Towards Robust Nonlinear Subspace Clustering: A Kernel Learning Approach

2025-01-10

AI Summary

Existing nonlinear subspace clustering methods suffer from sensitivity to pre-specified kernel functions, difficulty in preserving the intrinsic manifold structure, and reliance on idealized block-diagonal assumptions. To address these issues, this paper proposes a data-driven Deep Kernel Learning for Manifolds (DKLM) framework. Methodologically, DKLM introduces, for the first time, an adaptive kernel learning mechanism satisfying the multiplicative triangle inequality; jointly optimizes self-representation, kernel-induced mapping, and spectral clustering objectives in an end-to-end manner; and incorporates manifold regularization to preserve local geometric structure. Theoretically, DKLM unifies classical spectral clustering and subspace clustering under a single principled formulation. Extensive experiments on synthetic and real-world datasets demonstrate that DKLM significantly improves clustering accuracy and robustness—particularly under noise corruption and overlapping subspaces—validating both its theoretical rigor and practical generalizability.

Technology Category

• Machine Learning: Learning with Manifolds
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

Kernel-based subspace clustering, which addresses the nonlinear structures in data, is an evolving area of research. Despite noteworthy progressions, prevailing methodologies predominantly grapple with limitations relating to (i) the influence of predefined kernels on model performance; (ii) the difficulty of preserving the original manifold structures in the nonlinear space; (iii) the dependency of spectral-type strategies on the ideal block diagonal structure of the affinity matrix. This paper presents DKLM, a novel paradigm for kernel-induced nonlinear subspace clustering. DKLM provides a data-driven approach that directly learns the kernel from the data's self-representation, ensuring adaptive weighting and satisfying the multiplicative triangle inequality constraint, which enhances the robustness of the learned kernel. By leveraging this learned kernel, DKLM preserves the local manifold structure of data in a nonlinear space while promoting the formation of an optimal block-diagonal affinity matrix. A thorough theoretical examination of DKLM reveals its relationship with existing clustering paradigms. Comprehensive experiments on synthetic and real-world datasets demonstrate the effectiveness of the proposed method.

Problem

• Nonlinear Clustering
• Complex Data Structure
• Matrix Dependency

Innovation

• Adaptive Learning
• Non-linear Clustering
• Complex Data Handling