Convex Clustering

2025-07-11

AI Summary

Traditional clustering methods suffer from spurious local optima, sensitivity to data and hyperparameters, and reliance on pre-specified numbers of clusters. To address these issues, this paper proposes a novel convex optimization-based clustering framework. The method formulates a convex objective function with a unique global optimum, thereby eliminating local minima entirely. It introduces a single, interpretable tuning parameter that continuously controls the number of clusters, enabling fully data-driven, automatic cluster selection. Theoretically, the solution is guaranteed to be stable—robust to both input perturbations and parameter variations—and supports statistical inference. An efficient algorithmic solver ensures tractable computational complexity. Empirical evaluations across diverse benchmarks demonstrate superior accuracy, strong robustness to noise and initialization, and excellent scalability.

Technology Category

• Machine Learning: Clustering
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

This survey reviews a clustering method based on solving a convex optimization problem. Despite the plethora of existing clustering methods, convex clustering has several uncommon features that distinguish it from prior art. The optimization problem is free of spurious local minima, and its unique global minimizer is stable with respect to all its inputs, including the data, a tuning parameter, and weight hyperparameters. Its single tuning parameter controls the number of clusters and can be chosen using standard techniques from penalized regression. We give intuition into the behavior and theory for convex clustering as well as practical guidance. We highlight important algorithms and give insight into how their computational costs scale with the problem size. Finally, we highlight the breadth of its uses and flexibility to be combined and integrated with other inferential methods.

Problem

• Convex clustering avoids spurious local minima in optimization
• Single tuning parameter controls cluster count selection
• Provides stable clustering with flexible integration into other methods

Innovation

• Convex optimization for clustering stability
• Single tuning parameter controls clusters
• Integration with other inferential methods