Chebyshev polynomials, moment matching, and optimal estimation of the unseen

We consider the problem of estimating the support size of a discrete distribution whose minimum non-zero mass is at least $\frac{1}{k}$. Under the independent sampling model, we show that the minimax sample complexity to achieve an additive error of $\epsilon k$ with probability at least 0.5 is within universal constant factors of $\frac{k}{\log k}\log^2\frac{1}{\epsilon} ,$ which improves the state-of-the-art result $\frac{k}{\epsilon^2 \log k}$ due to Valiant and Valiant. The optimal procedure is a linear estimator based on the Chebyshev polynomial and its approximation-theoretic properties. We also study the closely related species problem where the goal is to estimate the number of distinct colors in an urn containing $k$ balls from repeated draws. While achieving an additive error proportional to $k$ still requires $\Omega(\frac{k}{\log k})$ samples, we show that with $\Theta(k)$ samples one can strictly outperform a general support size estimator using interpolating polynomials.