We consider the distinct elements problem, where the goal is to estimate the number of distinct colors in an urn containing balls based on samples drawn with replacements. Based on discrete polynomial approximation and interpolation, we propose an estimator with additive error guarantee that achieves the optimal sample complexity within factors, and in fact within constant factors for most cases. The estimator can be computed in time for an accurate estimation. The result also applies to sampling without replacement provided the sample size is a vanishing fraction of the urn size.
One of the key auxiliary results is a sharp bound on the minimum singular values of a real rectangular Vandermonde matrix, which might be of independent interest.
Cite this article
Yihong Wu, Pengkun Yang, Sample complexity of the distinct elements problem. Math. Stat. Learn. 1 (2018), no. 1, pp. 37–72DOI 10.4171/MSL/1-1-2