Results 1  10
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18
Gowers uniformity, influence of variables, and PCPs
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We pro ..."
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Cited by 51 (2 self)
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Gowers [Gow98, Gow01] introduced, for d ≥ 1, the notion of dimensiond uniformity U d (f) of a function f: G → C, where G is a finite abelian group. Roughly speaking, if a function has small Gowers uniformity of dimension d, then it “looks random ” on certain structured subsets of the inputs. We prove the following inverse theorem. Write G = G1 × · · · × Gn as a product of groups. If a bounded balanced function f: G1 × · · · Gn → C is such that U d (f) ≥ ε, then one of the coordinates of f has influence at least ε/2 O(d). Other inverse theorems are known [Gow98, Gow01, GT05, Sam05], and U 3 is especially well understood, but the properties of functions f with large U d (f), d ≥ 4, are not yet well characterized. The dimensiond Gowers inner product 〈{fS} 〉 U d of a collection {fS} S⊆[d] of functions is a related measure of pseudorandomness. The definition is such that if all the functions fS are equal to the same fixed function f, then 〈{fS} 〉 U d = U d (f). We prove that if fS: G1 × · · · × Gn → C is a collection of bounded functions such that 〈{fS} 〉 U d  ≥ ε and at least one of the fS is balanced, then there is a variable that has influence at least ε 2 /2 O(d) for at least four functions in the collection. Finally, we relate the acceptance probability of the “hypergraph longcode test ” proposed by Samorodnitsky and Trevisan to the Gowers inner product of the functions being tested and we deduce the following result: if the Unique Games Conjecture is true, then for every q ≥ 3 there is a PCP characterization of NP where the verifier makes q queries, has almost perfect completeness, and soundness at most 2q/2 q. For infinitely many q, the soundness is (q + 1)/2 q, which might be a tight result. Two applications of this results are that, assuming that the unique games conjecture is true, it is hard to approximate Max kCSP within a factor 2k/2 k ((k + 1)/2 k for infinitely many k), and it is hard to approximate Independent Set in graphs of degree D within a factor (log D) O(1) /D. 1
Discrepancy and the power of bottom fanin in depththree circuits
 In Proc. of the 48th Symposium on Foundations of Computer Science (FOCS
, 2007
"... We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty ‘Number on the Forehead ’ model. Our method is based on the notion of voting polynomial degree of functions and extends the DegreeDiscrepancy Lemma in the recent wor ..."
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Cited by 26 (3 self)
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We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty ‘Number on the Forehead ’ model. Our method is based on the notion of voting polynomial degree of functions and extends the DegreeDiscrepancy Lemma in the recent work of Sherstov [24]. Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fanin at the base layer i.e. circuits of type MAJ ◦ SYMM ◦ ANY O(1) cannot simulate the circuit class AC 0 in subexponential size. Further, even if the fanin of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC 0 in quasipolynomial size. This is in contrast to the classical result of Yao and BeigelTarui that shows that such circuits, having only MAJORITY gates, can simulate the class ACC 0 in quasipolynomial size when the bottom fanin is increased to polylogarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain [7] for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented by polynomials of small degree over Zm, when m, q ≥ 2 are coprime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of [7, 14]. It is known that such estimates imply that circuits of type MAJ ◦ MODm ◦ ANDɛ log n cannot compute the MODq function in subexponential size. It remains a major open question to determine if such circuits can simulate ACC 0 in polynomial size when the bottom fanin is increased to polylogarithmic size. 1
The sum of d smallbias generators fools polynomials of degree d
 In IEEE Conference on Computational Complexity
, 2007
"... We prove that the sum of d smallbias generators L: F s → F n fools degreed polynomials in n variables over a prime field F, for any fixed degree d and field F, including F = F2 = {0, 1}. Our result improves on both the work by Bogdanov and Viola (FOCS ’07) and the beautiful followup by Lovett (ST ..."
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Cited by 23 (2 self)
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We prove that the sum of d smallbias generators L: F s → F n fools degreed polynomials in n variables over a prime field F, for any fixed degree d and field F, including F = F2 = {0, 1}. Our result improves on both the work by Bogdanov and Viola (FOCS ’07) and the beautiful followup by Lovett (STOC ’08). The first relies on a conjecture that turned out to be true only for some degrees and fields, while the latter considers the sum of 2 d smallbias generators (as opposed to d in our result). Our proof builds on and somewhat simplifies the arguments by Bogdanov and Viola (FOCS ’07) and by Lovett (STOC ’08). Its core is a case analysis based on the bias of the polynomial to be fooled. 1
Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols
 In Proceedings of the 22nd Annual Conference on Computational Complexity. IEEE
, 2007
"... This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and GF (2) polynomials. The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely rela ..."
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Cited by 22 (7 self)
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This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and GF (2) polynomials. The key is the use of (known) norms on Boolean functions, which capture their proximity to each of these models (and are closely related to property testers of this proximity). The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most ɛ ≤ 1/2 with any of these models, then the correlation of the parity of its values on m independent instances drops exponentially with m. More specifically: • For GF (2) polynomials of degree d, the correlation drops to exp � −m/4 d �. No XOR lemma was known even for d = 2. • For cbit kparty protocols, the correlation drops to 2c · ɛm/2k. No XOR lemma was known for k ≥ 3 parties. Another contribution in this paper is a general derivation of direct product lemmas from XOR lemmas. In particular, assuming that f has correlation at most ɛ ≤ 1/2 with any of the above models, we obtain the following bounds on the probability of computing m independent instances of f correctly: • For GF (2) polynomials of degree d we again obtain a bound of exp � −m/4 d �. • For cbit kparty protocols we obtain a bound of 2 −Ω(m) in the special case when ɛ ≤ exp � −c · 2 k �. In this range of ɛ, our bound improves on a direct product lemma for twoparties by Parnafes, Raz, and Wigderson (STOC ’97). We also use the norms to give improved (or just simplified) lower bounds in these models. In particular we give a new proof that the Modm function on n bits, for odd m, has correlation at most exp(−n/4 d) with degreed GF (2) polynomials.
Inverse Conjecture for the Gowers norm is false
 In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing (STOC
, 2007
"... Let p be a fixed prime number, and N be a large integer. The ’Inverse Conjecture for the Gowers norm ’ states that if the ”dth Gowers norm ” of a function f: F N p → F is nonnegligible, that is larger than a constant independent of N, then f can be nontrivially approximated by a degree d − 1 poly ..."
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Cited by 13 (3 self)
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Let p be a fixed prime number, and N be a large integer. The ’Inverse Conjecture for the Gowers norm ’ states that if the ”dth Gowers norm ” of a function f: F N p → F is nonnegligible, that is larger than a constant independent of N, then f can be nontrivially approximated by a degree d − 1 polynomial. The conjecture is known to hold for d = 2, 3 and for any prime p. In this paper we show the conjecture to be false for p = 2 and for d = 4, by presenting an explicit function whose 4th Gowers norm is nonnegligible, but whose correlation any polynomial of degree 3 is exponentially small. Essentially the same result (with different correlation bounds) was independently obtained by Green and Tao [5]. Their analysis uses a modification of a Ramseytype argument of Alon and Beigel [1] to show inapproximability of certain functions by lowdegree polynomials. We observe that a combination of our results with the argument of Alon and Beigel implies the inverse conjecture to be false for any prime p, for d = p 2.
Optimal testing of ReedMuller codes
, 2009
"... We consider the problem of testing if a given function ..."
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Cited by 10 (7 self)
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We consider the problem of testing if a given function
New correlation bounds for GF(2) polynomials using Gowers uniformity
 Electronic Colloquium on Computational Complexity
, 2006
"... We study the correlation between lowdegree GF (2) polynomials p and explicit functions. Our main results are the following: I We prove that the Modm function on n bits has correlation at most exp � −Ω � n/4 d�� with any GF (2) polynomial of degree d, for any fixed odd integer m. This improves on th ..."
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Cited by 8 (4 self)
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We study the correlation between lowdegree GF (2) polynomials p and explicit functions. Our main results are the following: I We prove that the Modm function on n bits has correlation at most exp � −Ω � n/4 d�� with any GF (2) polynomial of degree d, for any fixed odd integer m. This improves on the previous exp � −Ω � n/8 d� � bound by Bourgain (C. R. Acad. Sci. Paris, 2005) and Green et al. (C. R. Acad. Sci. Paris, 2005). II We exhibit a polynomialtime computable function on n bits that has correlation at most exp � −Ω � n/2 d� � with any GF (2) polynomial of degree d. Previous to our work the best correlation bound for an explicit function was exp � −Ω � n / � d · 2 d�� �, which follows from (Chung and Tetali; SIAM J. Discrete Math., 1993). III We derive an ‘XOR Lemma ’ for lowdegree GF (2) polynomials: We show that if a function f has correlation at most 1 − 4 −d with any GF (2) polynomial of degree d (and Prx[f(x) = 1] ≈ 1/2) then the XOR of m independent copies of f has correlation at most exp � −Ω � m/4 d� � with any GF (2) polynomial of degree d. Our results rely on a measure of the ‘complexity ’ of a function due to Gowers (Geom.
Selected Results in Additive Combinatorics: An Exposition
, 2007
"... We give a selfcontained exposition of selected results in additive combinatorics over the group GF (2) n = {0, 1} n. In particular, we prove the celebrated theorems known as the BalogSzemerediGowers theorem (’94 and ’98) and the FreimanRuzsa theorem (’73 and ’99), leading to the remarkable resul ..."
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Cited by 6 (1 self)
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We give a selfcontained exposition of selected results in additive combinatorics over the group GF (2) n = {0, 1} n. In particular, we prove the celebrated theorems known as the BalogSzemerediGowers theorem (’94 and ’98) and the FreimanRuzsa theorem (’73 and ’99), leading to the remarkable result by Samorodnitsky (’07) that linear transformations are efficiently testable. No new result is proved here. However, we strip down the available proofs to the bare minimum needed to derive the efficient testability of linear transformations over {0, 1} n, thus hoping to provide a computer sciencefriendly introduction to the marvelous field of additive combinatorics.
The Littlewood–Gowers problem
"... Abstract. The paper has two main parts. To begin with suppose that G is a compact Abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions f ∈ L 2 (G). We prove an analogous result for functions f ∈ A(G), where A(G): = {f ∈ L 1 (G) : ‖ ̂ f‖1 < ∞} ..."
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Cited by 5 (1 self)
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Abstract. The paper has two main parts. To begin with suppose that G is a compact Abelian group. Chang’s Theorem can be viewed as a structural refinement of Bessel’s inequality for functions f ∈ L 2 (G). We prove an analogous result for functions f ∈ A(G), where A(G): = {f ∈ L 1 (G) : ‖ ̂ f‖1 < ∞} equipped with the norm ‖f ‖ A(G): = ‖ ̂ f‖1, and generalize this to the approximate Fourier transform on Bohr sets. As an application of the first part of the paper we improve a recent result of Green and Konyagin: Suppose that p is a prime number and A ⊂ Z/pZ has density bounded away from 0 and 1 by an absolute constant. Green and Konyagin have shown that ‖χA ‖ A(Z/pZ) ≫ε (log p) 1/3−ε, and we improve this to ‖χA ‖ A(Z/pZ) ≫ε (log p) 1/2−ε. To put this in context it is easy to see that if A is an arithmetic progression then ‖ ̂χA ‖ A(Z/pZ) ≪ log p. 1.
An application of a local version of a Chang’s theorem // http://www.arXiv:math.CA/0607668
"... Abstract. Suppose that G is a compact Abelian group. If A ⊂ G then how small can ‖χA ‖ A(G) be? In general there is no nontrivial lower bound. In [5] Green and Konyagin showed that if ̂ G has sparse small subgroup structure and A has density α with α(1 −α) ≫ 1 then ‖χA ‖ A(G) does admit a nontriv ..."
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Cited by 2 (1 self)
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Abstract. Suppose that G is a compact Abelian group. If A ⊂ G then how small can ‖χA ‖ A(G) be? In general there is no nontrivial lower bound. In [5] Green and Konyagin showed that if ̂ G has sparse small subgroup structure and A has density α with α(1 −α) ≫ 1 then ‖χA ‖ A(G) does admit a nontrivial lower bound. To complement this [11] addressed the case where ̂ G has rich small subgroup structure and further claimed a result for general compact Abelian groups. In this note we prove this claim by fusing the techniques of [5] and [11] in a straightforward fashion. 1. Notation and introduction We use the Fourier transform on compact Abelian groups, the basics of which may be found in Chapter 1 of Rudin [9]; we take a moment to standardize our notation. Suppose that G is a compact Abelian group. Write ̂ G for the dual group, that is the discrete Abelian group of continuous homomorphisms γ: G → S 1, where S 1: = {z ∈ C: z  = 1}. G may be endowed with Haar measure µG normalised so that µG(G) = 1 and as a consequence we may define the Fourier transform ̂.: L 1 (G) → ℓ ∞ ( ̂ G) which takes f ∈ L 1 (G) to We write