Feature selection by higher criticism thresholding achieves the optimal phase diagram

Abstract

We consider two-class linear classification in a high-dimensional, small-sample-size setting. Only a small fraction of the features are useful, these being unknown to us, and each useful feature contributes weakly to the classification decision. This was called the rare/weak (RW) model in our previous study (Donoho, D. & Jin, J. 2008 Proc. Natl Acad. Sci. USA 105, 14 790-14 795). We select features by thresholding feature Z-scores. The threshold is set by higher criticism (HC). For 1<or=i<or=N, let pi(i) denote the p-value associated with the ith Z-score and pi((i)) denote the ith order statistic of the collection of p-values. The HC threshold (HCT) is the order statistic of the Z-score corresponding to index i maximizing (i/N -- pi(i))/ (square root (i/N (1-i/N))). The ideal threshold optimizes the classification error. In that previous study, we showed that HCT was numerically close to the ideal threshold. We formalize an asymptotic framework for studying the RW model, considering a sequence of problems with increasingly many features and relatively fewer observations. We show that, along this sequence, the limiting performance of ideal HCT is essentially just as good as the limiting performance of ideal thresholding. Our results describe two-dimensional phase space, a two-dimensional diagram with coordinates quantifying 'rare' and 'weak' in the RW model. The phase space can be partitioned into two regions-one where ideal threshold classification is successful, and one where the features are so weak and so rare that it must fail. Surprisingly, the regions where ideal HCT succeeds and fails make exactly the same partition of the phase diagram. Other threshold methods, such as false (feature) discovery rate (FDR) threshold selection, are successful in a substantially smaller region of the phase space than either HCT or ideal thresholding. The FDR and local FDR of the ideal and HC threshold selectors have surprising phase diagrams, which are also described. Results showing the asymptotic equivalence of HCT with ideal HCT can be found in a forthcoming paper (Donoho, D. & Jin, J. In preparation).