
A linear lower bound on the unbounded error probabilistic communication complexity
Journal of Computer and System Sciences - Complexity 2001
Noise-tolerant learning, the parity problem, and the statistical query model
Journal of the ACM (JACM)
A Random Sampling based Algorithm for Learning the Intersection of Half-spaces
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Limitations of learning via embeddings in euclidean half spaces
The Journal of Machine Learning Research
Learning intersections and thresholds of halfspaces
Journal of Computer and System Sciences - Special issue on FOCS 2002
New lower bounds for statistical query learning
Journal of Computer and System Sciences - Special issue on COLT 2002
Theoretical Computer Science - Algorithmic learning theory(ALT 2002)
Cryptographic Hardness for Learning Intersections of Halfspaces
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
On Computation and Communication with Small Bias
CCC '07 Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity
Unconditional lower bounds for learning intersections of halfspaces
Machine Learning
Communication Complexity under Product and Nonproduct Distributions
CCC '08 Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity
Geometrical realization of set systems and probabilistic communication complexity
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
Complexity classes in communication complexity theory
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Complexity measures of sign matrices
Combinatorica
Learning complexity vs communication complexity
Combinatorics, Probability and Computing
Spectral norm in learning theory: some selected topics
ALT'06 Proceedings of the 17th international conference on Algorithmic Learning Theory
Hadamard tensors and lower bounds on multiparty communication complexity
ICALP'05 Proceedings of the 32nd international conference on Automata, Languages and Programming
COLT'05 Proceedings of the 18th annual conference on Learning Theory
SIAM Journal on Computing
Unbounded-error quantum query complexity
Theoretical Computer Science
SIAM Journal on Computing
The approximate rank of a matrix and its algorithmic applications: approximate rank
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Improved Approximation of Linear Threshold Functions
Computational Complexity
Hi-index | 0.00 |
We introduce the notion of a halfspace matrix, which is a sign matrix A with rows indexed by linear threshold functions f, columns indexed by inputs x 驴 {驴 1, 1} n , and the entries given by A f,x = f(x). We use halfspace matrices to solve the following problems.In communication complexity, we exhibit a Boolean function f with discrepancy 驴(1/n 4) under every product distribution but $$O(\sqrt{n}/2^{n/4})$$ under a certain non-product distribution. This partially solves an open problem of Kushilevitz and Nisan (1997).In learning theory, we give a short and simple proof that the statisticalquery (SQ) dimension of halfspaces in n dimensions is less than 2 $$(n + 1)^2$$ under all distributions. This improves on the $$n^{O(1)}$$ estimate from the fundamental paper of Blum et al. (1998). We show that estimating the SQ dimension of natural classes of Boolean functions can resolve major open problems in complexity theory, such as separating PSPACE $$^{cc}$$ and PH $$^{cc}$$ .Finally, we construct a matrix $$A \in \{-1, 1\}^{N \times N^{\text{log} N}}$$ with dimension complexity logN but margin complexity $$\Omega(N^{1/4}/\sqrt{\log N})$$ . This gap is an exponential improvement over previous work. We prove several other relationships among the complexity measures of sign matrices, omplementing work by Linial et al. (2006, 2007).