Random $\epsilon$-Cover on Compact Symmetric Space

Abstract

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.