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

• 作 者: ShiqiongHuang;JiashunJin;ZhigangYao;
• 机 构: Carnegie Mellon University and National University of Singapore,Singapore
• 出 处: The 24th International Workshop on Matrices and Statistics(第
• 发表日期: 2015年期

　　We propose Partial Correlation Screening (PCS) as a new row-by-row approach. To estimate the i-th row of Ω, 1 ≤ i ≤ p, PCS uses a Screen step and a Clean step. In the Screen step, PCS re-cruits a (small) subset of indices using a stage-wise algorithm, where in each stage, the algorithm updates the set of recruited indices by adding the index j that has the largest empirical partial correlation (in magnitude) with i, given the set of indices recruited so far. In the Clean step, PCS rst re-investigates all recruited indices in hopes of removing false positives, and then uses the resultant set of indices to reconstruct the i-th row of Ω.PCS is computationally efficient and modest in memory use: to estimate a row of Ω,it only needs a few rows (determined sequentially) of the empirical covariance matrix. This enables PCS to execute the estimation of a large precision ma-trix (e.g., p = 10K) in a few minutes, and open doors to estimating much larger precision matrices. We use PCS for classification. Higher Criticism Thresholding (HCT) is a recent classifier that enjoys opti-mality, but to exploit its full potential in practice, one needs a good estimate of the precision matrix. Combining HCT with any ap-proach to estimating Ω gives a new classifier: examples include HCT-PCS and HCT-glasso. We set up a general theoretical framework and show that in a broad context, PCS fully recovers the support of Ω and HCT-PCS yields optimal classification behavior. Our proofs shed interesting light on the behavior of stage-wise procedures.