Results 1  10
of
46
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 ..."
Abstract

Cited by 39 (3 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 34 (8 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 34 (4 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 33 (9 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 29 (9 self)
 Add to MetaCart
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
Some applications of hypercontractive inequalities in quantum information
, 2012
"... theory ..."
(Show Context)
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 ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
(Show Context)
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?
Simultaneously Satisfying Linear Equations Over F2: MaxLin2 and MaxrLin2 Parameterized Above Average
 IN FSTTCS 2011, LIPICS
, 2011
"... In the parameterized problem MAXLIN2AA[k], we are given a system with variables x1,..., xn consisting of equations of the form ∏i∈I x i = b, where x i, b ∈ {−1, 1} and I ⊆ [n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equa ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
(Show Context)
In the parameterized problem MAXLIN2AA[k], we are given a system with variables x1,..., xn consisting of equations of the form ∏i∈I x i = b, where x i, b ∈ {−1, 1} and I ⊆ [n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2 + k, where W is the total weight of all equations and k is the parameter (if k = 0, the possibility is assured). We show that MAXLIN2AA[k] has a kernel with at most O(k 2 log k) variables and can be solved in time 2 O(k log k) (nm) O(1). This solves an open problem of Mahajan et al. (2006). The problem MAXrLIN2AA[k, r] is the same as MAXLIN2AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove a theorem on MAXrLIN2AA[k, r] which implies that MAXrLIN2AA[k, r] has a kernel with at most (2k − 1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f: {−1, 1} n → R whose Fourier expansion (which is a multilinear polynomial) is of degree r. We show applicability of the lower bound by giving a new proof of the EdwardsErdős bound (each connected graph on n vertices and m edges has a bipartite subgraph with at least m/2 + (n − 1)/4 edges) and obtaining a generalization.
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 ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
(Show Context)
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.
Vertex sparsifiers and abstract rounding algorithms
 In 51st Annual Symposium on Foundations of Computer Science
, 2010
"... Abstract—The notion of vertex sparsification (in particular cutsparsification) is introduced in [18], where it was shown that for any graph G = (V, E) and any subset of k terminals K ⊂ V, there is a polynomial time algorithm to construct a graph H = (K, EH) on just the terminal set so that simultan ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Abstract—The notion of vertex sparsification (in particular cutsparsification) is introduced in [18], where it was shown that for any graph G = (V, E) and any subset of k terminals K ⊂ V, there is a polynomial time algorithm to construct a graph H = (K, EH) on just the terminal set so that simultaneously for all cuts (A, K − A), the value of the minimum cut in G separating A from K − A is approximately the same as the value of the corresponding cut in H. Then approximation algorithms can be run directly on H as a proxy for running on G. We give the first superconstant lower bounds for how well a cutsparsifier H can simultaneously approximate all minimum cuts in G. We prove a lower bound of Ω(log 1/4 k) – this is polynomiallyrelated to the known upper bound of O(logk/loglogk). Independently, a similar lower bound is given in [17]. This is an exponential improvement on the Ω(loglogk) bound given in [14] which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers. Despite this negative result, we show that for many natural optimization problems, we do not need to incur a multiplicative penalty for our reduction. Roughly, we show that any rounding algorithm which also works for the 0extension relaxation can be used to construct good vertexsparsifiers for which the optimization problem is easy. Using this, we obtain optimal O(logk)competitive Steiner oblivious routing schemes, which generalize the results in [20]. We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most O(logk) times the integrality gap restricted to trees. Lastly, we use our ideas to give an efficient construction for vertexsparsifiers that match the current best existential results – this was previously open. Our algorithm makes novel use of Earthmover constraints. Keywordsvertex sparsifier; approximation algorithms; A. Background I.