Constructive quantization: approximation by empirical measures

Dereich, Steffen; Scheutzow, Michael; Schottstedt, Reik

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1183-1203.

Dans cet article, nous étudions l’approximation d’une mesure de probabilité $\mu$ sur ${ℝ}^d$ par sa mesure empirique ${\widehat{\mu }}_N$, interprétée comme quantification aléatoire. Comme critère d’erreur, nous considérons une moyenne de métrique de Wasserstein d’ordre $p$. Dans le cas $2p<d$, nous établissons des bornes supérieures et inférieures améliorées pour l’erreur, une formule haute résolution. De plus, nous donnons une estimation universelle à base de moments, nomméee estimation du type Pierce. En particulier, nous prouvons que, sous de faibles hypothèses, la quantification par des mesures empiriques est d'ordre optimal.

In this article, we study the approximation of a probability measure $\mu$ on ${ℝ}^d$ by its empirical measure ${\widehat{\mu }}_N$ 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 fine upper and lower bounds for the error, a high resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

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