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.
@article{JOLT_2024_34_1_a6,
author = {S. Chakraborty},
title = {Random {\ensuremath{\varepsilon}-Cover} on {Compact} {Riemannian} {Symmetric} {Space
}},
journal = {Journal of Lie Theory},
pages = {137--169},
year = {2024},
volume = {34},
number = {1},
doi = {10.5802/jolt.1330},
zbl = {1554.43010},
language = {en},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1330/}
}
S. Chakraborty. Random ε-Cover on Compact Riemannian Symmetric Space. Journal of Lie Theory, Volume 34 (2024) no. 1, pp. 137-169. doi: 10.5802/jolt.1330
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