Boundary Point Selection: Methods & Applications

• Boundary point selection is the process of identifying points on or near the edges of a dataset, essential for clustering, manifold learning, and simulation.
• It employs methods such as geometric dispersion metrics, density estimation, and neural network classification to accurately delineate boundaries.
• Effective boundary selection enhances computational efficiency and accuracy in support estimation, mesh generation, and adversarial optimization.

Boundary point selection refers to the identification, estimation, or classification of points that reside on or near the boundary of a domain, class, manifold, or data structure. This process is fundamental in computational geometry, statistics, point-cloud analysis, machine learning, and adversarial optimization, where accurately detecting, representing, or exploiting boundaries is essential for decision-making, clustering, support estimation, PDE simulation, or the construction of efficient classification systems.

1. Mathematical Foundations and Definitions

The concept of a “boundary point” is context-dependent. In geometric and statistical settings, a point is considered a boundary point if it lies on the frontier between a domain (or cluster, or class) and its complement. For a compact domain $\Omega \subset \mathbb R^d$ with boundary $\partial\Omega$, boundary points are those with $\mathrm{dist}(x, \partial\Omega)\leq\epsilon$. In classification, a boundary (or relevant) point in a labeled point set $P\subset\mathbb R^d$ is one whose removal changes the Voronoi diagram and alters the classification of at least some queries (Flores-Velazco, 2022). In manifold learning, statistical, or PDE contexts, boundary points often control bias, convergence rates, and are pivotal in robust estimation (Girard et al., 2011, Calder et al., 2021).

The formal mathematical principles underlying boundary point selection include:

• Local geometric or statistical criteria (density, local covariance, neighbor dispersion);
• Decision-boundary or Voronoi/Delaunay adjacency conditions;
• Extreme value statistics in Poisson or other random point processes (Girard et al., 2011).

2. Algorithms and Methodologies

Boundary point selection admits diverse algorithms, each tuned to problem structure and data type:

a) Geometric and Statistical Estimation (Point Clouds and Manifolds)

• Local Direction Dispersion (LoDD): For each $x\in\mathbb R^d$, compute the eigenvalue spectrum of the k-NN neighborhood projected onto the unit sphere around $x$. The LoDD score $L^{(d)}(x)$ measures the isotropy of directions; low values indicate a locally “one-sided” (boundary) neighborhood (Peng et al., 2023).
• Normal Vector and Distance Estimators: Estimate the boundary normal via local mean direction, and the boundary distance as the maximal projection along the normal. Thresholding this projection identifies points near the boundary (Calder et al., 2021).
• Covariance and Local Statistics with Neural Networks: Extract multi-scale, per-point features (eigenvalues, vectorized distances) and classify using compact MLP architectures to distinguish boundary, edge, and interior points (Bode et al., 2022).

b) Classification and Decision Boundaries (Discreet Point Sets, Voronoi Structures)

• Border/Relevance Point Algorithms: Use geometric inversion and extreme point computations to identify points whose Voronoi regions share facets with points of other classes, revealing points that define classification boundaries. Output-sensitive algorithms achieve $O(nk^2)$ complexity, where $k$ is the number of boundary points (Flores-Velazco, 2022).

c) PDE and Finite Element Surrogates

• Shifted Boundary Method (SBM): On meshes (especially octrees), select surrogate boundaries by choosing the mesh interface with minimal average shift to the true domain boundary. This is optimally achieved by partitioning cells so that the average signed distance from the surrogate to the true boundary is $h/2$, where $h$ is mesh size. The enforcement of boundary conditions is then Taylor-corrected (Yang et al., 2023).

d) Statistical Extreme-Point and Projection Methods

• Projection-Extreme Estimators: Approximate the unknown frontier of a random set by expanding the frontier in an orthonormal basis (e.g., trigonometric), and estimating coefficients via maxima of observed process samples across partitioned cells. Systematic bias is addressed by collecting cell-wise minima and correcting estimators accordingly (Girard et al., 2011).

e) Adversarial Optimization (LLM Jailbreaking)

• Boundary Point Jailbreaking (BPJ): In black-box classification, boundary points are string modifications where a defender model (e.g., safety classifier) outputs mixed responses (flag/allow) within a candidate population. BPJ refines attacks via a curriculum of partially noised targets, but only evaluates on those boundary points giving maximal fitness gradient signal under one-bit feedback (Davies et al., 16 Feb 2026).

3. Theoretical Guarantees and Statistical Properties

Many boundary point selection algorithms provide rigorous theoretical guarantees:

• Consistency and Convergence: For Monte Carlo and kernel-based estimators, there are explicit rates for normal estimation ($O((\log n/n)^{1/(d+2)})$) and distance estimation ($O(r^2)$ bias with $r\sim(\log n/n)^{1/(d+2)}$), and high-probability guarantees for boundary strip detection (Calder et al., 2021).
• Error Bounds and Asymptotic Normality: For projection estimators, conditions on the basis, partition parameters, and kernel norms ensure mean-square and almost sure convergence, with pointwise asymptotic normality under normalization by $B_{n,2}(x)$ (Girard et al., 2011).
• Optimality in Surrogate Selection: When the average signed distance $\delta$ between surrogate and true boundary is $O(h)$ (with threshold at $(h/2)$), one recovers optimal $H^1$- and near-optimal $L^2$-convergence for finite element solvers (Yang et al., 2023).
• Price Equation and Query Efficiency (Adversarial Optimization): Only boundary point queries contribute nontrivial fitness variance in black-box settings, leading to substantially accelerated convergence and efficient search (Davies et al., 16 Feb 2026).

4. Practical Implementation and Computational Considerations

Implementation depends on the setting:

• kNN and Covariance-based Methods: k-NN search (via k-d or ball tree, $O((d+k)n\log n)$) dominates neighbor-based approaches; eigenvalue computations per point cost $O(d^3)$ (Peng et al., 2023, Bode et al., 2022).
• Extreme Point and Inversion Methods: For nearest neighbor border point extraction, repeated geometric inversion and convex hull computation are required. Advanced algorithms achieve $O(nk^2)$ (or $O(nk\log k)$ in low dimensions) (Flores-Velazco, 2022).
• Neural Models for 3D Clouds: Feature extraction is typically vectorized and hardware-efficient, with end-to-end classification for millions of points in seconds (Bode et al., 2022).
• Simulations in SBM: Surrogate boundary selection and assembly are mesh-parallelizable; thresholds can be chosen automatically (e.g., by setting $\lambda=0.5$ for cell classification) (Yang et al., 2023).
• Robust Parameter Selection: Adaptive heuristics (grid models for boundary fraction, cross-validation for radius, principal component smoothing) alleviate sensitivity to manual choices (Peng et al., 2023, Calder et al., 2021, Aaron et al., 2016).

5. Applications and Impact in Modern Research

Boundary point selection is central to:

• Support and Density Estimation: Accurate identification of sample support boundaries reduces edge-bias in smooth density or manifold learning (Girard et al., 2011, Calder et al., 2021).
• Clustering and Embedding: Pruning or weighting boundary points improves the stability and accuracy of unsupervised clustering (e.g., k-means, spectral cuts) and nonlinear embeddings (e.g., t-SNE/UMAP) (Peng et al., 2023).
• Efficient Classification: Minimally sufficient 1-NN classifiers require only the selected border points (“relevant set”), reducing storage and computational burden while preserving exact decision boundaries (Flores-Velazco, 2022).
• Meshing and Simulation: Immersed/shifted boundary methods leverage optimal surrogate boundary selection for error-minimizing, robust simulation in complex domains (Yang et al., 2023).
• Point Cloud Processing: Edge and boundary detection in 3D clouds improves downstream tasks in urban planning, CAD, and scene understanding (Bode et al., 2022).
• Machine Learning Robustness and Attack: Boundary point selection enables query-efficient attack algorithms against guarded models in adversarial contexts (Davies et al., 16 Feb 2026).

6. Empirical Performance and Benchmarks

Empirical studies confirm:

• Superiority of Directional Dispersion Metrics: LoDD exceeds geometry- or density-based methods on synthetic and real benchmarks, especially with nonconvex clusters or heterogenous densities (Peng et al., 2023).
• Practical and Asymptotic Accuracy: Bias-corrected projection estimators achieve negligible bias and variance matching theoretical predictions. MISE decays as $n^{-2/5}$ under appropriate parameter scaling (Girard et al., 2011).
• Neural Feature Classification Outperforms Competing Methods: In 3D point cloud classification, multi-scale statistical features and MLPs yield higher F1 and IoU, with significant computational efficiency (Bode et al., 2022).
• Algorithmic Scalability for Large Datasets: Output-sensitive border point selection enables the deployment of accelerated search structures (ANN, HNSW, FAISS) using only the core boundary set (Flores-Velazco, 2022).
• Boundary Point Sampling Drives Fast Adversarial Search: BPJ achieves an order-of-magnitude reduction in black-box query complexity vs. naïve alternatives for LLM jailbreaks (Davies et al., 16 Feb 2026).

7. Limitations and Ongoing Research Directions

Several limitations are highlighted:

• Algorithmic Sensitivity: Some approaches require careful selection of scale or neighbor count; very high dimensions incur both computational and estimation challenges (Peng et al., 2023, Girard et al., 2011).
• Edge vs. Boundary Separation: The distinction between sharp edges and true boundaries in geometrically complex point clouds requires multi-scale and orientation-aware methods (Bode et al., 2022).
• Curse of Dimensionality: Covariance and k-NN-based statistics grow more expensive and less discriminative as $d$ increases, often motivating the use of dimensionality reduction or approximate neighbor search (Peng et al., 2023).
• Parameter Tuning in Statistical Tests: Data-dependent heuristics or simulations are often needed to set bandwidth, sparsity, or decision thresholds for reliable detection (Calder et al., 2021, Aaron et al., 2016).
• Adversarial Adaptivity: In LLM security, boundary point selection for attack reveals intrinsic weaknesses in classifier-only defences, suggesting the need for batch or sequence-level monitoring (Davies et al., 16 Feb 2026).

Sophisticated boundary point detection remains an active area of research, integrating statistical, geometric, algorithmic, and machine learning advances across domains.