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

Shubhayan Pan, Saptarshi Chakraborty, Debolina Paul, Kushal Bose, Swagatam Das

TL;DR

This work introduces kernel convex clustering (KCC), a kernelized extension of convex clustering that operates in a reproducing kernel Hilbert space to handle non-linear data structures. By deriving a finite-dimensional embedding via a kernel factorization $K=Z^T Z$, the authors recast the problem as standard convex clustering on embedded points and solve it with an ADMM-based routine, followed by back-transformation to obtain centroids. They establish finite-sample guarantees under a RKHS noise model and demonstrate consistent estimation under mild weight-sparsity conditions. Empirically, KCC outperforms several baselines on synthetic and real datasets, validating its effectiveness for clustering non-linear and non-convex patterns and highlighting its potential for multi-kernel extensions and feature weighting.

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.

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

TL;DR

This work introduces kernel convex clustering (KCC), a kernelized extension of convex clustering that operates in a reproducing kernel Hilbert space to handle non-linear data structures. By deriving a finite-dimensional embedding via a kernel factorization , the authors recast the problem as standard convex clustering on embedded points and solve it with an ADMM-based routine, followed by back-transformation to obtain centroids. They establish finite-sample guarantees under a RKHS noise model and demonstrate consistent estimation under mild weight-sparsity conditions. Empirically, KCC outperforms several baselines on synthetic and real datasets, validating its effectiveness for clustering non-linear and non-convex patterns and highlighting its potential for multi-kernel extensions and feature weighting.

Abstract

Convex clustering is a well-regarded clustering method, resembling the similar centroid-based approach of Lloyd's -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.

Paper Structure

This paper contains 16 sections, 1 theorem, 37 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proposed Method
  3. A Motivating Example
  4. Problem Formulation
  5. A perspective from Convex Clustering
  6. Getting the optimal number of clusters
  7. Complexity Analysis
  8. Theoretical Guarantees
  9. Experiments
  10. Results on Synthetic dataset
  11. Performance on Real Benchmarks
  12. Conclusion and Future Works
  13. Necessary Assumptions
  14. Theoretical Proofs
  15. Sensitivity Analysis of Hyperparameters of Synthetic Dataset
  16. ...and 1 more sections

Key Result

Theorem 1

Let $\phi(\boldsymbol{x}_i)=\boldsymbol{u}_i+\boldsymbol{\epsilon}_i$ for all $i=1,\dots,n$, where $\boldsymbol{\epsilon}_i$ are i.i.d. mean zero sub Gaussian random variables in the RKHS $\mathcal{H}$, with respect to the operator $\Gamma$. Let $\hat{\boldsymbol{u}}_i$ be the solutions of opti_prob

Figures (7)

  • Figure 1: t-SNE plots of GLI85 dataset for (a) ground truth labels, (b) $k$-means clustering, (c) convex clustering, and (d) KCC are presented. Applying kernels improves performance over the Euclidean similarity measure.
  • Figure 2: Scatter plots of the synthetic dataset for (a) ground truth labels, (b) KCC, (c) convex clustering, and (d) spectral clustering are illustrated.
  • Figure 3: The impact on NMI with varying numbers of clusters is presented. Our method KCC performs consistently compared to other methods.
  • Figure 4: Elbow plot of Lymphoma dataset. The study reveals that the optimal number of clusters is 7. Though the data contains 9 clusters but some of them contain a very small number of points, and KCC merges them.
  • Figure 5: t-SNE plots of Lymphoma dataset for (a) ground truth labels, (b) KCC, (c) spectral clustering, and (d) $k$-means clustering, are presented.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • proof