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Bounded Independence Fools Halfspaces
 In Proc. 50th Annual Symposium on Foundations of Computer Science (FOCS), 2009
"... We show that any distribution on {−1, +1} n that is kwise independent fools any halfspace (a.k.a. linear threshold function) h: {−1, +1} n → {−1, +1}, i.e., any function of the form h(x) = sign ( ∑n i=1 wixi − θ) where the w1,..., wn, θ are arbitrary real numbers, with error ɛ for k = O(ɛ−2 log 2 ..."
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Cited by 46 (18 self)
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We show that any distribution on {−1, +1} n that is kwise independent fools any halfspace (a.k.a. linear threshold function) h: {−1, +1} n → {−1, +1}, i.e., any function of the form h(x) = sign ( ∑n i=1 wixi − θ) where the w1,..., wn, θ are arbitrary real numbers, with error ɛ for k = O(ɛ−2 log 2
Fooling Functions of Halfspaces under Product Distributions
, 2010
"... ... under a very broad class of product distributions. This class includes not only familiar cases such as the uniform distribution on the discrete cube, the uniform distribution on the solid cube, and the multivariate Gaussian distribution, but also includes any product of discrete distributions wi ..."
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Cited by 16 (3 self)
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(n)foolallmonotonefunctionsof(logn)/loglognhalfspaces,ourgeneratorrequires a seed of length just O(logn). Our second main result generalizes the work of Diakonikolas et al. [DGJ + 09] to show that bounded independence suffices to fool functions of halfspaces under product distributions. Assuming each coordinatesatisfiesacertainstrongermoment condition, we showthat
Bounded Independence Fools Degree2 Threshold Functions
"... Let x be a random vector coming from any kwise independent distribution over {−1, 1} n. For an nvariate degree2 polynomial p, we prove that E[sgn(p(x))] is determined up to an additive ε for k = poly(1/ε). This answers an open question of Diakonikolas et al. (FOCS 2009). Using standard constructi ..."
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Cited by 31 (9 self)
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functions is εfooled by poly(1/ε)wise independence. Our results also hold if the entries of x are kwise independent standard normals, implying for example that bounded independence derandomizes the GoemansWilliamson hyperplane rounding scheme. To achieve our results, we introduce a technique we dub
Fishing for Fools
, 2008
"... How big is the effect of a few fools on market outcomes? We argue that in auctions, even a small share of overbidding behavioral agents have a large effect, because the auction format “fishes” for the highestbidding behavioral buyers. Through this fishing mechanism, behavioral agents disproportiona ..."
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Cited by 3 (0 self)
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How big is the effect of a few fools on market outcomes? We argue that in auctions, even a small share of overbidding behavioral agents have a large effect, because the auction format “fishes” for the highestbidding behavioral buyers. Through this fishing mechanism, behavioral agents
The Upper Bound Conjecture for Arrangements of Halfspaces

, 1994
"... For an arbitrary arrangement of n open hemispheres of S d , it is conjectured that the number of vertices contained in at most k of the hemispheres attains its maximum for each k ! (n \Gamma d)=2 in case the hemispheres determine the dual of a spherical cyclic polytope. This would imply a sharp up ..."
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Cited by 6 (1 self)
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upper bound on the analogous numbers for arrangements of halfspaces in E d . The latter is proved here for d 4.
CROOKED HALFSPACES
, 2014
"... Dedicated to the memory of Robert Miner Abstract. We develop the Lorentzian geometry of a crooked halfspace in 2 + 1dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving an e ..."
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Dedicated to the memory of Robert Miner Abstract. We develop the Lorentzian geometry of a crooked halfspace in 2 + 1dimensional Minkowski space. We calculate the affine, conformal and isometric automorphism groups of a crooked halfspace, and discuss its stratification into orbit types, giving
Unconditional lower bounds for learning intersections of halfspaces
 Machine Learning
, 2007
"... We prove new lower bounds for learning intersections of halfspaces, one of the most important concept classes in computational learning theory. Our main result is that any statisticalquery algorithm for learning the intersection of √ n halfspaces in n dimensions must make 2 Ω( √ n) queries. This is ..."
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Cited by 20 (11 self)
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We prove new lower bounds for learning intersections of halfspaces, one of the most important concept classes in computational learning theory. Our main result is that any statisticalquery algorithm for learning the intersection of √ n halfspaces in n dimensions must make 2 Ω( √ n) queries
Testing ±1Weight Halfspaces
"... Abstract. We consider the problem of testing whether a Boolean function f: {−1, 1} n → {−1, 1} is a ±1weight halfspace, i.e. a function of the form f(x) =sgn(w1x1+ w2x2 + ···+ wnxn) where the weights wi take values in {−1, 1}. We show that the complexity of this problem is markedly different from t ..."
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Cited by 4 (1 self)
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the problem of testing whether f is a general halfspace with arbitrary weights. While the latter can be done with a number of queries that is independent of n [7], to distinguish whether f is a ±1weight halfspace versus ɛfar from all such halfspaces we prove that nonadaptive algorithms must make Ω(log n
Tight lower bounds for halfspace range searching
 In Symposium on Computational Geometry (2010
"... We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in R d, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting m denote the space ..."
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Cited by 3 (1 self)
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We establish two new lower bounds for the halfspace range searching problem: Given a set of n points in R d, where each point is associated with a weight from a commutative semigroup, compute the semigroup sum of the weights of the points lying within any query halfspace. Letting m denote the space
Results 1  10
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