Mass Distribution versus Density Distribution in the Context of Clustering

TL;DR

Density-based clustering often struggles with high-density bias and quadratic runtime due to density estimation and linking. This work reframes clustering through mass distribution using the Isolation Kernel (IK) and introduces Mass-Maximization Clustering (MMC), which represents clusters as distributions and maximizes the total mass across clusters to achieve linear-time clustering $O(n)$. The paper defines Isolation-induced Mass, τ-cohesive clusters, and a representative-sample property that enables robust discovery of arbitrary-shape clusters across varied densities, while identifying conditions where density-maximization fails. Extensive experiments on artificial, real, and Spatial Transcriptomics data show MMC consistently outperforms density-based methods and scales to large datasets, providing a practical, principled alternative to traditional density-based clustering.

Abstract

This paper investigates two fundamental descriptors of data, i.e., density distribution versus mass distribution, in the context of clustering. Density distribution has been the de facto descriptor of data distribution since the introduction of statistics. We show that density distribution has its fundamental limitation -- high-density bias, irrespective of the algorithms used to perform clustering. Existing density-based clustering algorithms have employed different algorithmic means to counter the effect of the high-density bias with some success, but the fundamental limitation of using density distribution remains an obstacle to discovering clusters of arbitrary shapes, sizes and densities. Using the mass distribution as a better foundation, we propose a new algorithm which maximizes the total mass of all clusters, called mass-maximization clustering (MMC). The algorithm can be easily changed to maximize the total density of all clusters in order to examine the fundamental limitation of using density distribution versus mass distribution. The key advantage of the MMC over the density-maximization clustering is that the maximization is conducted without a bias towards dense clusters.

Figures (10)

• Figure 1: Illustrations of the boundaries of $C^\tau_\beta$ (dense) and $C^\tau_\alpha$ (sparse) clusters due to different $\tau$ settings. Note that clusters $C^\tau$ with high $\tau$ cover smaller regions than those with low $\tau$: (a) The entire dense cluster is covered when using the Gaussian Kernel with $\tau=0.99$; and (b) both the dense and sparse clusters are merged when $\tau=0.81$. (c) This example denotes the lowest $\tau=0.7$ using the Isolation Kernel to cover the entire dense cluster, while lowering $\tau=0.35$ merges the two clusters is shown in (d). The plots of cohesiveness $\bar{S}_\kappa(C^\tau)$ versus $\tau$ for dense cluster $C_\beta$ & sparse cluster $C_\alpha$ are shown in (e) & (f). The cohesiveness is min-max normalized over values derived from all $\tau$ values.
• Figure 2: An illustration of the first condition in terms of the distribution of $\kappa(\mathbf{x}_{\imath},\mathbf{x}_{\imath+1})$. Moving averages with window sizes of 150 and 25 are used to produce the distributions of GK and IK, respectively.
• Figure 3: An example impact of the high-density bias in step 2 of DMC (where $Q_i$ obtained in step 1 is replaced with the ground-truth clusters).
• Figure 4: Total density or mass is a proxy for the goodness of a clustering outcome. The total density or mass versus AMI on the 3G dataset as the error correction process progresses. Beginning with completely random point assignments to all clusters. Then repeatedly randomly select a subset (of 50 points) and correct its individual point assignment if the initial assignment was incorrect, i.e., the error correction rate increases as more points are selected.
• Figure 5: Improvement due to post-processing in terms of percentage of AMI before post-processing (average results over 5 trials). The influence of post-processing is small or none on many datasets except for the first 4 datasets.

Theorems & Definitions (19)

• Definition 1
• Definition 2
• Lemma 1
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Proposition 1
• proof
• Proposition 2