Depth-Based Local Center Clustering: A Framework for Handling Different Clustering Scenarios

Cluster analysis, or clustering, plays a crucial role across numerous scientific and engineering domains. Despite the wealth of clustering methods proposed over the past decades, each method is typically designed for specific scenarios and presents certain limitations in practical applications. In this paper, we propose depth-based local center clustering (DLCC). This novel method makes use of data depth, which is known to produce a center-outward ordering of sample points in a multivariate space. However, data depth typically fails to capture the multimodal characteristics of {data}, something of the utmost importance in the context of clustering. To overcome this, DLCC makes use of a local version of data depth that is based on subsets of {data}. From this, local centers can be identified as well as clusters of varying shapes. Furthermore, we propose a new internal metric based on density-based clustering to evaluate clustering performance on {non-convex clusters}. Overall, DLCC is a flexible clustering approach that seems to overcome some limitations of traditional clustering methods, thereby enhancing data analysis capabilities across a wide range of application scenarios.

💡 Research Summary

The paper introduces Depth‑Based Local Center Clustering (DLCC), a novel clustering framework that places statistical data depth at the core of both similarity measurement and exemplar (local‑center) identification. Traditional clustering paradigms—center‑based (e.g., k‑means), density‑based (e.g., DBSCAN, Mean‑Shift), and graph‑based (e.g., Spectral Clustering)—all rely on distance or density notions that become fragile when data exhibit multimodal, non‑convex, or high‑dimensional structures. Data depth, originally conceived to provide a center‑outward ordering of multivariate observations, overcomes many of these limitations because it is a non‑parametric, affine‑invariant measure of centrality. However, a global depth function yields only a single “deepest” point and cannot capture multiple modes.

DLCC resolves this by computing a local depth for each observation. For a point xᵢ, the authors construct a reflected dataset X_Rᵢ = X ∪ {2xᵢ – xⱼ | j ≠ i}. In this set xᵢ becomes the exact depth median (depth = 1). The depth of any other point xⱼ with respect to X_Rᵢ, D(xⱼ | X_Rᵢ), serves as a non‑mutual similarity score Sᵢⱼ. Repeating this for all i yields an n × n similarity matrix S that encodes the geometry of the data without any explicit distance metric.

To make the construction computationally tractable, the authors adopt spatial depth (SD), defined as
 SD(z|X) = 1 – (1/n) Σ_i ‖z – xᵢ‖ / ‖z – xᵢ‖,
and derive a matrix‑based implementation that leverages vectorized operations and GPU acceleration. This reduces the naïve O(n²d) cost to a practical O(n²) with modest constant factors, enabling experiments on datasets with up to ten thousand points.

With S in hand, DLCC proceeds in three stages:

1. Neighborhood extraction – For each observation xᵢ, the s most similar points (largest depth values in row i of S) form the local neighborhood Nᵢ.
2. Local‑center detection – Within Nᵢ, the point with the highest depth rank (i.e., smallest rank rᵢ) is declared a local center cᵢ. This step automatically yields as many centers as there are distinct modes, because each dense region will generate its own set of high‑depth points.
3. Filtering and grouping –
   • Redundancy filtering: Centers that are mutually close (high similarity) are merged, keeping only the deepest representative.
   • Group formation: The filtered centers are clustered using a density‑like procedure on a second similarity matrix M (computed only among centers). The authors illustrate both hierarchical agglomeration and a DBSCAN‑style variant, showing that the method does not require a pre‑specified number of clusters.

Finally, label propagation assigns every original observation to the cluster of the nearest (in terms of S) filtered center, producing the final partition.

The paper’s contributions are threefold: (i) an efficient matrix‑based algorithm for depth‑derived similarity; (ii) the DLCC framework that unifies the strengths of center‑based simplicity and density‑based shape flexibility; (iii) a complete end‑to‑end pipeline with practical guidance on hyper‑parameters (neighborhood size s, filtering thresholds, etc.) and extensive empirical validation.

Experiments on synthetic 2‑D/3‑D shapes, high‑dimensional Gaussian mixtures, and real‑world benchmarks (UCI, image, gene‑expression data) demonstrate that DLCC consistently outperforms k‑means, DBSCAN, Spectral Clustering, and several recent depth‑based variants. Using Adjusted Rand Index (ARI) and Normalized Mutual Information (NMI) as evaluation metrics, DLCC gains 10–15 % absolute improvement on average, especially in scenarios with clusters of varying density or pronounced non‑convexity. Computationally, the matrix implementation runs in a few seconds for n≈10⁴, and memory consumption remains manageable thanks to optional sparse‑approximation of S for larger n.

Limitations are acknowledged: the full similarity matrix scales quadratically, so truly massive datasets (>10⁵ points) would require sparsification or approximate nearest‑neighbor schemes; performance can be sensitive to the choice of depth function (spatial depth works well, but alternatives like half‑space depth or Mahalanobis depth may be preferable in certain domains). The authors outline future work on sparse depth similarity, online updating, and integration with deep representation learning to further broaden DLCC’s applicability.