Regularized Weighted Chebyshev Approximations for Support Estimation

Abstract: We introduce a new method for estimating the support size of an unknown distribution which provably matches the performance bounds of the state-of-the-art techniques in the area and outperforms them in practice. In particular, we present both theoretical and computer simulation results that illustrate the utility and performance improvements of our method. The theoretical analysis relies on introducing a new weighted Chebyshev polynomial approximation method, jointly optimizing the bias and variance components of the risk, and combining the weighted minmax polynomial approximation method with discretized semi-infinite programming solvers. Such a setting allows for casting the estimation problem as a linear program (LP) with a small number of variables and constraints that may be solved as efficiently as the original Chebyshev approximation problem. Our technique is tested on synthetic data and used to address an important problem in computational biology - estimating the number of bacterial genera in the human gut. On synthetic datasets, for practically relevant sample sizes, we observe significant improvements in the value of the worst-case risk compared to existing methods. For the bioinformatics application, using metagenomic data from the NIH Human Gut and the American Gut Microbiome Projects, we generate a list of frequencies of bacterial taxa that allows us to estimate the number of bacterial genera to approximately 2300.