Constant depth circuits, Fourier transform, and learnability

Linial, Nathan; Mansour, Yishay; Nisan, Noam

doi:10.1145/174130.174138

Cited by 438 publications

(289 citation statements)

References 7 publications

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“…Lower bounds for pseudorandom function generators were studied in [14,16,13]. The only papers we are aware of that give impossibility results or lower bounds for "plain" pseudorandom generators as defined above, are the papers by Kharitonov et al [12] and Yu and Yung [19].…”

Section: Definitions and Backgroundmentioning

confidence: 99%

“…In particular, they showed how to construct a generator with a non-trivial stretch function (expanding n bits to n + Θ(log(n)) bits) in the complexity class AC 0 . This suggests that even rudimentary computational resources are sufficient for producing pseudorandomness (it is worth noting that Linial, Mansour and Nisan [14] proved that there are no pseudorandom function generators in AC 0 with very good security parameters).…”

Section: Introductionmentioning

confidence: 99%

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On Pseudorandom Generators in NC0

Cryan

Miltersen

2001

Lecture Notes in Computer Science

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Abstract. In this paper we consider the question of whether NC 0 circuits can generate pseudorandom distributions. While we leave the general question unanswered, we show• Generators computed by NC 0 circuits where each output bit depends on at most 3 input bits (i.e, NC 0 3 circuits) and with stretch factor greater than 4 are not pseudorandom.• A large class of "non-problematic" NC 0 generators with superlinear stretch (including all NC 0 3 generators with superlinear stretch) are broken by a statistical test based on a linear dependency test combined with a pairwise independence test.• There is an NC 0 4 generator with a super-linear stretch that passes the linear dependency test as well as k-wise independence tests, for any constant k.

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Section: Definitions and Backgroundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On Pseudorandom Generators in NC0

Cryan

Miltersen

2001

Lecture Notes in Computer Science

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“…which implies the following Lemma from [LMN93]. Roughly, the lemma says that in order to sample the random variable S ∩ A, one can first pick a random sample of ω A c , and then take a sample from the spectral sample of the function we get by plugging in these values for the bits in A c .…”

Section: The Spectral Sample In Generalmentioning

confidence: 99%

The Fourier spectrum of critical percolation

Garban

Pete

Schramm

2011

Selected Works of Oded Schramm

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Consider the indicator function f of a two-dimensional percolation crossing event. In this paper, the Fourier transform of f is studied and sharp bounds are obtained for its lower tail in several situations. Various applications of these bounds are derived. In particular, we show that the set of exceptional times of dynamical critical site percolation on the triangular grid in which the origin percolates has dimension 31/36 a.s., and the corresponding dimension in the half-plane is 5/9. It is also proved that critical bond percolation on the square grid has exceptional times a.s. Also, the asymptotics of the number of sites that need to be resampled in order to significantly perturb the global percolation configuration in a large square is determined.

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“…The goal of the learner is to construct a high-accuracy hypothesis function h, i.e., one which satisfies Pr[f (x) = h(x)] ≤ ǫ where the probability is with respect to the uniform distribution and ǫ is an error parameter given to the learning algorithm. Algorithms and hardness results in this framework have interesting connections with topics such as discrete Fourier analysis [Man94], circuit complexity [LMN93], noise sensitivity and influence of variables in Boolean functions [KKL88,BKS99,KOS04,OS07], coding theory [FGKP06], privacy [BLR08,KLN + 08], and cryptography [BFKL93,Kha95]. For these reasons, and because the model is natural and elegant in its own right, the uniform distribution learning model has been intensively studied for almost two decades.…”

Section: Background and Motivationmentioning

confidence: 99%

Optimal Cryptographic Hardness of Learning Monotone Functions

Dachman-Soled

Lee

Malkin

et al. 2008

Automata, Languages and Programming

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Abstract. A wide range of positive and negative results have been established for learning different classes of Boolean functions from uniformly distributed random examples. However, polynomial-time algorithms have thus far been obtained almost exclusively for various classes of monotone functions, while the computational hardness results obtained to date have all been for various classes of general (nonmonotone) functions. Motivated by this disparity between known positive results (for monotone functions) and negative results (for nonmonotone functions), we establish strong computational limitations on the efficient learnability of various classes of monotone functions. We give several such hardness results which are provably almost optimal since they nearly match known positive results. Some of our results show cryptographic hardness of learning polynomial-size monotone circuits to accuracy only slightly greater than 1/2 + 1/ √ n; this accuracy bound is close to optimal by known positive results (Blum et al., FOCS '98). Other results show that under a plausible cryptographic hardness assumption, a class of constant-depth, sub-polynomialsize circuits computing monotone functions is hard to learn; this result is close to optimal in terms of the circuit size parameter by known positive results as well (Servedio, Information and Computation '04). Our main tool is a complexitytheoretic approach to hardness amplification via noise sensitivity of monotone functions that was pioneered by O'Donnell (JCSS '04).

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Constant depth circuits, Fourier transform, and learnability

Cited by 438 publications

References 7 publications

On Pseudorandom Generators in NC0

On Pseudorandom Generators in NC0

The Fourier spectrum of critical percolation

Optimal Cryptographic Hardness of Learning Monotone Functions

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