Quantization dimension via quantization numbers.

Abstract

We give a characterization of the quantization dimension of Borel probability measures on $\mathbb{R}^{d}$ in terms of $\epsilon$-quantization numbers. Using this concept, we show that the upper rate distortion dimension is not greater than the upper quantization dimension of order one. We also prove that the upper quantization dimension of a product measure is not greater than the sum of that of its marginals. Finally, we introduce the notion of the $\epsilon$-essential radius for a given measure to construct an upper bound for its quantization dimension.