Generating an equidistributed net on a unit n-sphere using random rotations
Abstract
We develop a randomized algorithm (that succeeds with high probability) for generating an $\epsilon$-net in a sphere of dimension n. The basic scheme is to pick $O(n \ln(1/n) + \ln(1/\delta))$ random rotations and take all possible words of length $O(n \ln(1/\epsilon))$ in the same alphabet and act them on a fixed point. We show this set of points is equidistributed at a scale of $\epsilon$. Our main application is to approximate integration of Lipschitz functions over an n-sphere.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2018
- DOI:
- 10.48550/arXiv.1812.01845
- arXiv:
- arXiv:1812.01845
- Bibcode:
- 2018arXiv181201845C
- Keywords:
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- Mathematics - Probability;
- Computer Science - Computational Geometry
- E-Print:
- doi:10.1007/s00454-021-00333-0