Optimal construction of k-nearest neighbor graphs for identifying noisy clusters

TL;DR

This paper investigates how to optimally construct neighborhood graphs, specifically mutual and symmetric k-nearest neighbor graphs, for clustering data points into true underlying clusters, especially in noisy environments.

Contribution

It provides theoretical bounds on cluster detection success, revealing optimal parameter choices and differences between graph types for noisy clustering.

Findings

Choosing a high k (order n) improves cluster detection.

Mutual and symmetric graphs differ mainly when detecting the largest cluster.

Bounds on cluster identification success are derived using random geometric graph theory.

Abstract

We study clustering algorithms based on neighborhood graphs on a random sample of data points. The question we ask is how such a graph should be constructed in order to obtain optimal clustering results. Which type of neighborhood graph should one choose, mutual k-nearest neighbor or symmetric k-nearest neighbor? What is the optimal parameter k? In our setting, clusters are defined as connected components of the t-level set of the underlying probability distribution. Clusters are said to be identified in the neighborhood graph if connected components in the graph correspond to the true underlying clusters. Using techniques from random geometric graph theory, we prove bounds on the probability that clusters are identified successfully, both in a noise-free and in a noisy setting. Those bounds lead to several conclusions. First, k has to be chosen surprisingly high (rather of the order n than of the order log n) to maximize the probability of cluster identification. Secondly, the major difference between the mutual and the symmetric k-nearest neighbor graph occurs when one attempts to detect the most significant cluster only.