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${}^1$ 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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快速迭代收缩阈值算法对稀疏主成分分析的黎曼优化的扩展

稀疏主成分分析 (PCA) 是 PCA 的一个重要变体，它试图在进行降维时找到稀疏加载向量。本文考虑与ScoTLASS模型相关的非光滑黎曼优化问题${}^1$对于可以同时施加正交性和稀疏性的稀疏 PCA。Chen 等人的工作提出了黎曼近端方法。${}^9$为了有效地解决这个优化问题。本文介绍了两种加速方案。首先，我们将 FISTA 方法从欧几里得空间扩展到黎曼流形以解决稀疏 PCA，从而导致加速黎曼近端梯度方法。由于稀疏 PCA 的黎曼优化问题本质上是非凸的，因此在不牺牲快速收敛的情况下，采用重新启动技术来稳定加速方法。其次，针对黎曼近端子问题提出了一个对角预处理器，可以进一步加速黎曼近端方法的收敛。数值评估建立了所提出的方法相对于流形上现有的近端梯度方法的计算优势。此外，${}^9$ 可以显示黎曼近端方法的驻点收敛性。