An extension of fast iterative shrinkage‐thresholding algorithm to Riemannian optimization for sparse principal component analysis
Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the ScoTLASS model1 for sparse PCA which can impose orthogonality an...
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| Vydáno v: | Numerical linear algebra with applications Ročník 29; číslo 1 |
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| Abstract | Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the ScoTLASS model1 for sparse PCA which can impose orthogonality and sparsity simultaneously. A Riemannian proximal method is proposed in the work of Chen et al.9 for the efficient solution of this optimization problem. In this paper, two acceleration schemes are introduced. First and foremost, we extend the FISTA method from the Euclidean space to the Riemannian manifold to solve sparse PCA, leading to the accelerated Riemannian proximal gradient method. Since the Riemannian optimization problem for sparse PCA is essentially nonconvex, a restarting technique is adopted to stabilize the accelerated method without sacrificing the fast convergence. Second, a diagonal preconditioner is proposed for the Riemannian proximal subproblem which can further accelerate the convergence of the Riemannian proximal methods. Numerical evaluations establish the computational advantages of the proposed methods over the existing proximal gradient methods on a manifold. Additionally, a short result concerning the convergence of the Riemannian subgradients of a sequence is established, which, together with the result in the work of Chen et al.,9 can show the stationary point convergence of the Riemannian proximal methods. |
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| AbstractList | Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the ScoTLASS model for sparse PCA which can impose orthogonality and sparsity simultaneously. A Riemannian proximal method is proposed in the work of Chen et al. for the efficient solution of this optimization problem. In this paper, two acceleration schemes are introduced. First and foremost, we extend the FISTA method from the Euclidean space to the Riemannian manifold to solve sparse PCA, leading to the accelerated Riemannian proximal gradient method. Since the Riemannian optimization problem for sparse PCA is essentially nonconvex, a restarting technique is adopted to stabilize the accelerated method without sacrificing the fast convergence. Second, a diagonal preconditioner is proposed for the Riemannian proximal subproblem which can further accelerate the convergence of the Riemannian proximal methods. Numerical evaluations establish the computational advantages of the proposed methods over the existing proximal gradient methods on a manifold. Additionally, a short result concerning the convergence of the Riemannian subgradients of a sequence is established, which, together with the result in the work of Chen et al., can show the stationary point convergence of the Riemannian proximal methods. Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the ScoTLASS model1 for sparse PCA which can impose orthogonality and sparsity simultaneously. A Riemannian proximal method is proposed in the work of Chen et al.9 for the efficient solution of this optimization problem. In this paper, two acceleration schemes are introduced. First and foremost, we extend the FISTA method from the Euclidean space to the Riemannian manifold to solve sparse PCA, leading to the accelerated Riemannian proximal gradient method. Since the Riemannian optimization problem for sparse PCA is essentially nonconvex, a restarting technique is adopted to stabilize the accelerated method without sacrificing the fast convergence. Second, a diagonal preconditioner is proposed for the Riemannian proximal subproblem which can further accelerate the convergence of the Riemannian proximal methods. Numerical evaluations establish the computational advantages of the proposed methods over the existing proximal gradient methods on a manifold. Additionally, a short result concerning the convergence of the Riemannian subgradients of a sequence is established, which, together with the result in the work of Chen et al.,9 can show the stationary point convergence of the Riemannian proximal methods. |
| Author | Huang, Wen Wei, Ke |
| Author_xml | – sequence: 1 givenname: Wen orcidid: 0000-0001-8324-2416 surname: Huang fullname: Huang, Wen email: wen.huang@xmu.edu.cn organization: Xiamen University – sequence: 2 givenname: Ke surname: Wei fullname: Wei, Ke email: kewei@fudan.edu.cn organization: Fudan University |
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| SubjectTerms | Algorithms Convergence Euclidean geometry Euclidean space Iterative methods Manifolds (mathematics) Methods nonsmooth optimization on manifold Numerical methods Optimization Orthogonality Principal components analysis proximal gradient method Restarting Riemann manifold Riemannian optimization sparse PCA |
| Title | An extension of fast iterative shrinkage‐thresholding algorithm to Riemannian optimization for sparse principal component analysis |
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