Sparse Generalized Eigenvalue Problem Via Smooth Optimization


In this paper, we contemplate an $ell_0$-norm penalized formulation of the generalized eigenvalue downside (GEP), aimed at extracting the leading sparse generalized eigenvector of a matrix combine. The formulation involves maximization of a discontinuous nonconcave objective operate over a nonconvex constraint set, and is therefore computationally intractable. To tackle the problem, we have a tendency to first approximate the $ell_0$-norm by never-ending surrogate operate. Then an algorithm is developed via iteratively majorizing the surrogate operate by a quadratic separable operate, which at each iteration reduces to an everyday generalized eigenvalue downside. A preconditioned steepest ascent algorithm for locating the leading generalized eigenvector is provided. A systematic means primarily based on smoothing is proposed to house the “singularity issue” that arises when a quadratic operate is used to majorize the nondifferentiable surrogate perform. For sparse GEPs with special structure, algorithms that admit a closed-form answer 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.

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