This paper deals with the estimation of a high-dimensional covariance with a con-
ditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error
covariance matrix in an approximate factor model, we allow for the presence of some
cross-sectional correlation even after taking out common but unobservable factors.
We introduce the Principal Orthogonal complEment Thresholding (POET) method
to explore such an approximate factor structure with sparsity. The POET estimator
includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan,
and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive
thresholding estimator (Cai and Liu, 2011) as speci c examples. We provide mathe-
matical insights when the factor analysis is approximately the same as the principal
component analysis for high-dimensional data. The rates of convergence of the sparse
residual covariance matrix and the conditional sparse covariance matrix are studied
under various norms. It is shown that the impact of estimating the unknown factors
vanishes as the dimensionality increases. The uniform rates of convergence for the un-
observed factors and their factor loadings are derived. The asymptotic results are also
veri ed by extensive simulation studies. Finally, a real data application on portfolio
allocation is presented.