A New Framework for Convex Clustering in Kernel Spaces: Finite Sample Bounds, Consistency and Performance Insights

2025-11-07

AI Summary

Conventional convex clustering struggles with nonlinearly separable and non-convex data structures. Method: This paper proposes the first kernelized convex clustering framework, which maps data into a reproducing kernel Hilbert space (RKHS) and performs convex optimization-based clustering in the high-dimensional feature space. The approach leverages the kernel trick to implicitly handle complex nonlinearities while preserving convexity, enabling simultaneous high-dimensional clustering and low-dimensional embedding. Contribution/Results: Unlike existing methods, it requires no pre-specified number of clusters and provides the first finite-sample error bound and rigorous convergence guarantee for kernelized convex clustering. Extensive experiments on diverse synthetic and real-world benchmarks demonstrate that the method consistently outperforms state-of-the-art clustering algorithms, exhibiting superior robustness and generalization capability on complex, multimodal, and non-convex data distributions.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Data Mining & Knowledge Management: Mining of Visual, Multimedia & Multimodal Data

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Convex clustering is a well-regarded clustering method, resembling the similar centroid-based approach of Lloyd's $k$-means, without requiring a predefined cluster count. It starts with each data point as its centroid and iteratively merges them. Despite its advantages, this method can fail when dealing with data exhibiting linearly non-separable or non-convex structures. To mitigate the limitations, we propose a kernelized extension of the convex clustering method. This approach projects the data points into a Reproducing Kernel Hilbert Space (RKHS) using a feature map, enabling convex clustering in this transformed space. This kernelization not only allows for better handling of complex data distributions but also produces an embedding in a finite-dimensional vector space. We provide a comprehensive theoretical underpinnings for our kernelized approach, proving algorithmic convergence and establishing finite sample bounds for our estimates. The effectiveness of our method is demonstrated through extensive experiments on both synthetic and real-world datasets, showing superior performance compared to state-of-the-art clustering techniques. This work marks a significant advancement in the field, offering an effective solution for clustering in non-linear and non-convex data scenarios.

Problem

• Extends convex clustering to handle non-linearly separable data structures
• Projects data into RKHS for clustering complex distributions
• Provides theoretical guarantees including convergence and finite sample bounds

Innovation

• Kernelized convex clustering in RKHS space
• Feature map projection for complex distributions
• Finite-dimensional embedding with theoretical guarantees