Correctness and Discrepancy Metric (CDM)

• The paper introduces the CDM framework that unifies L^p-discrepancy theory with worst-case numerical integration error using Marcinkiewicz–Zygmund inequalities.
• CDM is defined as the sum of the worst-case integration error in function spaces and the discrepancy measure, providing explicit performance estimates for point sets.
• Applications on spheres, manifolds, and doubling spaces demonstrate CDM’s practical role in achieving optimal sampling and error rate analysis.

The Correctness and Discrepancy Metric (CDM) provides a comprehensive framework for evaluating point distributions in metric measure spaces, simultaneously quantifying both their uniformity with respect to a given family of test sets and their accuracy in numerical integration tasks. Developed by Brandolini, Chen, Colzani, Gigante, and Travaglini, the CDM paradigm systematically unifies $L^p$-discrepancy theory and worst-case numerical integration error over function classes defined by potentials or Besov-type regularity, employing the Marcinkiewicz–Zygmund (MZ) inequality as its central analytical foundation (Brandolini et al., 2013). The versatility of this approach enables sharp, universal bounds for point sets in general metric measure spaces—including spheres, manifolds, and spaces with doubling measures—by leveraging the geometry of partitions and the smoothness of testing or integration functionals.

1. Foundational Definitions

Given a metric measure space $(M, d, \mu)$, with $M$ equipped with a distance $d$ and a finite positive Borel measure $\mu$, the study assumes that $M$ admits, for each integer $N$, a partition $M = \bigsqcup_{j=1}^N X_j$ into $N$ measurable “cells” such that each $X_j$ has controlled mass and small diameter:

• $\mu(X_j) = w_j \simeq 1/N$
• $$\operatorname{diam} X_j = \delta_j \lesssim N^{-1/d}$$

Let $P = \{x_1, \ldots, x_N\}$ denote a weighted point set with weights $w_j > 0$.

$L^p$-Discrepancy

For a family $\mathcal{A}$ of measurable test-sets in $M$ (examples: metric balls, convex bodies), the local discrepancy for $A\in \mathcal{A}$ is

$$D[P, A] = \sum_{j=1}^N w_j \mathbb{1}_A(x_j) - \mu(A)$$

The $L^p$-discrepancy of $P$ relative to $\mathcal{A}$ (and possibly base measure $\nu$ on $\mathcal{A}$) is

$$\operatorname{disc}_p(P; \mathcal{A}) = \left( \int_{A \in \mathcal{A}} |D[P, A]|^p\, d\nu(A) \right)^{1/p}$$

Numerical Integration Error

For any integrable $f : M \to \mathbb{R}$, the quadrature error of $P$ with weights $w_j$ is defined as

$$E(f, P) = \sum_{j=1}^N w_j f(x_j) - \int_M f(x)\, d\mu(x)$$

The Correctness–Discrepancy Metric

The CDM unifies discrepancy and worst-case integration error. For a Banach function space $H$ (for example, a potential or Besov space),

$$\Vert E(\cdot, P) \Vert_{H^*} = \sup_{f \in H,\, \Vert f \Vert_H \leq 1} |E(f, P)|$$

A scalar CDM is defined by

$$\operatorname{CDM}(P; H, \mathcal{A}, p) := \Vert E(\cdot, P) \Vert_{H^*} + \operatorname{disc}_p(P; \mathcal{A})$$

or, equivalently, the pair $$(\Vert E(\cdot, P) \Vert_{H^*},\, \operatorname{disc}_p(P; \mathcal{A}))$$ assesses “correctness” and “discrepancy” separately.

2. Universal Inequalities and the MZ Principle

Central to the analysis is the Marcinkiewicz–Zygmund (MZ) inequality: for independent random variables with zero mean, and $1 < p < \infty$, there exist constants $A(p), B(p) > 0$ such that

$$A(p) \left( \sum_j \mathbb{E}\, |X_j|^2 \right)^{p/2} \leq \mathbb{E} \left| \sum_j X_j \right|^{p} \leq B(p) \left( \sum_j \mathbb{E}\, |X_j|^2 \right)^{p/2}$$

This foundation allows one to relate statistical moments to aggregated deviations in point sampling.

For the integration error $E(f, P)$, under mild integrability and smoothness hypotheses for the kernel $\varphi$, one sets $$X_j = w_j f(x_j) - \mathbb{E}_{x_j \in X_j}[w_j f(x_j)]$$ and obtains, for all $1 \leq p \leq \infty$,

$$\Vert E(\cdot, P) \Vert_{L^p_x} \leq B(q) \mathcal{V}_q, \qquad \geq A(q) \mathcal{V}_q, \quad (1/p + 1/q = 1)$$

where

$$\mathcal{V}_q = \left( \sum_{j=1}^N w_j \int_M |\varphi(x_j, y) - \varphi(z_j, y)|^q dy \right)^{1/q}$$

For typical geometric kernels $\varphi(x, y) \simeq d(x, y)^{a-d}$ (Riesz/Bessel), one achieves

$$\Vert E(\cdot, P) \Vert_{L^p} \lesssim \begin{cases} N^{-a/d}, & a < d/2 + 1 \ N^{-1/2 - 1/d} (\log N)^{1/2}, & a = d/2 + 1 \ N^{-1/2 - 1/d}, & a > d/2 + 1 \end{cases}$$

These provide optimal rates, with matching lower bounds under non-degeneracy conditions.

3. Function Spaces for Measuring Correctness

Potential Spaces $H_\varphi(M)$

Given a measurable kernel $\varphi(x, y)$ satisfying appropriate $L^q$ conditions, the potential space $H_\varphi(M)$ consists of functions expressible as

$f(x) = \int_M \varphi(x, y) g(y) d\mu(y)$

with $g \in L^p(M)$, $1/p + 1/q = 1$. The quasi-norm is

$$\Vert f \Vert_{H_\varphi} := \inf\{ \Vert g \Vert_{L^p} : f = \varphi * g \}$$

Two standard choices:

• On $\mathbb{R}^d$: $\varphi(x, y) = |x - y|^{a-d}$ yields the homogeneous Sobolev space $\dot{H}^a_p$.
• On a compact manifold: Bessel kernel $(1 + \Delta)^{-a/2}$.

Besov–Triebel–Lizorkin Spaces

For integrand error estimation, the (homogeneous) Hajłasz–Besov space $\dot{B}^\sigma_{p,\infty}(M)$ and Triebel–Lizorkin $\dot{F}^\sigma_{p,2}(M)$, defined via scales of “$p$-gradients” $\{g_j\}$ localized to dyadic scales $2^{-j}$, quantify function regularity:

• $f \in \dot{B}^\sigma_{p,\infty}(M)$ if $\sup_j 2^{j\sigma} \|g_j\|_{L^p} < \infty$
• $f \in \dot{F}^\sigma_{p,2}(M)$ if $$\| (\sum_j (2^{j\sigma} g_j)^2 )^{1/2} \|_{L^p} < \infty$$

Obtained bounds include:

$$\|E(\cdot, P)\|_{L^p_x} \lesssim N^{1/p-1-\sigma/d} \|f\|_{\dot{B}^\sigma_{p,\infty}} \quad (\sigma<1),$$

as well as specific scaling rates for $1 < p \leq 2$ and $2 \leq p < \infty$.

4. Existence and Construction of Good Point Sets

Stratified Random Sampling

Partitioning $M$ as above, and choosing $x_j$ uniformly at random in $X_j$, one shows that for each fixed $A$,

$$\mathbb{E}|D[P, A]|^p \lesssim N^{-3p/4} \quad (\text{p even integer}),$$

and more generally (for $1 < p < \infty$),

$$\mathbb{E}|D[P, A]|^p \lesssim N^{-p/2 - 1/(2d)} (\operatorname{diam} X_j)^{-d/p}.$$

Integrating over $A$ in $\mathcal{A}$ yields:

$$\mathbb{E} \operatorname{disc}_p(P; \mathcal{A}) \lesssim N^{-1/2 - \beta/(2d)}$$

where $\beta$ quantifies the boundary regularity: $\mathcal{V}_B(t) \lesssim t^\beta$.

Existence Results and VC Theory

Expectational bounds imply actual existence of point sets achieving these rates. For families $\mathcal{A}$ with finite VC-dimension, supremum discrepancy rates (for $p = \infty$) hold up to logarithmic factors.

5. Concrete Instances and Applications

Spheres, Manifolds, and Doubling Spaces

• On $S^d$, with $\mathcal{A} = \{$spherical caps$\}$ and $\beta = d - 1$, one recovers the Beck bound $N^{-1/2-1/(2d)}(\log N)^{1/2}$.
• On compact Riemannian manifolds, $\mathcal{A} = \{$geodesic balls$\}$, again with $\beta = d - 1$.
• In Ahlfors–regular metric spaces with boundary-regular families, the same exponents apply.

QMC-Designs and Optimality

For kernels of the form $\varphi =$ Bessel kernel on a compact manifold, the worst-case integration error in $H_\varphi(M)$ for equal-weight $N$-point rules decays as $N^{-a/d}$ when $a < d/2 + 1$ and $p=2$, giving rise to the concept of “QMC-designs of strength $a$.” This optimal rate, and its non-Euclidean analogues, hold for $p \neq 2$ and for general metric measure spaces.

6. Synthesis: The CDM Paradigm in Practice

The Correctness and Discrepancy Metric formalizes the trade-off between the two cardinal qualities of point sets:

• Discrepancy: $\operatorname{disc}_p(P; \mathcal{A})$ measures empirical measure uniformity across test sets. For properly stratified random samples, $\operatorname{disc}_p \sim N^{-1/2 - \beta/(2d)}$ is achievable.
• Correctness: $\Vert E(\cdot, P) \Vert_{H^*}$ captures the worst-case integration error for a class of integrands $H$, typically behaving as $N^{-a/d}$ or $N^{-1/2-1/d}(\log N)^{1/2}$, determined by the smoothness of the kernel.
• Interconnection: MZ-type inequalities ensure that two-sided $p$-norm bounds of the form $A(q) V_q \leq \|E(\cdot, P)\|_{L^p} \leq B(q) V_q$ hold, with $V_q$ quantifying cell-wise kernel variation.

The implementation of CDM is context-dependent: for a family $\mathcal{A}$ with boundary-regularity $\beta$, one chooses $P$ to achieve $$\operatorname{disc}_p(P; \mathcal{A}) \lesssim N^{-1/2-\beta/(2d)}$$; for a Sobolev-class with smoothness $a$, the target is $N^{-a/d}$. The CDM framework delivers both a fundamental analysis of randomized (or deterministic) point sets and explicit benchmarks for optimal sampling in numerical integration over general metric measure spaces (Brandolini et al., 2013).