Constructive quantization: Approximation by empirical measures

By Steffen Dereich, M. Scheutzow, Reik Schottstedt. 2011

227 citations recorded by the source corpus.

In this article, we study the approximation of a probability measure $\mu$ on $\mathbb{R}^{d}$ by its empirical measure $\hat{\mu}_{N}$ interpreted as a random quantization. As error criterion we consider an averaged $p$-th moment Wasserstein metric. In the case where $2p<d$, we establish refined upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a so-called Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

Subfields: Applied Mathematics, Finance, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/aaad392ac9e3bd74c7a3d5f966258188848b99c7

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In Applied Mathematics

In Finance

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Citations in the corpus

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, background): “4 below) relies on a sharp upper bound of W2 obtained in [19, 22] (see also [14, Section 5.”
Geometrical Insights for Implicit Generative Modeling (2017, background): “Comparable results for the 1-Wasserstein distance describe a convergence speed that quickly becomes considerably slower with the dimension d > 2 of the sample space X [50, 16, 18].”
On Optimal Matching of Gaussian Samples (2019, background): “Furthermore, for irregular laws, the decay can be faster (see [7, 15]), but we do not address this issue here.”
On the rate of convergence in Wasserstein distance of the empirical measure (2013, unknown): “On the contrary, the method of [16], exposed in details in the next section, is extremely simple, robust, and leads to the almost optimal results exposed here.”
The Master Equation and the Convergence Problem in Mean Field Games (2015, methodology): “For proving (ii), we use the the Lipschitz continuity of U and a result by Dereich, Scheutzow and Schottstedt [23] to deduce that, for d ě 3 and for any xi P Td, ˆ TdpN´1q |uN,ipt,xq ´ Upt, xi,m0q| ź j‰i m0pdxjq “ ˆ TdpN´1q |Upt, xi,mN,ix q ´ Upt, xi,m0q| ź j‰i m0pdxjq ď C ˆ TdpN´1q d1pmN,ix ,m0q ź ”
Weak quantitative propagation of chaos via differential calculus on the space of measures (2019, background): “pect to the Wasserstein distance, i.e, there exists a constant C > 0 such that |Φ(µ) −Φ(ν)| ≤ CW2(µ,ν), ∀µ,ν ∈ P2(Rd), one could bound |Φ(µ) − EΦ(µN)| by EW2(µ,µN). Consequently, following [15] or [14], the rate of convergence in the number of samples N deteriorates as the dimension d increases. ”
Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance (2017, methodology): “Nevertheless, the techniques employed in [16,21] are very similar to those employed in [7], and we follow the same approach.”
Convergence and concentration of empirical measures under Wasserstein distance in unbounded functional spaces (2018, methodology): “The following lemma is due to [12], which is the building block for the construction of optimal transport from μ to μ̂.”