Why does Greedy Search produce Optimal Clustering Outcomes? A Fixed-Core Assignment Theory

2026-07-27

AI Summary

Traditional clustering methods based on pairwise similarity often struggle to effectively identify clusters with irregular shapes or significant variations in density and scale. This work proposes the “Clusters as Distributions” (CaD) framework, which treats each cluster as an i.i.d. sample drawn from an unknown underlying distribution. The method employs a greedy search strategy to optimize the CaD objective and, for the first time, establishes a theoretical connection between this approach and partition matroids, proving that the greedy strategy achieves near-optimality when the embedded distributions closely approximate the true underlying distributions. Theoretical analysis demonstrates that the proposed method is capable of accurately recovering clusters of arbitrary shape, density, and size, and empirical results confirm its superior performance over conventional spectral clustering techniques.

Abstract

Many existing clustering methods are designed based on a set-oriented definition---a cluster is a set of similar points---relying a point-to-point similarity function to find similar points. This works well for compact clusters, but clustering performance can deteriorate badly when cluster shapes are irregular, and densities or sizes vary between clusters. Recent `Cluster-as-Distribution' (CaD) clustering has been shown to discover these generic types of clusters in practice by treating each cluster as a set of independent and identically distributed points generated from some unknown distribution via a greedy search, achieving a clustering objective equivalent to that of Spectral Clustering, but with better clustering outcomes without eigen-decomposition. However, a theoretical analysis of this phenomenon is still lacking. Our analyses are from two angles. First, we analyze the approximation error between the true and empirical distribution embeddings. Second, we show that the greedy search employed to achieve the CaD clustering objective can be mapped to a partition matroid---yielding greedy optimality. These yield a near-optimality guarantee for the CaD clustering objective, with regret controlled by the approximation error. This is the first analysis that explains why CaD clustering via greedy search can discover clusters of arbitrary shapes, densities and sizes (where all set-oriented clustering methods have failed to discover) when the estimated cluster embeddings faithfully approximate the underlying cluster distributions.

Problem

• clustering
• greedy search
• cluster-as-distribution
• arbitrary-shaped clusters
• theoretical analysis

Innovation

• Cluster-as-Distribution
• greedy search
• partition matroid
• distribution embedding
• clustering optimality