Absmean Quantization Function

• The absolute–mean quantization function is a framework that measures the discrepancy between a probability distribution and finite codebooks using the L1–norm, underpinning both static identifiability and dynamic convergence results.
• It leverages the convexity of the quantization error and is closely linked with Voronoi diagram constructions, ensuring robust geometric and probabilistic analyses.
• The framework has practical implications for designing efficient quantization schemes and facilitates convergence assessments via equivalence with the Wasserstein distance.

The absolute–mean quantization function, also called the $L^1$–quantization error function, is a functional framework for quantifying the discrepancy between a probability distribution $\mu$ on $\mathbb{R}^d$ and finite sets (called codebooks) under an arbitrary norm. Its key significance lies in characterizing probability distributions and convergence in the Wasserstein distance, with both static (identifiability) and dynamic (convergence) results. The concept is grounded in quantization theory, geometric constructions—especially Voronoi diagrams—and is central in the work of Liu and Pagès (Liu et al., 2018).

1. Definition and Formal Properties

Let $(\mathbb{R}^d, |\cdot|)$ be equipped with any norm. For $\mu \in \mathcal{P}_p(\mathbb{R}^d)$ (measures possessing finite $p$-th moment), and a measurable quantizer $q: \mathbb{R}^d \to \{x_1, \ldots, x_N\}$, the general $L^p$–quantization error is defined as

$$\|X - q(X)\|_p = \left( \mathbb{E}_{X \sim \mu} [|X - q(X)|^p] \right)^{1/p} .$$

The minimal quantization error over all codebooks $\Gamma$ of size $\le N$ is

$$e_{N,p}(\mu) = \inf_{|\Gamma| \le N} \left( \int_{\mathbb{R}^d} \min_{a \in \Gamma} |\xi - a|^p \, \mu(d\xi) \right)^{1/p}.$$

Specializing to $p=1$, the absolute–mean quantization error at level $N$ reads: $$e_{N,1}(\mu) = \inf_{|\Gamma| \le N} \int_{\mathbb{R}^d} \min_{a \in \Gamma} |\xi - a| \, \mu(d\xi).$$ For $N=1$, $$e_{1,1}(\mu; a) = \int_{\mathbb{R}^d} |\xi - a| \, \mu(d\xi)$$, with minimizer $a$ being any geometric median of $\mu$.

2. Analytical Expressions and Convexity

For a single code point $a \in \mathbb{R}^d$: $e_{1,1}(\mu; a) = \int |\xi - a| \, \mu(d\xi),$ and, when the Euclidean norm is used, its gradient is

$$\nabla_a e_{1,1}(\mu; a) = -\int \frac{\xi - a}{|\xi - a|} \, \mu(d\xi).$$

In dimension $d=1$, $e_{1,1}(\mu; x)$ is convex, and its derivative satisfies

$(e_{1,1})'(x) = -1 + 2\,\mu((-\infty, x]),$

which immediately implies the median characterization and uniqueness for probability laws given the absolute–mean quantization function.

3. Static Characterization: Identifiability of Measures

Identifiability via the absolute–mean quantization function rests on codebook cardinality. For any norm on $\mathbb{R}^d$, define

$$c(d, |\cdot|) = \min \{ k : \exists\, a_1, \ldots, a_k \in S(0,1),\; S(0,1) \subset \cup_{i=1}^k B(a_i, 1) \},$$

the minimal number of unit balls required to cover the unit sphere. If $N \geq c(d, |\cdot|) + 1$ (in Euclidean space, $N \geq d+2$), then the following holds:

Theorem (Static Characterization)

If $e_{N,1}(\mu;\, \cdot) = e_{N,1}(\nu; \, \cdot) + C$ for some constant $C$, then $\mu = \nu$ and $C=0$.

In $d=1$, it is established that $c(1,|\cdot|)=2$, and already $N=1$ provides identifiability, leading to the proposition:

Proposition (One–Dimensional Static Characterization)

If for all $x \in \mathbb{R}$, $e_{1,1}(\mu; x) = e_{1,1}(\nu; x) + C$, then $\mu = \nu$ and $C=0$.

4. Dynamic Characterization: Wasserstein Convergence

The $L^1$–Wasserstein distance on $\mathcal{P}_1(\mathbb{R}^d)$ is denoted $\mathcal{W}_1(\mu, \nu)$. The equivalence between convergence in Wasserstein distance and quantization error convergence is formalized as follows:

Theorem ($W_1$–Convergence)

For $\mu_n, \mu \in \mathcal{P}_1(\mathbb{R}^d)$ and fixed $N \geq c(d, |\cdot|)+1$, the following are equivalent:

1. $\mathcal{W}_1(\mu_n, \mu) \to 0$,
2. $$\sup_{x \in (\mathbb{R}^d)^N} |e_{N,1}(\mu_n; x) - e_{N,1}(\mu; x)| \to 0$$,
3. Pointwise: $e_{N,1}(\mu_n; x) \to e_{N,1}(\mu; x)$ for all $x \in (\mathbb{R}^d)^N$.

In $d=1$, $N=1$ suffices, yielding the efficient criterion: $$\mathcal{W}_1(\mu_n, \mu) \to 0 \;\;\Longleftrightarrow\;\; e_{1,1}(\mu_n; x) \to e_{1,1}(\mu; x), \quad \text{pointwise or uniformly in } x.$$

5. Quantization–Based Distances and Completeness

A quantization–based distance between $\mu, \nu \in \mathcal{P}_1(\mathbb{R}^d)$ is defined by

$$\mathcal{Q}_{N,1}(\mu, \nu) := \sup_{x \in (\mathbb{R}^d)^N} |e_{N,1}(\mu;x) - e_{N,1}(\nu;x)|.$$

It is always bounded above by the Wasserstein distance: $$\mathcal{Q}_{N,1}(\mu, \nu) \le \mathcal{W}_1(\mu, \nu)$$. By the static and dynamic characterizations, $\mathcal{Q}_{N,1}$ is a bona fide distance, topologically and Lipschitz equivalent to $\mathcal{W}_1$ for $N \geq c(d, |\cdot|)+1$. In $d=1$, $N=1$ already assures that $(\mathcal{P}_1(\mathbb{R}), \mathcal{Q}_{1,1})$ is complete.

6. Geometric Underpinning via Voronoi Diagrams

For codebook $\Gamma = \{x_1, \dots, x_N\} \subset \mathbb{R}^d$, Voronoi cells are given by

$$V_i = \{ \xi \in \mathbb{R}^d : |\xi - x_i| = \min_{1 \leq j \leq N} |\xi - x_j| \}.$$

Essential geometric features:

• Each cell $V_i$ is star-shaped around $x_i$.
• Existence of a codebook $\Gamma$ such that one Voronoi cell is nonempty and bounded (using a covering argument) allows construction of functions supported in that cell that serve as approximate identities. Specifically, $$\varphi(\xi) = \min_{j \ne i} |\xi - x_j| - \min_j |\xi - x_j|$$ is nonnegative, compactly supported in $V_i^\circ$, and, when normalized, functions as an approximate identity.
• This covering construction justifies the choice $N = c(d, |\cdot|)+1$.

7. Context, Significance, and Consequences

The absolute–mean quantization error function provides a principled link between quantization theory, transportation metrics, and geometric analysis. Its static and dynamic characterizations underpin identifiability and convergence results for probability measures in the Wasserstein framework, and its geometric foundation via Voronoi diagrams ensures the robustness of these characterizations across norms and dimensions. The minimality conditions on codebook cardinality have direct consequences for practical quantization schemes and theoretical studies concerning the completeness and topological properties of statistical metric spaces (Liu et al., 2018).