Random ε-Cover on Compact Riemannian Symmetric Space

Somnath Chakraborty¹

¹Fakultät für Mathematik, Ruhr-Universität Bochum, Bochum, Germany

Journal of Lie Theory, Tome 34 (2024) no. 1, pp. 137-169

Résumé

A randomized scheme that succeeds with probability $1-2\delta$ (for any $\delta>0$) has been devised to construct (1) an equidistributed $\epsilon$-cover, and (2) an approximate $(\lambda_r,2)$-design -- in a compact Riemannian symmetric space $\mathbb M$ of dimension $d_{\mathbb M}$ -- using $n(\epsilon,\delta)$-many Haar-random isometries of $\mathbb M$, where $$ n(\epsilon,\delta):={\mathcal O}_{\mathbb M} [d_{\mathbb M} (\ln (1/\epsilon) + \log_2 (1/\delta) ) ]\,, $$ and $\lambda_r=\mathcal O_{\mathbb M} (\epsilon^{-1-\frac{d_{\mathbb M}}2})$ 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 $\tilde {\mathcal O}_{\mathbb M}(\epsilon)$-error, as well as in comparing persistence homology computed from data cloud to that of a hypothetical data cloud sampled from the uniform measure.

Détail

Classification : 43A85, 53C30, 68W20
Mots-clés : Symmetric space, epsilon-cover, (lambda,2)-design, equidistributed cover, random isometries, Wasserstein distance, irreducible representations, Casimir operator, Laplace-Beltrami operator, Schrier graph, expander, spectral gap, Markov chain