# New Weak Error bounds and expansions for Optimal Quantization

## TL;DR

This paper introduces new weak error bounds and expansions for optimal quantization in one dimension, improving cubature formulas and proposing variance reduction methods, with extensions to higher dimensions.

## Contribution

It presents novel weak error bounds and expansions for optimal quantization, including higher-dimensional extensions and a new variance reduction technique.

## Key findings

- Enhanced convergence rates via Richardson-Romberg extrapolation.
- Extension of expansions to higher dimensions for the first time.
- A new variance reduction method based on 1D optimal quantizers.

## 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.