Local & Global Uniformity Scores

• Local and global uniformity scores are metrics that quantify dispersion and regularity in data, revealing structural defects and anomalies.
• Local scores focus on neighborhood-specific behavior using tests like the Kolmogorov–Smirnov statistic, while global scores aggregate these diagnostics for holistic assessment.
• These scoring methods are applied in experimental design, image quality evaluation, machine learning explanations, and anomaly detection to guide practical decisions.

Uniformity scores, encompassing both local and global variants, provide quantitative and visual diagnostics of how well points, representations, or predictions are distributed across a domain—be it a design space, a surface, a data manifold, or an outcome space. These scores assess dispersion, coverage, and regularity, revealing defects or non-uniformities that may impact statistical inference, optimization, machine learning, physical modeling, or evaluation. Local uniformity scores quantify regularity or anomaly at specific locations, subsets, or neighborhoods; global uniformity scores aggregate such measures over all relevant directions, samples, or features, capturing overarching structure or disorder.

1. Principles of Local and Global Uniformity Scores

Uniformity is a measure of how evenly distributed elements are within a domain. Local uniformity scores characterize regularity within restricted subsets, directions, or neighborhoods, enabling identification of "defects," clusters, or sparsity on a fine scale. In contrast, global uniformity scores capture aggregate regularity or irregularity across the entire domain, often by pooling local diagnostics or summarizing via a statistic sensitive to extremal deviations.

In experiment design for computer codes, for example, local uniformity may refer to the empirical cumulative distribution of projections of design points onto arbitrary lines, with the Kolmogorov–Smirnov statistic serving as the core local score at each direction (0802.2158). In machine learning, the uniformity score may reflect either the homogeneity of feature contributions to predictions across instances (local) or the overall aggregate importance (global) (Loecher et al., 2021). Other contexts include image quality (local vs. global distortion scores), manifold sampling, competitive learning topologies, and conformal prediction coverage.

Local scores are sensitive to anomalies, alignments, or clustering that would be missed by analyses restricted to margins or average statistics. Global scores, especially those defined as suprema, infima, or ratios of local scores, enable holistic assessment of space filling and uniform dispersion.

2. Methodologies for Computing Uniformity Scores

A. Projection-Based and Radar-Type Statistics

The radar-shaped uniformity statistic (0802.2158) computes, for each direction $a \in \mathbb{R}^d$, the maximum discrepancy between the empirical and theoretical cumulative distributions:

$D_N(a) = \sup_z \left| F_{N,a}(z) - F_a(z) \right|$

where $F_{N,a}$ is the ECDF of point projections onto direction $a$, and $F_a$ is the theoretical CDF under uniformity. The process is repeated for all directions, yielding a function (the "radar curve" or surface) from directions to local uniformity scores. Global uniformity is then assessed as

$G_N = \frac{\sup_{a} D_N(a)}{\inf_{a} D_N(a)}$

A high $G_N$ ratio signals severe local non-uniformity even if aggregate (marginal) projections appear well distributed. Similar approaches underpin maximal-projection uniformity tests on spheres, where statistics based on moments of the $\beta$th power of projections are maximized over all directions, with asymptotic distributions characterized via spherical harmonic expansions (Borodavka et al., 2023, García-Portugués et al., 2020).

B. Local vs. Global Distortion Measures in Vision

In full-reference image quality assessment, global uniformity is measured by saliency map correlation of the entire image (global perceptual importance), while local uniformity is quantified via RMS contrast and gradient differences within small neighborhoods (Saha et al., 2014). Final image quality scores result from saliency-weighted pooling of pixelwise local distortion maps, bridging local granularity with global perceptual structure.

C. Competitive Learning and Map Topology

Self-organizing maps require both local and global uniformity of map quality. While local updates are applied to the best-matching unit and its neighbors, global uniformity is stabilized via feedback mechanisms that adapt neighbor update strength in response to local quantization or topological errors. The learning rate for a neighbor $j$ of BMU $b$ is regulated as

$$F_j(i) = \bar{\alpha}_{b_i}(i) + f_j(i) - \bar{\alpha}_{b_i}(i) f_j(i)$$

where $f_j(i)$ captures the normalized ratio of local errors (Siddiqui et al., 2019). This balancing leads to uniform topological unfolding in the feature map even with strictly local interactions.

D. Machine Learning Explanations and Scores

Tree-based models use SHAP or Conditional Feature Contributions (CFCs) to decompose individual predictions. Local uniformity scores correspond to per-instance feature contributions, while global scores are aggregated across the dataset via summation or mean. Empirical studies reveal that for random forests, both local and global uniformity scores maintain high correlation and structure, justifying their use as proxies for predictive power (Loecher et al., 2021). In evaluation frameworks for NLP, global scores (e.g., accuracy, F1) are direct aggregations over examples, while local uniformity is probed via pairwise model comparisons; the Bradley-Terry model aggregates these comparisons into global rankings, revealing different facets of uniformity (Levtsov et al., 2 Jul 2025).

E. Adaptive Confidence Scores and Coverage

In conformal regression, global coverage guarantees require aggregate control over prediction intervals; local uniformity is enhanced by rescaling nonconformity scores using locally estimated error scales derived from calibration data:

$\sigma(X,y) = \frac{| \mu(X) - y |}{\hat{s}(X)}$

where $\hat{s}(X)$ is a Nadaraya-Watson kernel-regression estimator of the expected local error. The Jackknife+ framework ensures that local adaptivity in scores does not compromise global coverage, preserving exchangeability (Deutschmann et al., 2023).

F. Anomaly Detection: Dual Local and Global Scores

In anomaly detection, global normal scores reflect the degree of conformity to known clusters, whereas local sparsity scores quantify density (or isolation) within neighborhoods, partitioned via randomly generated subcubes. Aggregation yields a "GALScore" as

$GALScore(x) = LSS(x) - \mu \cdot GNS(x)$

where $LSS(x)$ is local sparsity and $GNS(x)$ is global normality (usually proximity to cluster centroids). Thresholding and weighting based on local/global scores improves detection even in one-class settings (Xu et al., 2023).

3. Visualization and Interpretability

The radar-type approach offers a visual means to scan uniformity defects directionally in high dimensions. 2D radar curves and 3D radar surfaces constructed via polar or spherical coordinates enable rapid identification of problematic directions, with fixed confidence-level circles or surfaces (e.g., the 95% KS threshold) superimposed for statistical rejection (0802.2158).

Heatmaps and bin histograms of sampling scores (e.g., in point cloud sampling (Wu et al., 28 Apr 2025)) are used to interpret local and global uniformity: naive high-score selection leads to clustered, non-uniform samples, whereas bin-based strategies with adaptive sampling ensure both edge detail preservation and global coverage.

Aggregation of local explanation scores (e.g., SHAP or CFCs) across the dataset can be used to assess uniformity of feature impact, guiding model trust and interpretation (Loecher et al., 2021).

4. Applications and Impact in Diverse Domains

Uniformity scores are essential in:

• Space-filling design: Assessing adequacy of experimental designs for surrogate modeling, uncertainty quantification, and variance reduction (0802.2158).
• Image and vision systems: Objective quality scores derived from local and global distortions correlate highly with subjective ratings, informing quality assessment in imaging pipelines (Saha et al., 2014).
• Anomaly detection: Integration of local and global scores in one-class settings enables robust outlier detection even when anomalies are not present in training (Xu et al., 2023).
• Manifold learning and self-organizing maps: Ensuring global topological coherence and uniformity through locally adaptive interactions and error correction improves scalability and model quality (Siddiqui et al., 2019).
• NLP evaluation: Choice between global uniformity (stable aggregated metrics) and local uniformity (pairwise comparison stability) informs benchmarking for emerging generative models (Levtsov et al., 2 Jul 2025).
• Medical and scientific imaging: Local uniformity priors (e.g., uni-gauss loss) improve downstream performance in segmentation and detection tasks by preventing collapse of local representations (Müller et al., 2022).
• Physical modeling and nematic order: Assessing both local and global quasi-uniformity of nematic line fields supports characterization of materials on curved surfaces, linking defect patterns to underlying geometry (Pedrini et al., 12 Jun 2025).

5. Extensions, Limitations, and Theoretical Insights

• Limitations of Marginal Uniformity: Designs such as Latin hypercubes and orthogonal arrays, while guaranteeing marginal uniformity, may fail in directions corresponding to non-axis-aligned projections, necessitating multi-directional uniformity assessment (0802.2158).
• Optimality and Theoretical Guarantees: Some uniformity test families (e.g., the projection-based and maximal-projection tests on spheres) achieve local asymptotic optimality and admit efficiency analyses via Bahadur slopes, linking power under contiguous alternatives to specific choices of test parameters (Borodavka et al., 2023, García-Portugués et al., 2020).
• Coverage Guarantees: Adaptive conformal prediction using locally rescaled scores preserves global coverage, with bounds on local (input-conditional) coverage quantified via mutual information between data and score (Deutschmann et al., 2023).
• Resolution and Sampling: Uniformity scores are sensitive to sample size, discretization, and binning strategies. Momentum-based boundary updates and probabilistic selection temper the bias towards high-detail regions, achieving a trade-off suited for shape-specific or data-specific applications (Wu et al., 28 Apr 2025).

6. Summary and Comparative Table

Uniformity scores, whether local or global, are foundational tools across statistical design, machine learning, anomaly detection, physical modeling, and benchmarking. They provide nuanced diagnostics for regularity, coverage, and defect identification in multidimensional and high-dimensional settings. The theoretical formulation, visualization, and empirical studies underpinning these scores inform the selection of design strategies, algorithms, and evaluation methods suitable for contemporary scientific and engineering problems.

Method/Domain	Local Uniformity Score	Global Uniformity Score
Radar Statistic	KS discrepancy in each direction ($D_N(a)$)	Ratio of max to min discrepancy ($G_N$)
Image Quality	Pixelwise contrast/gradient diff.	Saliency map correlation/aggregation
SOMs & Competitive	Local error/feedback adjustment	Uniformity of error/topology across map
Machine Learning	Instancewise SHAP/CFC contributions	Aggregate importance ranking across dataset
Anomaly Detection	Local sparsity via partitioning	Proximity to normal cluster centers
Conformal Regression	Locally rescaled nonconformity via kernel estimator	Coverage guarantee via conformal mechanism
Nematic Fields	Ratio/factor $f$ of splay and bend on surface	Deviation/integration of $f$ over surface

Uniformity scores thus unify modeling, evaluation, detection, and explanation frameworks by exposing regularity and defect structures non-trivially, with both local sensitivity and global oversight.