Results 1 
8 of
8
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) ..."
Abstract

Cited by 84 (28 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 49 (11 self)
 Add to MetaCart
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 p ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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.
Constructions of lowdegree and errorcorrecting ɛbiased generators
 Proceedings of the 21st Annual IEEE Conference on Computational Complexity (CCC
, 2006
"... In this work we give two new constructions of ɛbiased generators. Our first construction significantly extends a result of Mossel et al. (Random Structures and Algorithms 2006, pages 5681), and our second construction answers an open question of Dodis and Smith (STOC 2005, pages 654663). In parti ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
In this work we give two new constructions of ɛbiased generators. Our first construction significantly extends a result of Mossel et al. (Random Structures and Algorithms 2006, pages 5681), and our second construction answers an open question of Dodis and Smith (STOC 2005, pages 654663). In particular we obtain the following results: 1. For every k = o(log n) we construct an ɛbiased generator G: {0, 1} m → {0, 1} n that is implementable by degree k polynomials (namely, every output bit of the generator is a degree k polynomial in the input bits). For any constant k we get that n = Ω(m / log(1/ɛ)) k, which is nearly optimal. Our result also separates degree k generators from generators in NC0 k, showing that the stretch of the former can be much larger than the stretch of the latter. The problem of constructing degree k generators was introduced by Mossel et al. who gave a construction only for the case of k = 2. 2. We construct a family of asymptotically good binary codes such that the codes in our family are also ɛbiased sets for an exponentially small ɛ. Our encoding algorithm runs in polynomial time in the block length of the code. Moreover, these codes have a polynomial time decoding algorithm. This answers an open question of Dodis and Smith. The paper also contains an appendix by Venkatesan Guruswami that provides an explicit construction of a family of error correcting codes of rate 1/2 that has efficient encoding and decoding algorithms and whose dual codes are also good codes. 1
Robust PCPs of Proximity, Shorter PCPs
, 2004
"... Abstract We continue the study of the tradeoff between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size n): 1. We present PCPs of length exp ( ~O(log log n)2) \Delta n that can be verified by making o ..."
Abstract
 Add to MetaCart
Abstract We continue the study of the tradeoff between the length of PCPs and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size n): 1. We present PCPs of length exp ( ~O(log log n)2) \Delta n that can be verified by making o(log log n) Boolean queries. 2. For every &quot; ? 0, we present PCPs of length exp(log &quot; n) \Delta n that can be verified by making a constant number of Boolean queries. In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasipolylogarithmic in the first case (of query complexity o(log log n)), and 2(log n), for any &quot; ? 0, in the second case (of constant query complexity). In contrast, previous results required at least 2 plog n overhead in the length, even to get query complexity 2 plog n
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
 Add to MetaCart
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.