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Robust PCPs of Proximity, Shorter PCPs and Applications to Coding
 in Proc. 36th ACM Symp. on Theory of Computing
, 2004
"... We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) ..."
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Cited by 84 (28 self)
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We continue the study of the tradeo between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satis ability of circuits of size n): 1. We present PCPs of length exp( ~ O(log log n) ) n that can be veri ed by making o(log log n) Boolean queries.
Improved Time Bounds for NearOptimal Sparse Fourier Representations
 in Proc. SPIE Wavelets XI
, 2003
"... We study the problem of finding a Fourier representation R of B terms for a given discrete signal A of length N . The Fast Fourier Transform (FFT) can find the optimal Nterm representation in O(N log N) time, but our goal is to get sublinear algorithms for B ! N , typically, B N . Suppose kAk2 M ..."
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Cited by 49 (11 self)
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We study the problem of finding a Fourier representation R of B terms for a given discrete signal A of length N . The Fast Fourier Transform (FFT) can find the optimal Nterm representation in O(N log N) time, but our goal is to get sublinear algorithms for B ! N , typically, B N . Suppose kAk2 M kRoptk 2 , where Ropt is the optimal output. The previously best known algorithms output R such that kA \Gamma Rk poly(B; log(1=ffi); log N; log M; 1=ffl): Even though this is sublinear in the input size, the dominating term is the polynomial factor in B which is B . In our experience, this is a limitation in practice. Our main result is a significantly improved algorithm for this problem. Our algorithms output R such that kA \Gamma Rk B \Delta poly(log(1=ffi); log N; log M; 1=ffl): We also obtain improvements for higher dimensional Fourier transforms. We need two crucial ideas to achieve this bound: bulk sampling and estimation for multipoint polynomial evaluation using an unevenlyspaced Fourier tranform, and construction and use of arithmeticprogression independent random variables. Our improved algorithms are likely to find many applications. 1
EXPLICIT CONSTRUCTIONS OF RIP MATRICES AND RELATED PROBLEMS
"... Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε> 0, large N and any n satisfying N 1−ε ≤ n ≤ N, we construct RIP matrices of order k 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proo ..."
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Cited by 4 (1 self)
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Abstract. We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε> 0, large N and any n satisfying N 1−ε ≤ n ≤ N, we construct RIP matrices of order k 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proofs based on small coherence, which are used in all previous explicit constructions of RIP matrices. Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure. We also give a construction of sets of n complex numbers whose kth moments are uniformly small for 1 ≤ k ≤ N (Turán’s power sum problem), which improves upon known explicit constructions when (log N) 1+o(1) ≤ n ≤ (log N) 4+o(1). This latter construction produces elementary explicit examples of n × N matrices that satisfy RIP and whose columns constitute a new spherical code; for those problems the parameters closely match those of existing constructions in the range (log N) 1+o(1) ≤ n ≤ (log N) 5/2+o(1). 1.
EpsilonDiscrepancy Sets And Their Applications For Interpolation Of Sparse Polynomials
, 1995
"... We present a simple explicit construction of a probability distribution supported on (p \Gamma 1) 2 vectors in Z n p , where p n=" is a prime, for which the absolute value of each nontrivial Fourier coefficients is bounded by ". This construction is used to derandomize the algorithm of [Man92] t ..."
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Cited by 2 (0 self)
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We present a simple explicit construction of a probability distribution supported on (p \Gamma 1) 2 vectors in Z n p , where p n=" is a prime, for which the absolute value of each nontrivial Fourier coefficients is bounded by ". This construction is used to derandomize the algorithm of [Man92] that interpolates a sparse polynomial in polynomial time in the bit complexity model. 1 Introduction Given a set A ae Z n p , for each ff 2 Z n p define DISCA (ff) = 1 jAj fi fi fi fi fi X z2A ! !ff;z? fi fi fi fi fi ; where ! is the pth root of unity over the complex numbers, i.e. ! = e 2ßi=p . Definition 1 A set A ae Z n p is an " discrepancy set if for any ff 6= ~ 0, DISCA (ff) ". In this note we present a simple explicit construction as follows. Theorem 1.1 For any prime p and any n ? 1 there exists an explicit set A n p ae Z n p , such that jA n p j = (p \Gamma 1) 2 and A n p is an n\Gamma1 p\Gamma1 discrepancy set. The construction is a mod p variant of one of t...
Finding Significant Fourier Transform Coefficients Deterministically and Locally
, 2008
"... Computing the Fourier transform is a basic building block used in numerous applications. For data intensive applications, even the O(N logN) running time of the Fast Fourier Transform (FFT) algorithm may be too slow, and sublinear running time is necessary. Clearly, outputting the entire Fourier tr ..."
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Cited by 1 (0 self)
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Computing the Fourier transform is a basic building block used in numerous applications. For data intensive applications, even the O(N logN) running time of the Fast Fourier Transform (FFT) algorithm may be too slow, and sublinear running time is necessary. Clearly, outputting the entire Fourier transform in sublinear time is infeasible, nevertheless, in many applications it suffices to find only the τsignificant Fourier transform coefficients, that is, the Fourier coefficients whose magnitude is at least τfraction (say, 1%) of the energy (i.e., the sum of squared Fourier coefficients). We call algorithms achieving the latter SFT algorithms. In this work we present a deterministic algorithm that finds the τsignificant Fourier coefficients of functions f over any finite abelian group G in time polynomial in logG, 1/τ and L1 ( ̂f) (for L1 ( ̂f) denoting the sum of absolute values of the Fourier coefficients of f). Our algorithm is robust to random noise. Our algorithm is the first deterministic and efficient (i.e., polynomial in logG) SFT algorithm to handle functions over any finite abelian groups, as well as the first such algorithm to handle functions over ZN that are neither compressible nor Fouriersparse. Our analysis is the first to show robustness to noise in the context of deterministic SFT algorithms. Using our SFT algorithm we obtain (1) deterministic (universal and explicit) algorithms for sparse Fourier approximation, compressed sensing and sketching; (2) an algorithm solving the Hidden Number Problem with advice, with cryptographic bit security implications; and (3) an efficient decoding algorithm in the random noise model for polynomial rate variants of Homomorphism codes and any other concentrated & recoverable codes.
Electronic Colloquium on Computational Complexity, Report No. 102 (2008) Finding Significant Fourier Transform Coefficients Deterministically and Locally
, 2008
"... Computing the Fourier transform is a basic building block used in numerous applications. For data intensive applications, even the O(N logN) running time of the Fast Fourier Transform (FFT) algorithm may be too slow, and sublinear running time is necessary. Clearly, outputting the entire Fourier tr ..."
Abstract
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Computing the Fourier transform is a basic building block used in numerous applications. For data intensive applications, even the O(N logN) running time of the Fast Fourier Transform (FFT) algorithm may be too slow, and sublinear running time is necessary. Clearly, outputting the entire Fourier transform in sublinear time is infeasible, nevertheless, in many applications it suffices to find only the τsignificant Fourier transform coefficients, that is, the Fourier coefficients whose magnitude is at least τfraction (say, 1%) of the energy (i.e., the sum of squared Fourier coefficients). We call algorithms achieving the latter SFT algorithms. In this paper we present a deterministic algorithm that finds the τsignificant Fourier coefficients of functions f over any finite abelian group G in time polynomial in logG, 1/τ and L1(f) (for L1(f) denoting the sum of absolute values of the Fourier coefficients of f). Our algorithm is robust to random noise. Our algorithm is the first deterministic and efficient (i.e., polynomial in logG) SFT algorithm to handle functions over any finite abelian groups, as well as the first such algorithm to handle functions over ZN that are neither compressible nor Fouriersparse. Our analysis is the first to show robustness to noise in the context of deterministic SFT algorithms. Using our SFT algorithm we obtain (1) deterministic (universal and explicit) algorithms for sparse Fourier approximation, compressed sensing and sketching; (2) an algorithm solving the Hidden Number Problem with advice, with cryptographic bit security implications; and (3) an efficient decoding algorithm in the random noise model for polynomial rate variants of Homomorphism codes and any other concentrated & recoverable codes.