Consider two problems about an unknown probability distribution $p$: 1. How many samples from $p$ are required to test if $p$ is supported on $n$ elements or not? Specifically, given samples from $p$, determine whether it is supported on at most $n$ elements, or it is "$ε$-far" (in total variation distance) from being supported on $n$ elements. 2. Given $m$ samples from $p$, what is the largest lower bound on its support size that we can produce? The best known upper bound for problem (1) uses a general algorithm for learning the histogram of the distribution $p$, which requires $Θ(\tfrac{n}{ε^2 \log n})$ samples. We show that testing can be done more efficiently than learning the histogram, using only $O(\tfrac{n}{ε\log n} \log(1/ε))$ samples, nearly matching the best known lower bound of $Ω(\tfrac{n}{ε\log n})$. This algorithm also provides a better solution to problem (2), producing larger lower bounds on support size than what follows from previous work. The proof relies on an analysis of Chebyshev polynomial approximations outside the range where they are designed to be good approximations, and the paper is intended as an accessible self-contained exposition of the Chebyshev polynomial method.

翻译：考虑关于未知概率分布 $p$ 的两个问题：1. 需要从 $p$ 中抽取多少样本，才能检验 $p$ 是否支撑在 $n$ 个元素上？具体而言，给定从 $p$ 中抽取的样本，判断其支撑集大小是否至多为 $n$ 个元素，或者其与支撑在 $n$ 个元素上的分布之间的总变差距离至少为 $ε$（即“$ε$-远离”）。2. 给定从 $p$ 中抽取的 $m$ 个样本，我们能对其支撑集大小给出多大的下界？问题 (1) 的最佳已知上界使用了一种学习分布 $p$ 直方图的通用算法，该算法需要 $Θ(\tfrac{n}{ε^2 \log n})$ 个样本。我们证明，测试可以比学习直方图更高效地完成，仅需 $O(\tfrac{n}{ε\log n} \log(1/ε))$ 个样本，这几乎匹配已知最佳下界 $Ω(\tfrac{n}{ε\log n})$。该算法也为问题 (2) 提供了更好的解决方案，相比于先前工作，能够给出更大的支撑集大小下界。证明依赖于对切比雪夫多项式在其设计为良好近似范围之外的近似性质的分析，本文旨在作为切比雪夫多项式方法的自包含且易读的综述性阐述。