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New Weak Error bounds and expansions for Optimal Quantization

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 α-Hölder 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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Contributor : Thibaut Montes <>
Submitted on : Friday, May 1, 2020 - 12:29:50 PM
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Vincent Lemaire, Thibaut Montes, Gilles Pagès. New Weak Error bounds and expansions for Optimal Quantization. Journal of Computational and Applied Mathematics, Elsevier, In press, 371, pp.112670. ⟨10.1016/⟩. ⟨hal-02361644v3⟩



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