The Shannon entropy is a widely used summary statistic, for example, network traffic measurement, anomaly detection, neural computations, spike trains, etc. This study focuses on estimating Shannon entropy of data streams. It is known that Shannon entropy can be approximated by Reenyi entropy or Tsallis entropy, which are both functions of the p-th frequency moments and approach Shannon entropy as p->1. Compressed Counting (CC) is a new method for approximating the p-th frequency moments of data streams. Our contributions include: 1) We prove that Renyi entropy is (much) better than Tsallis entropy for approximating Shannon entropy. 2) We propose the optimal quantile estimator for CC, which considerably improves the previous estimators. 3) Our experiments demonstrate that CC is indeed highly effective approximating the moments and entropies. We also demonstrate the crucial importance of utilizing the variance-bias trade-off.
@cite_30 applied symmetric stable random projections to approximate the moments and Shannon entropy. The nice theoretical work @cite_8 @cite_5 provided the criterion to choose the @math so that Shannon entropy can be approximated with a guaranteed accuracy, using the @math th frequency moment.
@cite_30 @cite_5 @cite_8
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We give a method for estimating the empirical Shannon entropy of a distribution in the streaming model of computation. Our approach reduces this problem to the well-studied problem of estimating frequency moments. The analysis of our approach is based on new results which establish quantitative bounds on the rate of convergence of Renyi entropy towards Shannon entropy. Let F(x) be an absolutely continuous distribution having a density function f(x) with respect to the Lebesgue measure. The Shannon entropy is defined as H(f) = - f(x) f(x) dx . In this correspondence we propose, based on a random sample X_ 1 , , X_ n generated from F , a nonparametric estimate of H(f) given by H (f) = -(l n) i = 1 ^ n f (x) , where f (x) is the kernel estimate of f due to Rosenblatt and Parzen. Regularity conditions are obtained under which the first and second mean consistencies of H (f) are established. These conditions are mild and easily satisfied. Examples, such as Gamma, Weibull, and normal distributions, are considered. We give near-optimal sketching and streaming algorithms for estimating Shannon entropy in the most general streaming model, with arbitrary insertions and deletions. This improves on prior results that obtain suboptimal space bounds in the general model, and near-optimal bounds in the insertion-only model without sketching. Our high-level approach is simple: we give algorithms to estimate Tsallis entropy, and use them to extrapolate an estimate of Shannon entropy. The accuracy of our estimates is proven using approximation theory arguments and extremal properties of Chebyshev polynomials. Our work also yields the best-known and near-optimal additive approximations for entropy, and hence also for conditional entropy and mutual information. We study the problem of sampling a random signal with sparse support in frequency domain. Shannon famously considered a scheme that instantaneously samples the signal at equispaced times. He proved that the signal can be reconstructed as long as the sampling rate exceeds twice the bandwidth (Nyquist rate). Candes, Romberg, Tao introduced a scheme that acquires instantaneous samples of the signal at random times. They proved that the signal can be uniquely and efficiently reconstructed, provided the sampling rate exceeds the frequency support of the signal, times logarithmic factors. In this paper we consider a probabilistic model for the signal, and a sampling scheme inspired by the idea of spatial coupling in coding theory. Namely, we propose to acquire non-instantaneous samples at random times. Mathematically, this is implemented by acquiring a small random subset of Gabor coefficients. We show empirically that this scheme achieves correct reconstruction as soon as the sampling rate exceeds the frequency support of the signal, thus reaching the information theoretic limit.