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sparse generalized eigenvalue problem Via Smooth Optimization

IEEE TRANSACTIONS ON SIGNAL PROCESSING 2015年第7期63卷 1627-1642页

作者： Song, Junxiao Babu, Prabhu Palomar, Daniel P. Hong Kong Univ Sci & Technol Dept Elect & Comp Engn Hong Kong Hong Kong Peoples R China

In this paper, we consider an - 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 - 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.

关键词： Minorization-maximization sparse generalized eigenvalue problem sparse PCA smooth optimization

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A Fast Algorithm for sparse generalized eigenvalue problem 48

A Fast Algorithm for Sparse Generalized Eigenvalue Problem

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48h Asilomar Conference on Signals, Systems and Computers

作者： Song, Junxiao Babu, Prabhu Palomar, Daniel P. Hong Kong Univ Sci & Technol Dept Elect & Comp Engn Hong Kong Hong Kong Peoples R China

ISBN: (纸本)9781479982974

In this paper, we consider an l(0)-norm penalized formulation of the generalized eigenvalue problem, 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 and differentiable surrogate function. Then an algorithm is developed via iteratively majorizing the surrogate function by a separable quadratic function, which at each iteration reduces then to a regular generalized eigenvalue problem. The convergence of the proposed algorithm to a stationary point of an equivalent problem is proved. Numerical experiments show that the proposed algorithm outperforms existing algorithms in terms of both computational complexity and support recovery.

关键词： Minorization-maximization sparse generalized eigenvalue problem sparse PCA

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Efficient Contour Integral-based eigenvalue Computation Using an Iterative Linear Solver with Shift-Invert Preconditioning 2021

Efficient Contour Integral-based Eigenvalue Computation Usin...

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International Conference on High Performance Computing in Asia-Pacific Region (HPC Asia)

作者： Futamura, Yasunori Sakurai, Tetsuya Univ Tsukuba Tsukuba Ibaraki Japan

ISBN: (纸本)9781450388429

Contour integral-based (CI) eigenvalue solvers are one of the efficient and robust approaches for sparse eigenvalue problems. They have attracted attention owing to their inherent parallelism. For implementing a CI eigensolver, the inner linear systems arising in the algorithm need to be solved using an efficient method. One widely-used method is to use a sparse direct linear solver provided by a well-established numerical library;it is numerically robust and presents good load balancing of parallel execution of the CI eigensolver. However, owing to high total computational and memory cost, the performance of the direct solver approach is suboptimal. In this study, we propose an alternative method that utilizes a block Krylov iterative linear solver and shift-invert preconditioning that can take advantage of the shift-invariance of the block Krylov subspace. Our approach adaptively sets a preconditioning parameter according to the number of parallel processes to reduce the iteration counts. Several numerical examples confirm that our method outperforms the direct solver approach.

关键词： sparse generalized eigenvalue problem contour integral-based solver block Krylov subspace method shift-invert preconditioning

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FIRST-ORDER ALGORITHMS FOR A CLASS OF FRACTIONAL OPTIMIZATION problemS

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SIAM JOURNAL ON OPTIMIZATION 2022年第1期32卷 100-129页

作者： Zhang, Na Li, Qia South China Agr Univ Coll Math & Informat Dept Appl Math Guangzhou 510642 Peoples R China Sun Yat Sen Univ Sch Comp Sci & Engn Guangdong Prov Key Lab Computat Sci Guangzhou 510275 Peoples R China

In this paper, we consider a class of single-ratio fractional minimization problems, in which the numerator of the objective is the sum of a nonsmooth nonconvex function and a smooth nonconvex function while the denominator is a nonsmooth convex function. In this work, we first derive its first-order necessary optimality condition, by using the first-order operators of the three functions involved. Then we develop first-order algorithms, namely, the proximity-gradientsubgradient algorithm (PGSA), PGSA with monotone line search (PGSA ML), and PGSA with nonmonotone line search (PGSA NL). It is shown that any accumulation point of the sequence generated by them is a critical point of the problem under mild assumptions. Moreover, we establish global convergence of the sequence generated by PGSA or PGSA ML and analyze its convergence rate, by further assuming the local Lipschitz continuity of the nonsmooth function in the numerator, the smoothness of the denominator, and the Kurdyka--\Lojasiewicz (KL) property of the objective. The proposed algorithms are applied to the sparse generalized eigenvalue problem associated with a pair of symmetric positive semidefinite matrices, and the corresponding convergence results are obtained according to their general convergence theorems. We perform some preliminary numerical experiments to demonstrate the efficiency of the proposed algorithms.

关键词： fractional optimization first-order algorithms proximity algorithms sparse generalized eigenvalue problem KL property

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