Constrained quantization for probability distributions

Abstract

In this work, we extend the classical framework of quantization for Borel probability measures defined on normed spaces $\mathbb{R}^k$ by introducing and analyzing the notions of the $n$th constrained quantization error, constrained quantization dimension, and constrained quantization coefficient. These concepts generalize the well-established $n$th quantization error, quantization dimension, and quantization coefficient, traditionally considered in the unconstrained setting, and thereby broaden the scope of quantization theory. A key distinction between the unconstrained and constrained frameworks lies in the structural properties of optimal quantizers. In the unconstrained setting, if the support of $P$ contains at least $n$ elements, then the elements of an optimal set of $n$-points coincide with the conditional expectations over their respective Voronoi regions; this characterization does not, in general, persist under constraints. Moreover, it is known that if the support of $P$ contains at least $n$ elements, then any optimal set of $n$-points in the unconstrained case consists of exactly $n$ distinct elements. This property, however, may fail to hold in the constrained context. Further differences emerge in asymptotic behaviors. For absolutely continuous probability measures, the unconstrained quantization dimension is known to exist and equals the Euclidean dimension of the underlying space. In contrast, we show that this equivalence does not necessarily extend to the constrained setting. Additionally, while the unconstrained quantization coefficient exists and assumes a unique, finite, and positive value for absolutely continuous measures, we establish that the constrained quantization coefficient can exhibit significant variability and may attain any nonnegative value, depending critically on the specific nature of the constraint applied to the quantization process.