  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.
