Optimal Structured Principal Subspace Estimation: Metric Entropy and Minimax Rates

Driven by a wide range of applications, several 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 sparse PCA/SVD, non-negative 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 constraint 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 sparse SVD, non-negative PCA/SVD and subspace constrained PCA/SVD.

Automated summary

This paper presents a unified framework for the statistical analysis of various principal subspace estimation problems, revealing the interplay between the constraint set, signal-to-noise ratio, and dimensionality. The research results demonstrate interesting phase transition phenomena concerning the rates of convergence related to the signal-to-noise ratio and fundamental limit for consistent estimation.