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Manifold Clustering with Schatten p-norm Maximization

Fangfang Li, Quanxue Gao

TL;DR

This work tackles nonlinear clustering by unifying K-means with manifold learning in a label-guided framework. It constructs a manifold structure from the label matrix and optimizes clustering on that manifold while maximizing the Schatten $p$-norm of the label matrix $G$ to promote balanced classes, via a continuous relaxation with $\|G\|_{sp}^p$ and a trace-based objective. Theoretical contributions include linking K-means and manifold learning through a trace form and deriving a gradient for the Schatten p-norm using SVD, yielding an iterative update that preserves nonnegativity and sum-to-one constraints. Empirically, the approach with flexible distance metrics (Euclidean or KNN) improves clustering performance on synthetic nonlinear datasets and benchmark datasets (JAFFE, ORL) compared with several baselines, validating both its effectiveness and adaptability to nonlinear data.

Abstract

Manifold clustering, with its exceptional ability to capture complex data structures, holds a pivotal position in cluster analysis. However, existing methods often focus only on finding the optimal combination between K-means and manifold learning, and overlooking the consistency between the data structure and labels. To address this issue, we deeply explore the relationship between K-means and manifold learning, and on this basis, fuse them to develop a new clustering framework. Specifically, the algorithm uses labels to guide the manifold structure and perform clustering on it, which ensures the consistency between the data structure and labels. Furthermore, in order to naturally maintain the class balance in the clustering process, we maximize the Schatten p-norm of labels, and provide a theoretical proof to support this. Additionally, our clustering framework is designed to be flexible and compatible with many types of distance functions, which facilitates efficient processing of nonlinear separable data. The experimental results of several databases confirm the superiority of our proposed model.

Manifold Clustering with Schatten p-norm Maximization

TL;DR

This work tackles nonlinear clustering by unifying K-means with manifold learning in a label-guided framework. It constructs a manifold structure from the label matrix and optimizes clustering on that manifold while maximizing the Schatten -norm of the label matrix to promote balanced classes, via a continuous relaxation with and a trace-based objective. Theoretical contributions include linking K-means and manifold learning through a trace form and deriving a gradient for the Schatten p-norm using SVD, yielding an iterative update that preserves nonnegativity and sum-to-one constraints. Empirically, the approach with flexible distance metrics (Euclidean or KNN) improves clustering performance on synthetic nonlinear datasets and benchmark datasets (JAFFE, ORL) compared with several baselines, validating both its effectiveness and adaptability to nonlinear data.

Abstract

Manifold clustering, with its exceptional ability to capture complex data structures, holds a pivotal position in cluster analysis. However, existing methods often focus only on finding the optimal combination between K-means and manifold learning, and overlooking the consistency between the data structure and labels. To address this issue, we deeply explore the relationship between K-means and manifold learning, and on this basis, fuse them to develop a new clustering framework. Specifically, the algorithm uses labels to guide the manifold structure and perform clustering on it, which ensures the consistency between the data structure and labels. Furthermore, in order to naturally maintain the class balance in the clustering process, we maximize the Schatten p-norm of labels, and provide a theoretical proof to support this. Additionally, our clustering framework is designed to be flexible and compatible with many types of distance functions, which facilitates efficient processing of nonlinear separable data. The experimental results of several databases confirm the superiority of our proposed model.
Paper Structure (12 sections, 3 theorems, 23 equations, 3 figures, 1 table)

This paper contains 12 sections, 3 theorems, 23 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Unified Framework
  3. Methodology
  4. Our Model
  5. Optimization
  6. Experiments
  7. Experiments of Artificial Dataset
  8. Experimental on Benchmark Datasets
  9. Results
  10. Parameters setting and analysis
  11. Discussion of the value of p
  12. Conclusion

Key Result

Theorem 1

Suppose data matrix ${\mathbf{X}}=[\mathbf{x}_1\textrm{, }\ldots\textrm{ , }\mathbf{x}_N]\in \mathbb{R}^{d\times N}$, where $N$ is the number of samples, $d$ is the feature dimension. The label matrix is defined as ${{\mathbf{G}}} = {\left[ {{\mathbf{g}}_1, \ldots ,{\mathbf{g}}_K} \right]} \in \math where the manifold structure ${\mathbf{M}}$ represents the cluster structure in the data, ${\mathbf

Figures (3)

  • Figure : (a)K-means
  • Figure : (a)K-means
  • Figure : (a)Our

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • proof