自适应学习核函数，提升非线性子空间聚类的鲁棒性
Towards Robust Nonlinear Subspace Clustering: A Kernel Learning Approach
- 基于数据自表示直接学习核函数，避免预设核的限制
- 保留非线性空间中的局部流形结构，促进最优块对角相似矩阵形成
- 理论证明与实验验证，适用于复杂结构数据聚类
基于核的子空间聚类能够捕捉数据中的非线性结构，是当前研究热点。然而，现有方法普遍存在三大局限：(i) 预定义核对模型性能影响大；(ii) 难以保持非线性空间中原始流形结构；(iii) 谱类策略依赖理想块对角相似矩阵结构。本文提出DKLM，一种新型核诱导的非线性子空间聚类范式。DKLM通过数据自表示直接学习核函数，实现自适应加权，并满足乘法三角不等式约束，增强核的鲁棒性。利用该学习核，DKLM在非线性空间中有效保留数据局部流形结构，同时促进最优块对角相似矩阵的生成。理论分析揭示其与已有聚类范式的联系。在合成与真实数据集上的大量实验表明该方法具有显著有效性。
原文摘要 · Abstract (English)
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.
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