Quantization for probability measures with respect to the geometric mean error

Abstract

Consider $e_n = \inf \exp \int \log \parallel x - f(x) \parallel dP(x)$, where $p$ is a probability measure on $\real^d$ and the infimum is taken over all measurable maps $f{:}\ \real^d \rightarrow \real^d$ with $| f(\real^d)| \leq n$. We study solutions $f$ of this minimization problem. For absolutely continuous distributions and for self-similar distributions we derive the exact rates of convergence to zero of the $n$th quantization error $e_n$ as $n \rightarrow \infty$. We establish a relationship between the quantization dimension that rules the rates and the Hausdorff dimension of $P$.