Abstract
In this paper, we extend the APG method to solve matrix l_{2,1}-norm minimization problem in multi-task feature learning. We investigate that the resulting inner subproblem has closed-form solution which can be easily determined by taking the problem's favorable structures. Under suitable conditions, we can establish a comprehensive convergence result for the proposed method. Furthermore, we present three different inexact APG algorithms by using the Lipschitz constant, the eigenvalue of Hessian matrix and the Barzilai and Borwein parameter in the inexact model, respectively. Numerical experiments on simulated data and real data set are reported to show the efficiency of proposed method.
Highlights
Consider the following matrix l2,1-norm minimization problem min X ∈Rn×t 1 ∥AX − b∥22 + μ∥X ∥2,1, (1)ISSN 2310-5070 ISSN 2311-004XCopyright ⃝c 2014 International Academic Press where the matrix l2,1-norm ∥X∥2,1 is defined by the sum of l2-norm of each row ∑n ∥X∥2,1 =
In this paper, we extend the implementable Accelerated proximal gradient (APG) method to solve the matrix l2,1-norm minimization problem arising in multi-task feature learning
We investigate that the resulting inner subproblem has closed-form solution which can be determined by taking the problem’s favorable structures
Summary
Copyright ⃝c 2014 International Academic Press where the matrix l2,1-norm ∥X∥2,1 is defined by the sum of l2-norm of each row. The extension of the APG algorithm to the matrix l2,1-norm minimization problem in multi-task feature learning is interesting in terms of practical perspective, because it takes computational advantage over alternative algorithms for solving the problem (1). We solve the matrix l2,1-norm minimization problem by using some inexact APG algorithms, establish the iteration complexities, and present numerical results to demonstrate the efficiency of our proposed algorithms. We present inexact APG version with three different choices of the self-adjoint positive definite linear operator Hk. In Section 4 we conduct some preliminary numerical experiments to evaluate the practical performance of our proposed inexact APG algorithms for solving matrix l2,1-norm minimization problems arising from simulated data and real data set, compare it with the existing IADM-MFL method.
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