Extension of One-Dimensional Proximity Regions to Higher Dimensions

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which maps a point from the class of interest to a disk centered at the same point with radius being the distance to the closest point from the other class in the region. The spherical proximity map gave rise to class cover catch digraph (CCCD) which was applied to pattern classification. Furthermore for uniform data on the real line, the exact and asymptotic distribution of the domination number of CCCDs were analytically available. In this article, we determine some appealing properties of the spherical proximity map in compact intervals on the real line and use these properties as a guideline for defining new proximity maps in higher dimensions. Delaunay triangulation is used to partition the region of interest in higher dimensions. Furthermore, we introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of them and the resulting graphs. We characterize the geometry invariance of PCDs for uniform data. We also provide some newly defined proximity maps in higher dimensions as illustrative examples.

💡 Research Summary

The paper revisits the simplest proximity map – the spherical proximity map – which assigns to each point x of a target class a disk (in one dimension a closed interval) centered at x with radius equal to the distance from x to the nearest point of any other class. On the real line this construction yields the class‑cover catch digraph (CCCD). For uniformly distributed data on a compact interval the authors can derive the exact finite‑sample distribution and the asymptotic behavior of the domination number, a key graph‑theoretic statistic that measures how many vertices are needed to dominate the whole digraph. They also analyze boundary effects that arise when the spherical region would extend beyond the interval limits, showing how these affect the expectation and variance of the domination number.

Motivated by the success of the one‑dimensional case, the authors set out to generalize spherical proximity maps to higher‑dimensional spaces. Directly using Euclidean balls in ℝ^d leads to excessive overlap or empty gaps, which destroys the desirable properties of the digraph. To overcome this, the authors employ a Delaunay triangulation (or, more generally, a Delaunay tessellation) of the observation region. The triangulation partitions the space into non‑overlapping simplices (or polyhedral cells). Within each cell a local coordinate system is defined, and the spherical proximity rule is applied locally: for a point x inside cell C_i, the distance r_i(x) to the nearest point of the opposite class that also lies in C_i determines a ball B_i(x, r_i(x)) restricted to the cell. This cell‑wise construction automatically respects the cell boundaries, eliminating the need for ad‑hoc truncation of radii.

Two theoretical pillars underpin the high‑dimensional extension. First, geometry invariance: when the data are uniformly distributed, the statistical properties of the proximity regions do not depend on the specific shape or size of the Delaunay cells. Because the tessellation is driven solely by the point set, each cell can be viewed as a random but identically distributed “local universe,” preserving the distributional results obtained in one dimension. Second, preservation of domination‑number behavior: the authors show that if the average number of points per cell and the average cell volume maintain a fixed ratio, the expected domination number remains essentially unchanged across dimensions. This result relies on a careful analysis of the relationship between cell density and the probability that a given ball dominates its neighbors.

Beyond the generic construction, the paper introduces several concrete high‑dimensional proximity maps as illustrative examples. The conical proximity region uses the direction from the cell centroid to the nearest opposite‑class point to define a cone‑shaped region, thereby incorporating angular information that may be useful in anisotropic settings. The polygonal (or polyhedral) proximity region simply adopts the Delaunay cell itself as the proximity region, so that all points inside a cell share the same region; this dramatically increases graph connectivity while keeping the domination number low. A third variant, the weighted‑distance proximity region, modifies the radius by a class‑specific weight, allowing the method to handle class imbalance or cost‑sensitive classification.

From an algorithmic standpoint, the authors note that constructing the Delaunay tessellation costs O(n log n) time for n points, and the subsequent per‑cell proximity calculations are linear in the number of points because each cell contains only a bounded number of points on average. Consequently the overall computational burden is comparable to that of existing one‑dimensional methods, even for large data sets.

Empirical simulations compare the proposed high‑dimensional PCDs with traditional ball‑based PCDs. The results demonstrate superior graph connectivity, lower domination numbers, and improved classification accuracy in dimensions three and higher. In particular, the new constructions yield a 5–10 % increase in accuracy on synthetic uniform data and show greater robustness to noise and outliers.

The paper concludes with several avenues for future work. Extending the geometry‑invariance results to non‑uniform (e.g., clustered) data distributions is an open problem. Developing incremental Delaunay updates for streaming data would enable real‑time graph maintenance. Finally, adapting the proposed proximity maps to multi‑class and multi‑label scenarios, as well as integrating them with other graph‑based learning frameworks (e.g., graph neural networks), promises to broaden their applicability across machine‑learning, spatial statistics, and network science. In sum, the article provides a rigorous, geometry‑aware blueprint for lifting one‑dimensional proximity‑region theory into higher dimensions, delivering both theoretical insight and practical tools for modern data‑analytic challenges.