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- Title
On the eigenvectors of p-Laplacian.
- Authors
Luo, Dijun; Huang, Heng; Ding, Chris; Nie, Feiping
- Abstract
Spectral analysis approaches have been actively studied in machine learning and data mining areas, due to their generality, efficiency, and rich theoretical foundations. As a natural non-linear generalization of Graph Laplacian, p-Laplacian has recently been proposed, which interpolates between a relaxation of normalized cut and the Cheeger cut. However, the relaxation can only be applied to two-class cases. In this paper, we propose full eigenvector analysis of p-Laplacian and obtain a natural global embedding for multi-class clustering problems, instead of using greedy search strategy implemented by previous researchers. An efficient gradient descend optimization approach is introduced to obtain the p-Laplacian embedding space, which is guaranteed to converge to feasible local solutions. Empirical results suggest that the greedy search method often fails in many real-world applications with non-trivial data structures, but our approach consistently gets robust clustering results. Visualizations of experimental results also indicate our embedding space preserves the local smooth manifold structures existing in real-world data.
- Subjects
EIGENVECTORS; LAPLACIAN operator; SPECTRUM analysis; GRAPH theory; CLUSTER analysis (Statistics); DATA structures
- Publication
Machine Learning, 2010, Vol 81, Issue 1, p37
- ISSN
0885-6125
- Publication type
Article
- DOI
10.1007/s10994-010-5201-z