Sparse Generalized Eigenvalue Problem Via Smooth Optimization

In this paper, we consider an ℓ 0 -norm penalized formulation of the generalized eigenvalue problem (GEP), aimed at extracting the leading sparse generalized eigenvector of a matrix pair. The formulation involves maximization of a discontinuous nonconcave objective function over a nonconvex constrai...

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Bibliographic Details
Published in:IEEE transactions on signal processing Vol. 63; no. 7; pp. 1627 - 1642
Main Authors: Junxiao Song, Babu, Prabhu, Palomar, Daniel P.
Format: Journal Article
Language:English
Published: IEEE 01.04.2015
Subjects:
Approximation methods
Eigenvalues and eigenfunctions
Minorization-maximization
Principal component analysis
Signal processing algorithms
smooth optimization
sparse generalized eigenvalue problem
Sparse matrices
sparse PCA
Tin
Vectors
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Summary:In this paper, we consider an ℓ 0 -norm penalized formulation of the generalized eigenvalue problem (GEP), aimed at extracting the leading sparse generalized eigenvector of a matrix pair. The formulation involves maximization of a discontinuous nonconcave objective function over a nonconvex constraint set, and is therefore computationally intractable. To tackle the problem, we first approximate the ℓ 0 -norm by a continuous surrogate function. Then an algorithm is developed via iteratively majorizing the surrogate function by a quadratic separable function, which at each iteration reduces to a regular generalized eigenvalue problem. A preconditioned steepest ascent algorithm for finding the leading generalized eigenvector is provided. A systematic way based on smoothing is proposed to deal with the "singularity issue" that arises when a quadratic function is used to majorize the nondifferentiable surrogate function. For sparse GEPs with special structure, algorithms that admit a closed-form solution at every iteration are derived. Numerical experiments show that the proposed algorithms match or outperform existing algorithms in terms of computational complexity and support recovery.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2015.2394443