论文标题
最佳结构性主空间估计：度量熵和最小值速率
Optimal Structured Principal Subspace Estimation: Metric Entropy and Minimax Rates
论文作者
论文摘要
在广泛的应用范围内，许多主要的子空间估计问题已在不同的结构约束下进行了单独研究。本文提出了一个统一的框架，用于对通用结构性主空间估计问题进行统计分析，该估计问题包括特殊情况非负PCA/SVD，稀疏PCA/SVD，子空间约束PCA/SVD和光谱群集。建立了一般最小值下限和上限，以表征主要子空间的结构集的信息几何复杂性，信噪比（SNR）（SNR）和维度。结果产生了有关收敛速率与SNR的函数以及一致估计的基本限制有关的有趣的相变现象。将一般结果应用于特定设置可为这些问题带来最小收敛速率，包括以前的非负PCA/SVD的最佳最佳速率，稀疏SVD和子空间约束PCA/SVD。
Driven by a wide range of applications, many principal subspace estimation problems have been studied individually under different structural constraints. This paper presents a unified framework for the statistical analysis of a general structured principal subspace estimation problem which includes as special cases non-negative PCA/SVD, sparse PCA/SVD, subspace constrained PCA/SVD, and spectral clustering. General minimax lower and upper bounds are established to characterize the interplay between the information-geometric complexity of the structural set for the principal subspaces, the signal-to-noise ratio (SNR), and the dimensionality. The results yield interesting phase transition phenomena concerning the rates of convergence as a function of the SNRs and the fundamental limit for consistent estimation. Applying the general results to the specific settings yields the minimax rates of convergence for those problems, including the previous unknown optimal rates for non-negative PCA/SVD, sparse SVD and subspace constrained PCA/SVD.
