Results 1  10
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50
Expander Graphs and their Applications
, 2003
"... Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . ..."
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Cited by 186 (5 self)
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Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Derandomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Derandomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
COMBINING GEOMETRY AND COMBINATORICS: A UNIFIED APPROACH TO SPARSE SIGNAL RECOVERY
"... Abstract. There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix Φ and then uses linear programming to decode information about x from Φx. The combinatorial approach constru ..."
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Cited by 77 (12 self)
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Abstract. There are two main algorithmic approaches to sparse signal recovery: geometric and combinatorial. The geometric approach starts with a geometric constraint on the measurement matrix Φ and then uses linear programming to decode information about x from Φx. The combinatorial approach constructs Φ and a combinatorial decoding algorithm to match. We present a unified approach to these two classes of sparse signal recovery algorithms. The unifying elements are the adjacency matrices of highquality unbalanced expanders. We generalize the notion of Restricted Isometry Property (RIP), crucial to compressed sensing results for signal recovery, from the Euclidean norm to the ℓp norm for p ≈ 1, and then show that unbalanced expanders are essentially equivalent to RIPp matrices. From known deterministic constructions for such matrices, we obtain new deterministic measurement matrix constructions and algorithms for signal recovery which, compared to previous deterministic algorithms, are superior in either the number of measurements or in noise tolerance. 1.
Extractors: Optimal up to Constant Factors
 STOC'03
, 2003
"... This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)ent ..."
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Cited by 52 (12 self)
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This paper provides the first explicit construction of extractors which are simultaneously optimal up to constant factors in both seed length and output length. More precisely, for every n, k, our extractor uses a random seed of length O(log n) to transform any random source on n bits with (min)entropy k, into a distribution on (1 − α)k bits that is ɛclose to uniform. Here α and ɛ can be taken to be any positive constants. (In fact, ɛ can be almost polynomially small). Our improvements are obtained via three new techniques, each of which may be of independent interest. The first is a general construction of mergers [22] from locally decodable errorcorrecting codes. The second introduces new condensers that have constant seed length (and retain a constant fraction of the minentropy in the random source). The third is a way to augment the “winwin repeated condensing” paradigm of [17] with error reduction techniques like [15] so that the our constant seedlength condensers can be used without error accumulation.
Extracting Randomness via Repeated Condensing
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... On an input probability distribution with some (min)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution ( ..."
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Cited by 43 (16 self)
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On an input probability distribution with some (min)entropy an extractor outputs a distribution with a (near) maximum entropy rate (namely the uniform distribution). A natural weakening of this concept is a condenser, whose output distribution has a higher entropy rate than the input distribution (without losing much of the initial entropy). In this paper we construct efficient explicit condensers. The condenser constructions combine (variants or more efficient versions of) ideas from several works, including the block extraction scheme of [NZ96], the observation made in [SZ94, NT99] that a failure of the block extraction scheme is also useful, the recursive "winwin" case analysis of [ISW99, ISW00], and the error correction of random sources used in [Tre99]. As a natural byproduct, (via repeated iterating of condensers), we obtain new extractor constructions. The new extractors give significant qualitative improvements over previous ones for sources of arbitrary minentropy; they...
Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δsource if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the fol ..."
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Cited by 43 (13 self)
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We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δsource if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the following poly(n)time computable functions: 2source disperser: D: ({0, 1} n) 2 → {0, 1} such that for any two independent δsources X1, X2 we have that the support of D(X1, X2) is {0, 1}. Bipartite Ramsey graph: Let N = 2 n. A corollary is that the function D is a 2coloring of the edges of KN,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N δ by N δ is not monochromatic. 3source extractor: E: ({0, 1} n) 2 → {0, 1} such that for any three independent δsources X1, X2, X3 we have that E(X1, X2, X3) is (o(1)close to being) an unbiased random bit. No previous explicit construction was known for either of these, for any δ < 1/2 and these results constitute major progress to longstanding open problems. A component in these results is a new construction of condensers that may be of independent
Extractors for a constant number of polynomially small minentropy independent sources
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded ex ..."
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Cited by 39 (10 self)
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We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded extractors in simple ways. We introduce a new technique to condense independent somewhererandom sources which looks like a useful way to manipulate independent sources. Our techniques are different from those used in recent work [BIW04, BKS + 05, Raz05, Bou05] for this problem in the sense that they do not rely on any results from additive number theory. Using Bourgain’s extractor [Bou05] as a black box, we obtain a new extractor for 2 independent blocksources with few blocks, even when the minentropy is as small as polylog(n). We also show how to modify the 2 source disperser for linear minentropy of Barak et al. [BKS + 05] and the 3 source extractor of Raz [Raz05] to get dispersers/extractors with exponentially small error and linear output length where previously both were constant. In terms of Ramsey Hypergraphs, for every constant 1> γ> 0 our construction gives a family of explicit O(1/γ)uniform hypergraphs on N vertices that avoid cliques and independent sets of (log N)γ size 2.
2source dispersers for subpolynomial entropy and Ramsey graphs beating the FranklWilson construction
 Proceedings of STOC06
, 2006
"... The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = n o(1). Put differently, setting N = 2 n and K = 2 k, we construct explicit N × N Boolean matrices for which no K × K submatrix is monochromatic. Viewed as adjacency matrices of bipartit ..."
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Cited by 27 (6 self)
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The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = n o(1). Put differently, setting N = 2 n and K = 2 k, we construct explicit N × N Boolean matrices for which no K × K submatrix is monochromatic. Viewed as adjacency matrices of bipartite graphs, this gives an explicit construction of KRamsey bipartite graphs of size N. This greatly improves the previous bound of k = o(n) of Barak, Kindler, Shaltiel, Sudakov and Wigderson [4]. It also significantly improves the 25year record of k = Õ( √ n) on the special case of Ramsey graphs, due to Frankl and Wilson [9]. The construction uses (besides ”classical ” extractor ideas) almost all of the machinery developed in the last couple of years for extraction from independent sources, including: • Bourgain’s extractor for 2 independent sources of some entropy rate < 1/2 [5] • Raz’s extractor for 2 independent sources, one of which has any entropy rate> 1/2 [18] • Rao’s extractor for 2 independent blocksources of entropy n Ω(1) [17]
Compressive Sensing
, 2010
"... Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many poten ..."
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Cited by 24 (8 self)
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Compressive sensing is a new type of sampling theory, which predicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main feature, efficient algorithms such as ℓ1minimization can be used for recovery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing.
Dense error correction via ℓ1 minimization
, 2009
"... This paper studies the problem of recovering a nonnegative sparse signal x ∈ Rn from highly corrupted linear measurements y = Ax + e ∈ Rm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper prov ..."
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Cited by 17 (5 self)
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This paper studies the problem of recovering a nonnegative sparse signal x ∈ Rn from highly corrupted linear measurements y = Ax + e ∈ Rm, where e is an unknown error vector whose nonzero entries may be unbounded. Motivated by an observation from face recognition in computer vision, this paper proves that for highly correlated (and possibly overcomplete) dictionaries A, any nonnegative, sufficiently sparse signal x can be recovered by solving an ℓ1minimization problem: min ‖x‖1 + ‖e‖1 subject to y = Ax + e. More precisely, if the fraction ρ of errors is bounded away from one and the support of x grows sublinearly in the dimension m of the observation, then as m goes to infinity, the above ℓ1minimization succeeds for all signals x and almost all signandsupport patterns of e. This result suggests that accurate recovery of sparse signals is possible and computationally feasible even with nearly 100 % of the observations corrupted. The proof relies on a careful characterization of the faces of a convex polytope spanned together by the standard crosspolytope and a set of iid Gaussian vectors with nonzero mean and small variance, which we call the “crossandbouquet ” model. Simulations and experimental results corroborate the findings, and suggest extensions to the result.
Revisiting the Efficiency of Malicious TwoParty Computation
 In Eurocrypt ’07, SpringerVerlag (LNCS 4515
, 2006
"... In a recent paper Mohassel and Franklin study the e#ciency of secure twoparty computation in the presence of malicious behavior. Their aim is to make classical solutions to this problem, such as zeroknowledge compilation, more practical. The authors provide several schemes which are the most e# ..."
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Cited by 12 (0 self)
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In a recent paper Mohassel and Franklin study the e#ciency of secure twoparty computation in the presence of malicious behavior. Their aim is to make classical solutions to this problem, such as zeroknowledge compilation, more practical. The authors provide several schemes which are the most e#cient to date. We propose a modification to their main scheme using expanders.