An Inverse Power Method for Nonlinear Eigenproblems with Applications in 1-Spectral Clustering and Sparse PCA (1012.0774v1)

Abstract: Many problems in machine learning and statistics can be formulated as (generalized) eigenproblems. In terms of the associated optimization problem, computing linear eigenvectors amounts to finding critical points of a quadratic function subject to quadratic constraints. In this paper we show that a certain class of constrained optimization problems with nonquadratic objective and constraints can be understood as nonlinear eigenproblems. We derive a generalization of the inverse power method which is guaranteed to converge to a nonlinear eigenvector. We apply the inverse power method to 1-spectral clustering and sparse PCA which can naturally be formulated as nonlinear eigenproblems. In both applications we achieve state-of-the-art results in terms of solution quality and runtime. Moving beyond the standard eigenproblem should be useful also in many other applications and our inverse power method can be easily adapted to new problems.

• The paper presents a generalized inverse power method (IPM) for solving nonlinear eigenproblems, extending the technique beyond traditional linear applications using nonsmooth analysis.
• Applying the method to the graph 1-Laplacian improves the quality of cuts in spectral clustering compared to standard spectral methods.
• The method solves sparse PCA, reformulated with an L1 penalty, achieving state-of-the-art performance for obtaining interpretable principal components.

A Comprehensive Examination of the Inverse Power Method for Nonlinear Eigenproblems and Its Application in Machine Learning

The paper presented by Hein and Bühler introduces an innovative extension of the inverse power method (IPM) catered explicitly to nonlinear eigenproblems, a significant departure from the classical linear scope. This extension embodies a transformative approach to the fields of spectral clustering and sparse principal component analysis (PCA), providing both theoretical advancements and computational efficacy.

Overview and Theoretical Contributions

Eigenvalue problems permeate numerous domains within machine learning and statistics, traditionally constrained within the field of symmetric and positive semi-definite matrices. Here, eigenvectors arise as critical points of quadratic functionals constrained in a similar quadratic manner. The authors challenge this conventional paradigm by conceptualizing a framework where p-homogeneous functions replace quadratic forms, consequently leading to a generalized class of nonlinear eigenproblems.

The paper's primary theoretical contribution is the development of a generalized inverse power method. This method diverges from typical linear eigenproblems, proposing an efficient iterative process guaranteed to converge towards a nonlinear eigenvector. The underpinning theory is grounded in nonsmooth analysis, addressing the complexities that arise from using non-quadratic objective functions within constrained optimization problems. Key theoretical insights involve leveraging the Euler identity to articulate critical points through subdifferentials, ensuring the preservation of necessary properties for reliable eigenvector computation.

Applications in Spectral Clustering and Sparse PCA

Two significant applications unveil the utility of this generalized IPM. Firstly, in the domain of spectral clustering, this method addresses limitations observed in traditional spectral relaxations associated with Cheeger cuts. By employing the graph p-Laplacian, specifically focusing on the 1-Laplacian, the paper substantiates improvements in cut quality. This extension is corroborated by empirical results, which illustrate enhanced performance compared to established spectral methods.

The second application applies the IPM to sparse PCA, a model reformulated to support sparsity in linear projections. This novel formulation utilizes a penalty based on the L1 norm, operationalized through the generalized IPM to resolve the sparse PCA as a nonlinear eigenproblem. The results reported align with current benchmarks, indicating state-of-the-art performance in obtaining interpretable principal components.

Implications and Future Directions

This research opens pathways for broader applications of nonlinear eigenproblems in various fields such as image segmentation, community detection, and dimensionality reduction techniques. The design and convergence of this IPM provide methodologies apt for complex datasets where linear approximations falter, particularly in capturing manifold structures and underlying data sparsity.

Theoretically, the paper suggests potential explorations into inverse methods applicable to other forms of nonlinear operators, namely those emerging in non-symmetric setups or with nonpositive definite structures. Practically, adapting this framework for distributed computing or real-time analysis scenarios could immensely benefit large-scale and high-dimension datasets typical in modern machine learning challenges.

Conclusion

Hein and Bühler’s paper delivers a significant leap forward in the methodology for solving eigenvalue problems beyond conventional boundaries. Through a deft combination of advanced mathematical rigor and practical adaptability, the IPM for nonlinear eigenproblems stands as a robust tool with noticeable implications for applied computing and theoretical research alike. Moving beyond basic spectral clustering and PCA, this method positions itself as a versatile instrument for next-generation solutions that demand both precision and computational efficiency.