We study a principal component analysis problem under the spiked Wishart model in which the structure in the signal is captured by a class of union-of-subspace models. This general class includes vanilla sparse PCA as well as its variants with graph sparsity. With the goal of studying these problems under a unified statistical and computational lens, we establish fundamental limits that depend on the geometry of the problem instance, and show that a natural projected power method exhibits local convergence to the statistically near-optimal neighborhood of the solution. We complement these results with end-to-end analyses of two important special cases given by path and tree sparsity in a general basis, showing initialization methods and matching evidence of computational hardness. Overall, our results indicate that several of the phenomena observed for vanilla sparse PCA extend in a natural fashion to its structured counterparts.

翻译：我们研究尖峰Wishart模型下的主成分分析问题，其中信号的结构由一类子空间并集模型刻画。该通用类包含经典稀疏PCA及其图稀疏变体。为在统一的统计与计算视角下研究这些问题，我们建立了依赖于问题实例几何特性的基本极限，并证明自然投影幂方法能在统计近优解邻域内实现局部收敛。我们进一步通过一般基下的路径与树稀疏这两个重要特例的端到端分析（包括初始化方法及计算难度的匹配证据）补充了上述结果。总体而言，我们的研究表明，经典稀疏PCA中观测到的若干现象能以自然方式推广至结构化对应版本。