Sketching and Streaming Entropy via Approximation Theory

TL;DR

This paper introduces near-optimal sketching and streaming algorithms for estimating Shannon entropy in general streaming models, improving space bounds and employing approximation theory techniques with Chebyshev polynomials.

Contribution

It presents the first near-optimal algorithms for entropy estimation in general streaming models, using approximation theory to improve accuracy and space efficiency.

Findings

Achieves near-optimal space bounds for entropy estimation

Provides the best-known additive approximations for entropy and related measures

Introduces a novel use of Chebyshev polynomials in approximation algorithms

Abstract

We conclude a sequence of work by giving 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 Renyi and 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, a technique which may be useful for other problems. Our work also yields the best-known and near-optimal additive approximations for entropy, and hence also for conditional entropy and mutual information.