We derandomize G. Valiant's [J. ACM 62 (2015) Art. 13] subquadratic-time algorithm for finding outlier correlations in binary data. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant's randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders in Reingold, Vadhan, and Wigderson [Ann. of Math. 155 (2002) 157--187]. We say that a function @math is a correlation amplifier with threshold @math , error @math , and strength @math an even positive integer if for all pairs of vectors @math it holds that (i) @math implies @math ; and (ii) @math implies @math .
cor:lightbulb extends to parity functions of larger (constant) weight in the presence of noise (cf. @cite_24 @cite_15 @cite_27 ). This generalized version of the problem is as follows.
@cite_24 @cite_27 @cite_15
2026103826 1983749341 2170021087 1915272284
We address well-studied problems concerning the learnability of parities and halfspaces in the presence of classification noise. Learning of parities under the uniform distribution with random classification noise, also called the noisy parity problem is a famous open problem in computational learning. We reduce a number of basic problems regarding learning under the uniform distribution to learning of noisy parities. We show that under the uniform distribution, learning parities with adversarial classification noise reduces to learning parities with random classification noise. Together with the parity learning algorithm of [5], this gives the first nontrivial algorithm for learning parities with adversarial noise. We show that learning of DNF expressions reduces to learning noisy parities of just logarithmic number of variables. We show that learning of k-juntas reduces to learning noisy parities of k variables. These reductions work even in the presence of random classification noise in the original DNF or junta. We then consider the problem of learning halfspaces over Q ^n with adversarial noise or finding a halfspace that maximizes the agreement rate with a given set of examples. We prove an essentially optimal hardness factor of 2- , improving the factor of 85 84 - due to Bshouty and Burroughs [8]. Finally, we show that majorities of halfspaces are hard to PAC-learn using any representation, based on the cryptographic assumption underlying the Ajtai-Dwork cryptosystem. Applications of compressed sensing motivate the possibility of using different operators to encode and decode a signal of interest. Since it is clear that the operators cannot be too different, we can view the discrepancy between the two matrices as a perturbation. The stability of l 1 -minimization and greedy algorithms to recover the signal in the presence of additive noise is by now well-known. Recently however, work has been done to analyze these methods with noise in the measurement matrix, which generates a multiplicative noise term. This new framework of generalized perturbations (i.e., both additive and multiplicative noise) extends the prior work on stable signal recovery from incomplete and inaccurate measurements of Candes, Romberg and Tao using Basis Pursuit (BP), and of Needell and Tropp using Compressive Sampling Matching Pursuit (CoSaMP). We show, under reasonable assumptions, that the stability of the reconstructed signal by both BP and CoSaMP is limited by the noise level in the observation. Our analysis extends easily to arbitrary greedy methods. The authors derive the Cramer-Rao lower bound (CRLB) for complex signals with constant amplitude and polynomial phase, measured in additive Gaussian white noise. The exact bound requires numerical inversion of an ill-conditioned matrix, while its O(N sup -1 ) approximation is free of matrix inversion. The approximation is tested for several typical parameter values and is found to be excellent in most cases. The formulas derived are of practical value in several radar applications, such as electronic intelligence systems (ELINT) for special pulse-compression radars, and motion estimation from Doppler measurements. Consequently, it is of interest to analyze the best possible performance of potential estimators of the phase coefficients, as a function of signal parameters, the signal-to-noise ratio, the sampling rate, and the number of measurements. This analysis is carried out. > We consider the problem of learning sparse parities in the presence of noise. For learning parities on r out of n variables, we give an algorithm that runs in time poly (log 1 δ, 1 1-2η)n(1+(2η)2+o(1))r 2 and uses only r log(n δ)ω(1) (1-2η)2 samples in the random noise setting under the uniform distribution, where η is the noise rate and δ is the confidence parameter. From previously known results this algorithm also works for adversarial noise and generalizes to arbitrary distributions. Even though efficient algorithms for learning sparse parities in the presence of noise would have major implications to learning other hypothesis classes, our work is the first to give a bound better than the brute-force O(nr). As a consequence, we obtain the first nontrivial bound for learning r-juntas in the presence of noise, and also a small improvement in the complexity of learning DNF, under the uniform distribution.