## Low-degree tests at large distances (2007)

Venue: | In Proceedings of the 39th Annual ACM Symposium on Theory of Computing |

Citations: | 36 - 2 self |

### BibTeX

@INPROCEEDINGS{Samorodnitsky07low-degreetests,

author = {Alex Samorodnitsky},

title = {Low-degree tests at large distances},

booktitle = {In Proceedings of the 39th Annual ACM Symposium on Theory of Computing},

year = {2007},

pages = {506--515}

}

### OpenURL

### Abstract

Abstract We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and thenumber of queries. In particular, we show that functions with small Gowers uniformity norms behave "ran-domly " with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem forthe third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficientlythe distance from the second-order Reed-Muller code on inputs lying far beyond its listdecoding radius.

### Citations

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Citation Context ...inguishes between linear functions and functions which are far from quadratic polynomials with optimal soundness of s ≤ 2Ω(q 1/3 ) 2q . 3 Second-Order Reed-Muller Codes A binary error-correcting code =-=[19]-=- of length N and normalized distance δ is a subset of {0,1} N in which any two distinct elements disagree on at least δ-fraction of the domain (the coordinates). This allows for error-correction: a co... |

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Citation Context ...in distance λ from the corrupted codeword. To the best of our knowledge there are no such algorithms for binary RM codes of order larger than 1. Another useful property of a code is local testability =-=[10]-=-. A code is locally testable if there exists an efficient randomized algorithm (test) which, given an access to a putative codeword 6sf ∈ {0,1} N , examines a finite number of coordinates of f and dec... |

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Citation Context ...adratic Fourier analysis, a function is approximated by a combination of quadratic polynomials. This approach has proven to be quite effective in additive number theory [13, 14] and in ergodic theory =-=[16, 28]-=-, in situations in which classical Fourier analysis fails. Theorems 2.3 and 3.3 can be viewed as an application of quadratic Fourier analysis on Z n 2 to boolean functions. We suggest that this tool m... |

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Citation Context ...ly many in N) codewords at distance λ from a corrupted codeword. We call maximal λ with this property the list-decoding radius of the code. For many codes there are efficient list-decoding algorithms =-=[25]-=- that, for any λ smaller than list-decoding radius, recover all the codewords within distance λ from the corrupted codeword. To the best of our knowledge there are no such algorithms for binary RM cod... |

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Citation Context ...oundness of the PCP, namely the probability s of accepting an encoding of a false proof. It is easy to see that, unless P = NP, the lower bound s ≥ 1/2q must hold. Stronger lower bounds were given in =-=[15, 26]-=-. The best known lower bound [7] is s ≥ Ω � q 2q � . From the other direction, the PCP theorem shows that we can achieve s ≤ 1 2O(q), and it was shown in [8], following [23], that s ≤ 2√ 2q 2 q . In [... |

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Citation Context .... Stronger lower bounds were given in [15, 26]. The best known lower bound [7] is s ≥ Ω � q 2q � . From the other direction, the PCP theorem shows that we can achieve s ≤ 1 2O(q), and it was shown in =-=[8]-=-, following [23], that s ≤ 2√ 2q 2 q . In [24], assuming the Unique Games Conjecture [18], the upper bound was improved to s ≤ q/2 q−1 , which is of course (conditionally) best possible, up to constan... |

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Citation Context ...large third uniformity norm is somewhat close to an n-variate quadratic polynomial over F2. Similar results for finite Abelian groups of cardinality indivisible by 6 have been independently proved in =-=[14]-=-. – We show that functions on which the hypergraph linearity tests defined in [23] fail with non-negligible probability have large uniformity norms. – We observe that functions with small uniformity n... |

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Citation Context ...timal.) A technical ingredient of this result has a natural interpretation in the framework of errorcorrecting codes. We give a tight analysis of the acceptance probability of a natural local test of =-=[1]-=- for the second-order Reed-Muller code at distances near the covering radius of this code. As a consequence, it turns out to be possible to estimate efficiently the distance from this code on inputs l... |

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Citation Context ...eed-Muller code, also known as the Hadamard code. The distance of this code is 1/2, and therefore its unique-decoding radius is 1/4. However, it is efficiently list-decodable for any distance λ < 1/2 =-=[9]-=-. The Hadamard code is also locally testable. In fact, the basic linearity test of [6] is a good 3-query local test for this code. [4] studies the dependence of the probability this test accepts an in... |

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Citation Context ...d on ǫ is also necessary [5]. If both G and H are powers of Z2, the lower bound on ǫ was relaxed to ǫ > 83/128 [4]. Our results We show the following theorem to be a simple consequence of two results =-=[11, 22]-=- in additive number theory. Theorem 4.1: Let p be a prime number, and let ǫ > 0. Let G be a p-group of order r and let H be a power of Zp. Let φ : G → H such that Prx,y∈G (φ(x) + φ(y) = φ(x + y)) ≥ ǫ.... |

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Citation Context ...efer to both as linearity testing problems. 3 We observe that to transform this test to an affinity (degree-1) test, it suffices to replace 1 with f(0) in the definition of the test. 3sIt is shown in =-=[4]-=- that if this test accepts f with probability 1 1 2 + δ then f is 2 − 2δ close to a linear function. Therefore, according to our definition, this test has soundness s = 1 2 . Independent repetition of... |

2 | Montréal notes on quadratic Fourier analysis
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(Show Context)
Citation Context ...aining good bounds on arithmetic progressions of length 4 in subsets of Z n 5 (and in general finite Abelian groups). In this context, Green and Tao introduce the notion of quadratic Fourier analysis =-=[12]-=-. According to this point of view, the subject of classical Fourier analysis is to represent a function as a combination of several linear functions (elements of the Fourier basis) it has non-negligib... |

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Citation Context ...ction f : {0,1} n → {−1,1} and we want to determine whether 1. The function f can be represented by a degree-d polynomial 2. It is 1 2 − ǫ far from any function with such representation. 1 See, e.g., =-=[27]-=- for a precise definition 1sThe distance between two functions is a fraction of points in which they disagree. Low-degree tests we consider have perfect completeness, namely in case (1) they always ac... |

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Personal communication to the authors of [6
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Citation Context ...(x)) ≥ ǫ ′ . The question is whether ρ can be lower bounded in terms of a function of ǫ that is independent of |G|. In [6] this is shown to be true if ǫ > 7/9. This lower bound on ǫ is also necessary =-=[5]-=-. If both G and H are powers of Z2, the lower bound on ǫ was relaxed to ǫ > 83/128 [4]. Our results We show the following theorem to be a simple consequence of two results [11, 22] in additive number ... |

1 |
personal communication to the authors of [24
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Citation Context ...ity s of accepting an encoding of a false proof. It is easy to see that, unless P = N P , the lower bound s >= 1/2q must hold. Stronger lower bounds were given in [15, 26]. The best known lower bound =-=[7]-=- is s >= \Omegas\Gammasq2q \Delta . From the other direction, the PCP theorem shows that we can achieve s <= 12O(q) , and it was shown in [8], following [23], that s <= 2 p2q 2q . In [24], assuming th... |

1 | Testing low-degree polynomials over
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(Show Context)
Citation Context ...timal.) A technical ingredient of this result has a natural interpretation in the framework of errorcorrecting codes. We give a tight analysis of the acceptance probability of a natural local test of =-=[1]-=- for the second-order Reed-Muller code at distances near the covering radius of this code. As a consequence, it turns out to be possible to estimate efficiently the distance from this code on inputs l... |