# Sparse Generalized Eigenvalue Problem via Smooth Optimization

arXiv: https://arxiv.org/abs/1408.6686 · HTML: https://mlsanity.com/paper/1408.6686/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Aug 2014.

## Built on by (most cited first, 6 of 6)

- [RU-Net: Regularized Unrolling Network for Scene Graph Generation](https://mlsanity.com/paper/2205.01297/) (2022)
- [Sparse Quadratic Optimisation over the Stiefel Manifold with Application to Permutation Synchronisation](https://mlsanity.com/paper/2110.00053/) (2021)
- [Generalized Eigenvalue Problems with Generative Priors](https://mlsanity.com/paper/2411.01326/) (2024)
- [An iterative coordinate descent algorithm to compute sparse low-rank approximations](https://mlsanity.com/paper/2107.14608/) (2021)
- [Penalized Principal Component Analysis Using Smoothing](https://mlsanity.com/paper/2309.13838/) (2023)
- [Optimizing Sparse Generalized Singular Vectors for Feature Selection in Proximal Support Vector Machines with Application to Breast and Ovarian Cancer Detection](https://mlsanity.com/paper/2410.03978/) (2024)

## Abstract (arXiv)

In this paper, we consider an ℓ0-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 ℓ0-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.