Local Dimensionality Assessments

Updated 12 October 2025

Local dimensionality assessments are methods to quantify the effective number of latent dimensions in localized regions, capturing spatial, relational, or contextual complexity.
They employ techniques such as k-NN estimators, neural density methods, and Fisher separability to reveal data heterogeneity and guide algorithmic design.
Applications span manifold learning, outlier detection, robust representation learning, and physical systems analysis, offering actionable insights for theory and practice.

Local dimensionality assessments are approaches that characterize the effective number of degrees of freedom or "intrinsic dimension" present in localized regions of data, networks, representations, or physical systems. Unlike global dimensionality measures that provide a single value for entire datasets or systems, local assessments explicitly account for spatial, relational, or contextual heterogeneity, making them particularly powerful for understanding complex, high-dimensional, or structurally diverse data. Research across fields such as manifold learning, neural networks, outlier detection, self-supervised representation learning, complex networks, and the physics of condensed matter systems has led to a variety of methodologies and applications rooted in the rigorous quantification of local intrinsic dimensionality.

1. Definitions and Theoretical Foundations

Local intrinsic dimensionality (LID) quantifies, for a given point or region, the effective number of latent variables that explain the behavior of the surrounding data. Mathematically, for a reference point $x$ with neighborhood distances governed by a cumulative density function $F(r)$ , the local intrinsic dimension at (small) scale $r$ is given by

$\mathrm{LID}_F(r) = \frac{r F'(r)}{F(r)}$

with the asymptotic LID being defined as $r \rightarrow 0$ .

The concept generalizes to diverse domains:

In data analysis, LID describes the "space-filling capability" locally, as opposed to global PCA-based rank or correlation-dimension estimators (Ma et al., 2018).
For networks, local dimension $D_i(r)$ quantifies the growth of the neighborhood (number of nodes within graph distance $r$ of node $i$ ) via $B_i(r) \sim r^{D_i(r)}$ (Silva et al., 2012).
In physical and quantum systems, local dimensionality is linked to local correlation structures, e.g. via von Neumann entropy or local operator spectra (Doerfler et al., 2021).

These measures are fundamental to understanding phenomena such as the curse of dimensionality, robustness of representations, and the behavior of generative or discriminative models under adversarial or outlier perturbations.

2. Methodological Approaches

A variety of techniques have been developed for local dimensionality estimation and assessment:

Distance-Based Nearest Neighbor Estimators: Empirical estimates via maximum likelihood over $k$ -nearest neighbor distances have become standard, notably the Levina–Bickel/Hill estimator:

$\widehat{\mathrm{LID}}(x) = - \left( \frac{1}{k} \sum_{i=1}^k \log\frac{r_i(x)}{r_k(x)} \right)^{-1}$

and extended variants such as tight locality estimators that exploit all intra-neighborhood pairwise distances for stability at small $n$ (Amsaleg et al., 2022), as well as CrossLID for cross-distribution comparisons (Barua et al., 2019).

Likelihood and Density-Based Methods: LIDL directly infers local intrinsic dimension using modern neural density estimation methods. It perturbs the dataset at various Gaussian noise scales and observes the scaling of log-likelihood to extract the slope $(d-D)$ , so that local dimension $d$ is recovered by linear regression:

$\log\rho_\delta(x) = (d - D) \log\delta + \mathcal{O}(1)$

(Tempczyk et al., 2022).

Fisher Separability and Concentration of Measure: Using the phenomenon that high-dimensional data points become nearly linearly separable, Fisher separability-based estimators use inseparability probability $p_\alpha(x)$ to estimate dimension via inversion to the reference uniform n-sphere law (Bac et al., 2020).
Surrogate Modeling for Interpretability: Methods such as LXDR fit local linear surrogates to explain non-linear dimensionality reduction outputs, constructing interpretable mappings from the original space to the locally reduced space (Bardos et al., 2022). SLISEMAP jointly optimizes local explanations and global embeddings for supervised models (Björklund et al., 2022).
Entropy and Operator Analysis: In signal-processing or quantum harmonic analysis, local dimension is related to the spectral properties of localized covariance (or data) operators, with the von Neumann entropy of such operators capturing “local effective dimensionality” (Doerfler et al., 2021).
Regularization and Learning Objectives: LDReg and similar methods directly inject a regularization term into learning algorithms to increase the local intrinsic dimensionality, leveraging analytic properties (e.g., via the Fisher–Rao metric on parameterized distance distributions) (Huang et al., 19 Jan 2024).

3. Applications Across Domains

Local dimensionality assessments underpin modern research and applications in several areas:

Robustness and Adversarial Detection: Local intrinsic dimensionality is a sensitive detector of adversarial manipulation in deep neural networks. Perturbed samples typically manifest higher LID, which can be detected using threshold-based or statistical tests (Ma et al., 2018, Weerasinghe et al., 2021).
Outlier and Anomaly Detection: DAO is a nonparametric outlier detection method that explicitly incorporates local LID into its density ratio computations:

$\mathrm{DAO}_k(q) = \frac{1}{k} \sum_{o\in \mathcal{N}_k(q)} \left(\frac{\mathrm{kdist}(q)}{\mathrm{kdist}(o)}\right)^{\widehat{\mathrm{LID}}(o)}$

(Anderberg et al., 10 Jan 2024). Accounting for local dimension yields improved performance over traditional Local Outlier Factor approaches, especially in data with heterogeneous local complexities.

Manifold Learning and Dimensionality Reduction: Full correlation integral (FCI) estimators and related multiscale methods yield robust dimension estimates even for highly curved or locally sparse data (e.g., image manifolds after geometric transformations). The discovery of locally varying dimension reveals critical heterogeneity and guides the design of dimensionality reduction algorithms (Erba et al., 2019).
Representation Learning: LDReg demonstrates that representation spaces learned without explicit LID regularization may suffer from "local collapse" despite seemingly high global ranks. By maximizing local dimension via regularization, downstream linear evaluation and transfer learning tasks are consistently improved (Huang et al., 19 Jan 2024).
Complex Networks and Physics: In physically embedded networks, local dimension reliably distinguishes planar and non-planar (with long-range connections) topologies, revealing both local heterogeneity and border effects (Silva et al., 2012). In condensed matter materials, the decay law of defect-induced displacements (e.g., $\propto 1/r^2$ , $1/r$, or constant for 3D/2D/1D) is governed by the system’s local dimension, as reflected in fine-structure spectroscopy (Furrer, 2015).
Interpretable Machine Learning: LXDR and SLISEMAP bridge black-box representations and interpretability by aligning local explanations (via interpretable surrogates) with the embedding structure, enabling deeper insight into model behavior and decision boundaries (Bardos et al., 2022, Björklund et al., 2022).

4. Quantitative Properties and Comparative Analyses

A spectrum of comparative and empirical results highlight the impact and sensitivity of local dimensionality assessments:

Estimator Bias and Variance: Advanced estimators (e.g., tight local MLE, Fisher separability-based, neural density estimators) systematically reduce bias and variance relative to traditional global or k-NN methods, enabling reliable dimension estimates with as few as 20 samples per neighborhood (Amsaleg et al., 2022).
Sensitivity to Data Complexity: Local LID correlates strongly with algorithmic difficulty. In nearest neighbor search benchmarks, queries with high local dimension are consistently more challenging, requiring more computation for equivalent recall (Aumüller et al., 2019).
Impact on Downstream Algorithms: Incorporating LID improves or stabilizes outlier detection performance in data with variable local complexity; the performance of DAO remains robust as LID varies across clusters, in sharp contrast to generic LOF or k-NN-based methods (Anderberg et al., 10 Jan 2024).
GAN Evaluation and Mode Collapse: CrossLID is more sensitive than Inception Score or FID to local mismatches between real and generated data, showing monotonic decrease as GAN training improves and strong correlation with mode coverage (Barua et al., 2019).

5. Extensions, Limitations, and Open Problems

While local dimensionality assessments have demonstrated significant impact, several methodological and practical issues persist:

Estimation in High Dimensions: Scalability is limited for nearest neighbor-based estimators due to the curse of dimensionality. LIDL sidesteps this via neural density estimators, but accuracy is contingent on the parametric model’s fidelity (Tempczyk et al., 2022).
Interpretability and Surrogate Explanations: For non-linear DR methods without inverse mappings, locality-based surrogates (LXDR) provide meaningful instance-level explanations but suffer from increased computational cost in high-dimensional or large-scale regimes (Bardos et al., 2022).
Regularization Tuning and Overfitting: Local dimensionality regularization (LDReg) may itself trigger global or local mode collapse if the regularization weight is not properly calibrated, underscoring the need for careful cross-validation and analytic guidelines (Huang et al., 19 Jan 2024).
Complexity of the Neighborhood Definition: Choice of the neighborhood size ( $k$ ) or scale ( $\delta$ ) directly impacts bias–variance tradeoff and estimator sensitivity, with no universally optimal strategy across data types.

6. Future Directions

Advances in local dimensionality assessment continue on several fronts:

Hybrid Methods: Integrating full local–global structure preservation (e.g., DREAMS combines t‑SNE and PCA or MDS regularization) yields embeddings that support both fine-scale pattern discovery and global organization, addressing a persistent challenge in data visualization (Kury et al., 19 Aug 2025).
Adaptive and Multimodal Estimation: Extensions to heterogeneous, multimodal, or non-Euclidean data—such as in networks, physical systems, or deep generative models—require localized, flexible, and potentially multi-scale estimators.
Inference for Downstream Task Design: As local dimensionality increasingly informs model selection, data augmentation, and algorithmic design, developing theoretical prescriptions that directly leverage LID or related metrics for automatic tuning remains a core challenge.
Explainability and Trustworthiness: Model-agnostic, locality-focused surrogates for dimension reduction and classification models suggest a path toward interpretable AI in unsupervised or semi-supervised contexts, enhancing trust in high-stakes domains.

7. Representative Mathematical Formalisms

Principle	Formula	Context/Usage
LID (distance CDF-based)	$\mathrm{LID}_F(r) = \frac{rF'(r)}{F(r)}$	Pointwise or local neighborhood dimension estimation
MLE LID estimator	$\widehat{\mathrm{LID}}(x) = - \left( \frac{1}{k} \sum_{i=1}^k \log\frac{r_i(x)}{r_k(x)} \right)^{-1}$	Sample-based estimation (k-NN distances)
LIDL scaling	$\log\rho_\delta(x) = (d - D)\log\delta + O(1)$	Likelihood-based dimension inference via noise perturbation
Fisher–Rao local metric	$d_{\mathrm{FR}} = \|\log(\theta_2/\theta_1)\|$	Comparing LID across data points in SSL (Huang et al., 19 Jan 2024)
Power-law node growth	$B_i(r) \sim r^{D_i(r)}$	Local dimension in graphs/networks (Silva et al., 2012)
DAO outlier score	$\mathrm{DAO}_k(q) = \frac{1}{k} \sum_{o\in\mathcal{N}_k(q)} \left(\frac{\mathrm{kdist}(q)}{\mathrm{kdist}(o)}\right)^{\widehat{\mathrm{LID}}(o)}$	Density–ratio based local outlier detection (Anderberg et al., 10 Jan 2024)

Through these mathematical frameworks and empirical validations, local dimensionality assessments have established themselves as critical instruments for analyzing, interpreting, and understanding the complexity, robustness, and interpreted structure of high-dimensional and heterogeneous data. The ongoing convergence of theory, scalable computation, and actionable applications signals continued expansion and innovation within this domain.