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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Bibliographic Details
Published in:Numerical linear algebra with applications Vol. 29; no. 1
Main Authors: Huang, Wen, Wei, Ke
Format: Journal Article
Language:English
Published: Oxford Wiley Subscription Services, Inc 01.01.2022
Subjects:
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
ISSN:1070-5325, 1099-1506
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Summary: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.
Bibliography:Funding information
National Natural Science Foundation of China, 12001455; NSFC, 11801088; Shanghai Sailing Program, 18YF1401600; The Fundamental Research Funds for the Central Universities, 20720190060
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ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2409