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

TL;DR

DLCC presents a depth-based clustering framework that leverages spatial depth to construct a depth-based similarity matrix and identify local centers from depth-driven neighborhoods. By employing two grouping strategies (Min and Max) and a subsequent labeling step via depth-based classifiers, DLCC handles convex, nonconvex, and multimodal clustering without assuming rigid shapes. The approach achieves competitive results on synthetic and real datasets, with explicit discussion of complexity ($O(n^3)$ for similarity construction) and practical guidance for parameter selection. Limitations include scalability to very large datasets and reliance on neighborhood geometry, with future work proposing approximate search, sampling, adaptive neighborhoods, and exploring alternative depth notions.

Abstract

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.

Figures (32)

• Figure 1: Flowchart for the DLCC algorithm, detailing the sections that include the key contents represented in the nodes.
• Figure 2: Running time comparison (log scale) between the proposed method and the traditional construction.
• Figure 3: Local center example ($s=50$) in the two–PC visualization of the Iris dataset (see Section \ref{['sec:ee']} for dataset description). A focus point (green triangle), its neighbors (pink points), and the resulting local center (blue diamond) are highlighted.
• Figure 4: t-SNE visualization van2008visualizing of the Yale B dataset in two-dimensional space (see Section \ref{['sec:ee']} for dataset description). Colors indicate the temporary labels of the points, i.e., membership in $\{\mathbf{T}_k\}_{k=1}^K$. From left to right, the panels illustrate the progression from the initial state (unique neighbors) to the temporary clusters after Algorithm \ref{['alg:utc']}. Points enclosed by diamond-shaped boxes denote filtered centers.
• Figure 5: A toy example illustrating the hierarchical structure in grouping filtered centers, alongside the relationship between the threshold $\delta$ and the number of groups $G$. This process generates two lists, $\boldsymbol{G}$ and $\boldsymbol{\delta}$, which store the number of groups and the corresponding threshold values at each stage, respectively. The elements of these lists are displayed in the top right.

Theorems & Definitions (1)

• Proposition 1: Symmetry