A randomized scheme that succeeds with probability $1-\delta$ (for any $\delta>0$) has been devised to construct (1) an equidistributed $\epsilon$-cover of a compact Riemannian symmetric space $\mathbb M$ of dimension $d_{\mathbb M}$ and antipodal dimension $\bar{d}_{\mathbb M}$, and (2) an approximate $(\lambda_r,2)$-design, using $n(\epsilon,\delta)$-many Haar-random isometries of $\mathbb M$, where \begin{equation}n(\epsilon,\delta):=O_{\mathbb M}\left(d_{\mathbb M}\ln \left(\frac 1\epsilon\right)+\log\left(\frac 1\delta\right)\right)\,,\end{equation} and $\lambda_r$ is the $r$-th smallest eigenvalue of the Laplace-Beltrami operator on $\mathbb M$. The $\epsilon$-cover so-produced can be used to compute the integral of 1-Lipschitz functions within additive $\tilde O(\epsilon)$-error, as well as in comparing persistence homology computed from data cloud to that of a hypothetical data cloud sampled from the uniform measure.

翻译：本文设计了一种能以概率$1-\delta$（对任意$\delta>0$）成功的随机方案，用以构造：(1) 紧致黎曼对称空间$\mathbb M$（其维数为$d_{\mathbb M}$，对径维数为$\bar{d}_{\mathbb M}$）的一个均匀分布的$\epsilon$-覆盖，以及(2) 一个近似$(\lambda_r,2)$-设计。该方案利用$n(\epsilon,\delta)$个$\mathbb M$上的Haar随机等距映射，其中 \begin{equation}n(\epsilon,\delta):=O_{\mathbb M}\left(d_{\mathbb M}\ln \left(\frac 1\epsilon\right)+\log\left(\frac 1\delta\right)\right)\,,\end{equation} 而$\lambda_r$是$\mathbb M$上Laplace-Beltrami算子的第$r$小特征值。由此生成的$\epsilon$-覆盖可用于在加性$\tilde O(\epsilon)$误差范围内计算1-Lipschitz函数的积分，也可用于比较从数据云计算的持久同调与假设从均匀测度采样的数据云的持久同调。