The Fantope-constrained sparse principal subspace estimation problem is initially proposed by Vu et al. (Vu et al., 2013). This paper investigates a semismooth Newton based proximal point () algorithm for solving the equivalent form of this problem, where a semismooth Newton () method is utilized to optimize the inner problems involved in the algorithm. Under standard conditions, the algorithm is proven to achieve global convergence and an asymptotic superlinear convergence rate. Computationally, we derive nontrivial expressions for the Fantope projection and its generalized Jacobian, which are key ingredients for the algorithm. Some numerical results on synthetic and real data sets are presented to illustrate the effectiveness of the proposed algorithm for large-scale problems and superiority over the alternating direction method of multipliers (ADMM).

中文翻译：

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Fantope约束稀疏主子空间估计问题的高效算法

Fantope约束的稀疏主子空间估计问题最初由Vu等人提出。 （Vu 等人，2013）。本文研究了一种基于半光滑牛顿的近端点（）算法来求解该问题的等价形式，其中利用半光滑牛顿（）方法来优化算法中涉及的内部问题。在标准条件下，该算法被证明能够实现全局收敛和渐近超线性收敛速度。通过计算，我们推导出 Fantope 投影及其广义雅可比行列式的重要表达式，这是该算法的关键要素。给出了合成数据集和真实数据集上的一些数值结果，以说明所提出的算法对于大规模问题的有效性以及相对于乘子交替方向法（ADMM）的优越性。