A fast algorithm for sparse generalized eigenvalue problem

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