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Optimal quantization for a probability measure on a nonuniform stretched Sierpiński triangle

Megha Pandey, Mrinal Kanti Roychowdhury

TL;DR

This paper analyzes optimal quantization for a Borel measure P supported on a nonuniform stretched Sierpiński triangle generated by three contractive mappings with ratios 1/4, 1/4, and 1/2 and probabilities p=(1/5, 1/5, 3/5). It derives an induction formula (Th1) to construct optimal sets of n-means for all n≥2, leveraging self-similarity through E() and E(1,2) to express distortion; it explicitly determines the optimal 2-mean and 3-mean configurations with V_2= rac{117}{1408} and V_3= rac{189}{7040}, and demonstrates how the optimal sets occupy the base subtriangles with Voronoi regions aligned to fractal structure. The framework yields a constructive, iterative procedure to obtain m(n) for increasing n, accompanied by tree-diagram representations and sample enumerations, though a closed-form formula for general n in this nonuniform setting remains unavailable. The results extend the quantization analysis to nonuniform fractal measures and suggest potential applicability to broader classes of self-similar distributions.

Abstract

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. In this paper, we have considered a Borel probability measure $P$ on $\mathbb R^2$, which has support a nonuniform stretched Sierpiński triangle generated by a set of three contractive similarity mappings on $\mathbb R^2$. For this probability measure, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.

Optimal quantization for a probability measure on a nonuniform stretched Sierpiński triangle

TL;DR

This paper analyzes optimal quantization for a Borel measure P supported on a nonuniform stretched Sierpiński triangle generated by three contractive mappings with ratios 1/4, 1/4, and 1/2 and probabilities p=(1/5, 1/5, 3/5). It derives an induction formula (Th1) to construct optimal sets of n-means for all n≥2, leveraging self-similarity through E() and E(1,2) to express distortion; it explicitly determines the optimal 2-mean and 3-mean configurations with V_2= rac{117}{1408} and V_3= rac{189}{7040}, and demonstrates how the optimal sets occupy the base subtriangles with Voronoi regions aligned to fractal structure. The framework yields a constructive, iterative procedure to obtain m(n) for increasing n, accompanied by tree-diagram representations and sample enumerations, though a closed-form formula for general n in this nonuniform setting remains unavailable. The results extend the quantization analysis to nonuniform fractal measures and suggest potential applicability to broader classes of self-similar distributions.

Abstract

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. In this paper, we have considered a Borel probability measure on , which has support a nonuniform stretched Sierpiński triangle generated by a set of three contractive similarity mappings on . For this probability measure, we investigate the optimal sets of -means and the th quantization errors for all positive integers .

Paper Structure

This paper contains 6 sections, 11 theorems, 49 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Delineation
  3. Significance of the work
  4. Basic definitions and lemmas
  5. Optimal sets of $n$-means for all $n\geq 2$
  6. Some results and observations

Key Result

Proposition 1.2

Let $\alpha$ be an optimal set of $n$-means and $a\in \alpha$. Then, $(i)$$P(M(a|\alpha))>0$, $(ii)$$P(\partial M(a|\alpha))=0$, $(iii)$$a=E(X : X \in M(a|\alpha))$, and $(iv)$$P$-almost surely the set $\{M(a|\alpha) : a \in \alpha\}$ forms a Voronoi partition of $\mathbb{R}^k$.

Figures (5)

  • Figure 1: Some basic triangles with their vertices that construct the stretched Sierpiński triangle.
  • Figure 2: Optimal configuration of $n$ points for $1\leq n\leq 6$.
  • Figure 3: Optimal configuration of $n$ points for $n=7$.
  • Figure 4: Optimal configuration of $n$ points for $n=8$.
  • Figure 5: Tree diagram of the optimal sets from $\alpha_8$ to $\alpha_{21}$.

Theorems & Definitions (23)

  • Definition 1.1
  • Proposition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • ...and 13 more