New Weak Error bounds and expansions for Optimal Quantization

Vincent Lemaire; Thibaut Montes; Gilles Pagès

doi:10.1016/j.cam.2019.112670

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arXiv · 1903.10330

New Weak Error bounds and expansions for Optimal Quantization

Vincent Lemaire·Thibaut Montes·Gilles Pagès

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Abstract

We propose new weak error bounds and expansion in dimension one for optimal quantization-based cubature formula for different classes of functions, such that piecewise affine functions, Lipschitz convex functions or differentiable function with piecewise-defined locally Lipschitz or $\alpha$-H\"older derivatives. This new results rest on the local behaviors of optimal quantizers, the $L^r$-$L^s$ distribution mismatch problem and Zador's Theorem. This new expansion supports the definition of a Richardson-Romberg extrapolation yielding a better rate of convergence for the cubature formula. An extension of this expansion is then proposed in higher dimension for the first time. We then propose a novel variance reduction method for Monte Carlo estimators, based on one dimensional optimal quantizers.

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Vincent Lemaire, Thibaut Montes, Gilles Pagès. 2019-03-25. New Weak Error bounds and expansions for Optimal Quantization. https://doi.org/10.1016/j.cam.2019.112670

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