Constructive quantization: Approximation by empirical measures

Abstract

In this article, we study the approximation of a probability measure µ on R d by its empirical measureμN 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.

On Optimal Matching of Gaussian Samples

“…For Gaussian samples, it is extended up to p < 2 in dimension 3 in [8]. It is also known from the works [6,7] that…”

“…The result extends (1.6) from p < d 2 to p < d. This might look as only a small step, but it overcomes the 1 √ n rate and, as the proof will amply demonstrate, the amount of work to reach this conclusion is rather significant. Due to the results in [6,7,8] when 1 ≤ p < 2 mentioned above, only the values 2 ≤ p < d have to be considered.…”

Iterative multilevel particle approximation for McKean–Vlasov SDEs

“…. Consequently, following [16] or [15], the rate of convergence in the number of samples N deteriorates as the dimension d increases. We also refer the reader to recent works [1,37,23] that study the problem from the perspective of Monge-Ampére PDEs.…”

Behavior of the empirical Wasserstein distance in ${\mathbb R}^d$ under moment conditions

“…There are many ways to see that the upper bounds obtained in the present paper are optimal in some sense, by considering the special cases d = 1, p = 1, p = 2, or by following the general discussion in [16], and we shall make some comments about this question all along the paper. However, the optimality for large d is only a kind of minimax optimality: one can see that the rates are exact for compactly supported measures which are not singular with respect to the Lebesgue measure on R d (by using, for instance, Theorem 2 in [12]).…”