Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
On the Fourier spectrum of monotone functions
Journal of the ACM (JACM)
Recursive reconstruction on periodic trees
Random Structures & Algorithms
Uniform-distribution attribute noise learnability
COLT '99 Proceedings of the twelfth annual conference on Computational learning theory
Hardness amplification within NP
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Using nondeterminism to amplify hardness
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
On uniform amplification of hardness in NP
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Coin flipping from a cosmic source: On error correction of truly random bits
Random Structures & Algorithms
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
Bounding the average sensitivity and noise sensitivity of polynomial threshold functions
Proceedings of the forty-second ACM symposium on Theory of computing
Discrete Applied Mathematics
A stronger LP bound for formula size lower bounds via clique constraints
Theoretical Computer Science
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It is known that for all monotone functions f : {0, 1}n → {0, 1}, if x ∈ {0, 1}n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ε = n-α then P[f(x) ≠ f(y)] -α+1/2, for some c P[fn(x) ≠ fn(y)] ≥ δ, where 0 c(δ)n-α, where α = 1 - ln 2/ln 3 = 0.36907..., and c(δ) 0. We improve this result by achieving for every 0 P[fn(x) ≠ fn(y)] ≥ δ, with: • ε = c(δ)n-α for any α k = k(α); • ε = c(δ)n-1/2logtn for t = log2√π/2 = .3257 ..., using an explicit recursive majority function with increasing arities; and • ε = c(δ)n-1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.We also study the problem of achieving the best dependence on δ in the case that the noise rate ε is at least a small constant; the results we obtain are tight to within logarithmic factors.