Estimating Entropy of Data Streams Using Compressed Counting

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

On Practical Algorithms for Entropy Estimation and the Improved Sample Complexity of Compressed Counting

Estimating the p-th frequency moment of data stream is a very heavily studied problem. The problem is actually trivial when p = 1, assuming the strict Turnstile model. The sample complexity of our proposed algorithm is essentially O(1) near p=1. This is a very large improvement over the previously believed O(1/eps^2) bound. The proposed algorithm makes the long-standing problem of entropy estimation an easy task, as verified by the experiments included in the appendix