Constrained quantization for probability distributions

TL;DR

This work extends classical quantization to constrained settings by introducing the constrained $n$th quantization error $V_{n,r}(P)$, along with constrained quantization dimension and coefficient, and demonstrates how the presence of a constraint $S$ fundamentally alters optimal configurations and asymptotics. It provides explicit results for several canonical supports, including line segments, circles, and chords, showing that optimal $n$-point sets may fail to have $n$ elements and that the optimal elements need not be conditional expectations on their Voronoi regions. The authors establish explicit formulas for optimal point placements and distortion in these geometries, and they show that the constrained dimension and coefficient can take a variety of values depending on $S$, with some limits remaining strictly positive, others approaching zero, and some remaining bounded away from zero. Overall, the paper broadens quantization theory by systematically analyzing how geometric constraints shape both finite-n and asymptotic distortion behavior, offering a framework applicable to constrained data representations.

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.

Figures (3)

• Figure 1: Support of the probability distribution $P$ is the closed interval joining the points $(a, 0)$ and $(b, 0)$; $A_i (a_i, ma_i+c)$ are the elements in an optimal set of $n$-points lying on the line $y=mx+c$ between the two points $(d, md+c)$ and $(e, me+c)$; $B_i ((m^2+1) a_i+mc, 0)$ are the points where the perpendiculars through $A_i$ on the line $y=mx+c$ intersect the support of $P$.
• Figure 2: Optimal configuration of $n$ elements for $1\leq n\leq 9$.
• Figure :

Theorems & Definitions (54)

• Proposition 1.1
• proof
• Proposition 1.2
• proof
• Definition 1.3
• Proposition 1.4
• proof
• Corollary 1.5
• Corollary 1.6
• Proposition 1.7