Core Concepts
A novel linear integration rule called control neighbors is proposed that achieves the optimal O(n^(-1/2)n^(-s/d)) convergence rate for integrating Hölder functions of regularity s over metric spaces with intrinsic dimension d, where n is the number of function evaluations.
Abstract
The content presents a novel Monte Carlo integration method called "control neighbors" that leverages nearest neighbor estimates as control variates to speed up the convergence rate of standard Monte Carlo integration.
Key highlights:
- The control neighbors estimate achieves the optimal O(n^(-1/2)n^(-s/d)) convergence rate for integrating Hölder functions of regularity s over metric spaces with intrinsic dimension d, where n is the number of function evaluations.
- This rate matches the known lower bound for integration over the unit cube with the uniform measure and Lipschitz integrands.
- The method can be applied to general metric spaces, including Riemannian manifolds like the sphere and orthogonal group, not just Euclidean spaces.
- The approach is post-hoc and can be applied after sampling the particles, independent of the sampling mechanism.
- Theoretical results include root mean squared error bounds and high-probability concentration inequalities for the proposed estimator.
- Numerical experiments validate the complexity bounds and demonstrate the good performance of the control neighbors estimator compared to standard Monte Carlo.
Stats
The content does not contain any explicit numerical data or statistics to support the key claims. The theoretical results are stated in terms of asymptotic convergence rates.