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We propose Partial Correlation Screening (PCS) as a new row-by-row approachto estimating a large precision matrix $\Omega$. To estimate the $i$-th row of$\Omega$, $1 \leq i \leq p$, PCS uses a Screen step and a Clean step. In theScreen step, PCS recruits a (small) subset of indices using a stage-wisealgorithm, where in each stage, the algorithm updates the set of recruitedindices by adding the index $j$ that has the largest (in magnitude) empiricalpartial correlation with $i$. In the Clean step, PCS re-investigates allrecruited indices and use them to reconstruct the $i$-th row of $\Omega$.  PCS is computationally efficient and modest in memory use: to estimate a rowof $\Omega$, it only needs a few rows (determined sequentially) of theempirical covariance matrix. This enables PCS to execute the estimation of alarge precision matrix (e.g., $p=10K$) in a few minutes, and open doors toestimating much larger precision matrices.  We use PCS for classification. Higher Criticism Thresholding (HCT) is arecent classifier that enjoys optimality, but to exploit its full potential inpractice, one needs a good estimate of the precision matrix $\Omega$. CombiningHCT with any approach to estimating $\Omega$ gives a new classifier: examplesinclude HCT-PCS and HCT-glasso. We have applied HCT-PCS to two large microarraydata sets ($p = 8K$ and $10K$) for classification, where it not onlysignificantly outperforms HCT-glasso, but also is competitive to the SupportVector Machine (SVM) and Random Forest (RF). The results suggest that PCS givesmore useful estimates of $\Omega$ than the glasso.  We set up a general theoretical framework and show that in a broad context,PCS fully recovers the support of $\Omega$ and HCT-PCS yields optimalclassification behavior. Our proofs shed interesting light on the behavior ofstage-wise procedures.

Partial Correlation Screening for Estimating Large Precision Matrices, with Applications to Classification