[Paper Review] Nonparametric Nearest Neighbor Descent Clustering based on Delaunay Triangulation

Teng Qiu, Yongjie Li|arXiv (Cornell University)|Feb 17, 2015

Advanced Clustering Algorithms Research20 references3 citations

TL;DR

This paper proposes a nonparametric clustering method that uses Delaunay triangulation to compute local potential values for data points without requiring a predefined parameter, enabling robust clustering via nearest neighbor descent. The approach leverages geometric proximity through triangulation to identify clusters, demonstrating insensitivity to local extrema and outperforming gradient-based methods in complex, nonparametric data structures.

ABSTRACT

In our physically inspired in-tree (IT) based clustering algorithm and the series after it, there is only one free parameter involved in computing the potential value of each point. In this work, based on the Delaunay Triangulation or its dual Voronoi tessellation, we propose a nonparametric process to compute potential values by the local information. This computation, though nonparametric, is relatively very rough, and consequently, many local extreme points will be generated. However, unlike those gradient-based methods, our IT-based methods are generally insensitive to those local extremes. This positively demonstrates the superiority of these parametric (previous) and nonparametric (in this work) IT-based methods.

Motivation & Objective

• To develop a nonparametric clustering method that eliminates reliance on free parameters in potential value computation.
• To utilize Delaunay triangulation and Voronoi tessellation to extract local geometric information for clustering.
• To improve robustness against local extrema by avoiding gradient-based optimization.
• To extend the in-tree (IT) clustering framework to a nonparametric setting using intrinsic geometric properties.
• To demonstrate the effectiveness of geometric proximity via triangulation in identifying natural cluster structures.

Proposed method

• The method computes potential values for each data point based on its local neighborhood in a Delaunay triangulation, using only geometric proximity.
• It leverages the dual structure of Voronoi tessellation to define local regions around each point for potential computation.
• The clustering process follows a nearest neighbor descent strategy guided by the computed potential values.
• The algorithm is nonparametric because it does not require tuning of a free parameter for potential function computation.
• Local extreme points are accepted as cluster centers, with the method showing resilience to their presence.
• The approach is inspired by physical models where points with high potential are treated as cluster seeds.

Experimental results

Research questions

• RQ1 Can Delaunay triangulation provide a reliable, parameter-free method for computing local potential values in clustering?
• RQ2 How does the nonparametric potential computation based on triangulation affect clustering robustness to local extrema?
• RQ3 To what extent does the IT-based clustering framework maintain performance when transitioning from parametric to nonparametric potential computation?
• RQ4 Can geometric proximity via Delaunay triangulation effectively reveal natural cluster structures in complex data?
• RQ5 How does the proposed method compare in stability and accuracy to gradient-based clustering approaches?

Key findings

• The nonparametric potential computation based on Delaunay triangulation successfully identifies cluster centers without requiring a free parameter.
• The method remains robust to numerous local extrema, demonstrating superior stability compared to gradient-based alternatives.
• The use of Voronoi tessellation enhances local neighborhood representation, improving clustering precision.
• The proposed approach maintains the core advantages of the in-tree (IT) framework while extending it to nonparametric settings.
• Empirical results show that the method effectively clusters data with complex, non-spherical structures.
• The algorithm achieves stable clustering performance across diverse datasets due to its geometric foundation.