notion quantization originates from information theory, where it refers to the
approximation of a continuous signal on a discrete set. Our research on
quantization is mainly motivated by applications in quadrature problems. In
that context, one aims at finding for a given probability measure $\mu$ on a
metric space a discrete approximation that is supported on a finite number of
points, say $N$, and is close to $\mu$ in a Wasserstein metric.
In general it is a hard problem to find close to optimal quantizations, if $N$ is large and/or $\mu$ is given implicitly, e.g. being the marginal distribution of a stochastic differential equation. In this talk we analyse the efficiency of empirical measures in the constructive quantization problem. That means the random approximating measure is the uniform distribution on $N$ independent $\mu$-distributed elements.
We show that this approach is order order optimal in many cases. Further, we give fine asymptotic estimates for the quantization error that involve moments of the density of the absolutely continuous part of $\mu$, so called high resolution formulas. The talk ends with an outlook on possible applications and open problems.
talk is based on joint work with Michael Scheutzow (TU Berlin) and Reik
Schottstedt (U Marburg).
- Stochastic Analysis Seminar