Matrix analysis
Probabilistic communication complexity
Journal of Computer and System Sciences
On the degree of polynomials that approximate symmetric Boolean functions (preliminary version)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
The probabilistic communication complexity of set intersection
SIAM Journal on Discrete Mathematics
On the distributional complexity of disjointness
Theoretical Computer Science
Threshold circuits of bounded depth
Journal of Computer and System Sciences
Toward Efficient Agnostic Learning
Machine Learning - Special issue on computational learning theory, COLT'92
Boosting a weak learning algorithm by majority
Information and Computation
On the degree of Boolean functions as real polynomials
Computational Complexity - Special issue on circuit complexity
Perceptrons, PP, and the polynomial hierarchy
Computational Complexity - Special issue on circuit complexity
On the computational power of depth-2 circuits with threshold and modulo gates
Theoretical Computer Science
Communication complexity
Quantum vs. classical communication and computation
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Computing Boolean functions by polynomials and threshold circuits
Computational Complexity
Interaction in quantum communication and the complexity of set disjointness
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Complexity measures and decision tree complexity: a survey
Theoretical Computer Science - Complexity and logic
Improved Quantum Communication Complexity Bounds for Disjointness and Equality
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
New degree bounds for polynomial threshold functions
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Bounds for Small-Error and Zero-Error Quantum Algorithms
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
Las Vegas is better than determinism in VLSI and distributed computing (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Some complexity questions related to distributive computing(Preliminary Report)
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Communication Complexity Lower Bounds by Polynomials
CCC '01 Proceedings of the 16th Annual Conference on Computational Complexity
Lower Bounds for Quantum Communication Complexity
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Informational Complexity and the Direct Sum Problem for Simultaneous Message Complexity
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
The Quantum Communication Complexity of Sampling
SIAM Journal on Computing
Journal of Computer and System Sciences - STOC 2001
Quantum lower bounds for the collision and the element distinctness problems
Journal of the ACM (JACM)
Learning intersections and thresholds of halfspaces
Journal of Computer and System Sciences - Special issue on FOCS 2002
Perceptrons: An Introduction to Computational Geometry
Perceptrons: An Introduction to Computational Geometry
Toward Attribute Efficient Learning of Decision Lists and Parities
The Journal of Machine Learning Research
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
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
Lower Bounds for Quantum Communication Complexity
SIAM Journal on Computing
Computational Complexity
Lattices, mobius functions and communications complexity
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
SFCS '93 Proceedings of the 1993 IEEE 34th Annual Foundations of Computer Science
Improved Separations between Nondeterministic and Randomized Multiparty Communication
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Learning complexity vs communication complexity
Combinatorics, Probability and Computing
Lower bounds in communication complexity based on factorization norms
Random Structures & Algorithms
Separating ${AC}^0$ from Depth-2 Majority Circuits
SIAM Journal on Computing
Multiparty Communication Complexity and Threshold Circuit Size of AC^0
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
A lower bound for agnostically learning disjunctions
COLT'07 Proceedings of the 20th annual conference on Learning theory
A uniform lower bound on weights of perceptrons
CSR'08 Proceedings of the 3rd international conference on Computer science: theory and applications
A note on quantum algorithms and the minimal degree of ε-error polynomials for symmetric functions
Quantum Information & Computation
CSR'07 Proceedings of the Second international conference on Computer Science: theory and applications
Composition theorems in communication complexity
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
Strong direct product theorems for quantum communication and query complexity
Proceedings of the forty-third annual ACM symposium on Theory of computing
On quantum-classical equivalence for composed communication problems
Quantum Information & Computation
From information to exact communication
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Communication lower bounds using directional derivatives
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Dual lower bounds for approximate degree and markov-bernstein inequalities
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
Faster private release of marginals on small databases
Proceedings of the 5th conference on Innovations in theoretical computer science
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We develop a novel technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function $f: \{0,1\}^n \to\{0,1\}$, and let $A_f$ be the matrix whose columns are each an application of $f$ to some subset of the variables $x_1,x_2,\ldots,x_{4n}.$ We prove that $A_f$ has bounded-error communication complexity $\Omega(d),$ where $d$ is the approximate degree of $f.$ This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov's breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of $A_f$ in terms of well-studied analytic properties of $f,$ broadly generalizing several recent results on small-bias communication and agnostic learning. The method of this paper has also enabled important progress in multiparty communication complexity.