Optimal Structured Principal Subspace Estimation: Metric Entropy and Minimax Rates
Optimal Structured Principal Subspace Estimation: Metric Entropy and Minimax Rates
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发表时间：
2020-02
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通讯作者：
T. Cai;Hongzhe Li;Rong Ma
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作者：
T. Cai;Hongzhe Li;Rong Ma
在广泛应用的驱动下，许多主子空间估计问题在不同的结构约束下被单独研究。本文给出了一般结构化主子空间估计问题统计分析的统一框架，其特例包括非负PCA/SVD、稀疏PCA/SVD、子空间约束的PCA/SVD和谱聚类。建立了广义极小极大下界和上界，刻画了主子空间结构集的信息几何复杂性、信噪比和维度之间的相互作用。结果产生了有趣的相变现象，涉及收敛速度作为信噪比的函数和一致性估计的基本极限。将一般结果应用于特定的设置，得到了这些问题的极小极大收敛速度，包括先前未知的非负PCA/SVD、稀疏SVD和子空间约束的PCA/SVD的最优收敛速度。
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.