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33
A hypercontractive inequality for matrixvalued functions with applications to quantum computing and LDCs
"... The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and ..."
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Cited by 28 (2 self)
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The BonamiBeckner hypercontractive inequality is a powerful tool in Fourier analysis of realvalued functions on the Boolean cube. In this paper we present a version of this inequality for matrixvalued functions on the Boolean cube. Its proof is based on a powerful inequality by Ball, Carlen, and Lieb. We also present a number of applications. First, we analyze maps that encode n classical bits into m qubits, in such a way that each set of k bits can be recovered with some probability by an appropriate measurement on the quantum encoding; we show that if m<0.7n, then the success probability is exponentially small in k. This result may be viewed as a direct product version of Nayak’s quantum random access code bound. It in turn implies strong direct product theorems for the oneway quantum communication complexity of Disjointness and other problems. Second, we prove that errorcorrecting codes that are locally decodable with 2 queries require length exponential in the length of the encoded string. This gives what is arguably the first “nonquantum” proof of a result originally derived by Kerenidis and de Wolf using quantum information theory.
A brief introduction to Fourier analysis on the Boolean cube
 Theory of Computing Library– Graduate Surveys
, 2008
"... Abstract: We give a brief introduction to the basic notions of Fourier analysis on the ..."
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Cited by 27 (3 self)
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Abstract: We give a brief introduction to the basic notions of Fourier analysis on the
Simple and Practical Algorithm for Sparse Fourier Transform
"... We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, an ..."
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Cited by 26 (10 self)
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We consider the sparse Fourier transform problem: given a complex vector x of length n, and a parameter k, estimate the k largest (in magnitude) coefficients of the Fourier transform of x. The problem is of key interest in several areas, including signal processing, audio/image/video compression, and learning theory. We propose a new algorithm for this problem. The algorithm leverages techniques from digital signal processing, notably Gaussian and DolphChebyshev filters. Unlike the typical approach to this problem, our algorithm is not iterative. That is, instead of estimating “large ” coefficients, subtracting them and recursing on the reminder, it identifies and estimates the k largest coefficients in “one shot”, in a manner akin to sketching/streaming algorithms. The resulting algorithm is structurally simpler than its predecessors. As a consequence, we are able to extend considerably the range of sparsity, k, for which the algorithm is faster than FFT, both in theory and practice. 1
Property Testing Lower Bounds Via Communication Complexity
, 2011
"... We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexit ..."
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Cited by 25 (6 self)
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We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is klinear (a parity function on k variables), we achieve a lower bound of Ω(k) queries, even for adaptive algorithms with twosided error, thus confirming a conjecture of Goldreich [25]. The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as kjuntas. For some classes, such as the class of monotone functions and the class of ssparse GF(2) polynomials, we significantly strengthen the best known bounds.
Nearly Optimal Sparse Fourier Transform
"... We consider the problem of computing the ksparse approximation to the discrete Fourier transform of an ndimensional signal. We show: • An O(k log n)time algorithm for the case where the input signal has at most k nonzero Fourier coefficients, and • An O(k log n log(n/k))time algorithm for gener ..."
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Cited by 21 (10 self)
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We consider the problem of computing the ksparse approximation to the discrete Fourier transform of an ndimensional signal. We show: • An O(k log n)time algorithm for the case where the input signal has at most k nonzero Fourier coefficients, and • An O(k log n log(n/k))time algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k = o(n). Further, they are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly ksparse case is optimal for any k = n Ω(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log(n/k) / log log n) signal samples, even if it is allowed The discrete Fourier transform (DFT) is one of the most important and widely used computational tasks. Its applications are broad and include signal processing, communications, and audio/image/video compression. Hence, fast algorithms for DFT are highly valuable. Currently, the fastest such algorithm is the Fast Fourier
Approximability and proof complexity
, 2012
"... This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Pa ..."
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Cited by 11 (5 self)
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This work is concerned with the proofcomplexity of certifying that optimization problems do not have good solutions. Specifically we consider boundeddegree “Sum of Squares ” (SOS) proofs, a powerful algebraic proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor, Lasserre, and Parrilo shows that this proof system is automatizable using semidefinite programming (SDP), meaning that any nvariable degreed proof can be found in time n O(d). Furthermore, the SDP is dual to the wellknown Lasserre SDP hierarchy, meaning that the “d/2round Lasserre value ” of an optimization problem is equal to the best bound provable using a degreed SOS proof. These ideas were exploited in a recent paper by Barak et al. (STOC 2012) which shows that the known “hard instances ” for the UniqueGames problem are in fact solved close to optimally by a constant level of the Lasserre SDP hierarchy. We continue the study of the power of SOS proofs in the context of difficult optimization problems. In particular, we show that the BalancedSeparator integrality gap instances proposed by Devanur et al. can have their optimal value certified by a degree4 SOS proof. The key ingredient is an SOS proof of the KKL Theorem. We also investigate the extent to which the Khot–Vishnoi MaxCut integrality gap instances can have their optimum value certified by an SOS proof. We show they can be certified to within a factor.952 (>.878) using a constantdegree proof. These investigations also raise an interesting mathematical question: is there a constantdegree SOS proof of the Central Limit Theorem?
Reconstruction and Clustering in Random Constraint Satisfaction Problems
, 2009
"... Random instances of Constraint Satisfaction Problems (CSP’s) appear to be hard for all known algorithms, when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we for ..."
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Cited by 7 (4 self)
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Random instances of Constraint Satisfaction Problems (CSP’s) appear to be hard for all known algorithms, when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we formulate a set of technical conditions on a large family of random CSP’s, and prove bounds on three most interesting thresholds for the density of such an ensemble: namely, the satisfiability threshold, the threshold for clustering of the solution space, and the threshold for an appropriate reconstruction problem on the CSP’s. The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges. The families are general enough to include commonly studied problems such as, random instances of NotAllEqualSAT, kXOR formulae, hypergraph 2coloring, and graph kcoloring. An important new ingredient is a condition involving the Fourier expansion of clauses, which characterizes the class of problems with a similar threshold structure.
Systems of linear equations over F2 and problems parameterized above average
 Proc. SWAT 2010
, 1999
"... In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over F2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the ..."
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Cited by 7 (5 self)
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In the problem Max Lin, we are given a system Az = b of m linear equations with n variables over F2 in which each equation is assigned a positive weight and we wish to find an assignment of values to the variables that maximizes the excess, which is the total weight of satisfied equations minus the total weight of falsified equations. Using an algebraic approach, we obtain a lower bound for the maximum excess. Max Lin Above Average (Max Lin AA) is a parameterized version of Max Lin introduced by Mahajan et al. (Proc. IWPEC’06 and J. Comput. Syst. Sci. 75, 2009). In Max Lin AA all weights are integral and we are to decide whether the maximum excess is at least k, where k is the parameter. It is not hard to see that we may assume that no two equations in Az = b have the same lefthand side and n = rankA. Using our maximum excess results, we prove that, under these assumptions, Max Lin AA is fixedparameter
The Fourier Entropy–Influence Conjecture for certain classes of Boolean functions
"... entropy of f, I[f] is the total influence of f, and C is a universal constant. In this work we verify the conjecture for symmetric functions. More generally, we verify it for functions with symmetry group Sn1 × · · ·×Sn d where d is constant. We also verify the conjecture for functions computable b ..."
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entropy of f, I[f] is the total influence of f, and C is a universal constant. In this work we verify the conjecture for symmetric functions. More generally, we verify it for functions with symmetry group Sn1 × · · ·×Sn d where d is constant. We also verify the conjecture for functions computable by readonce decision trees.
NearOptimal and Explicit Bell Inequality Violations
"... Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new twoplayer games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Match ..."
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Cited by 4 (1 self)
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Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new twoplayer games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work. The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by BarYossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPRpairs), while we show that the winning probability of any classical strategy differs from 1 2 by at most O(log n/ √ n). The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here ndimensional entanglement allows to win the game with probability 1/(logn) 2, while the best winning probability without entanglement is 1/n. This nearlinear ratio (“Bell inequality violation”) is nearoptimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players.