Feature selection by Higher Criticism thresholding: optimal phase diagram

Abstract

We consider two-class linear classification in a high-dimensional, low-sample size setting. Only a small fraction of the features are useful, the useful features are unknown to us, and each useful feature contributes weakly to the classification decision -- this setting was called the rare/weak model (RW Model). We select features by thresholding feature $z$-scores. The threshold is set by {\it higher criticism} (HC). Let $\pee_i$ denote the $P$-value associated to the $i$-th $z$-score and $\pee_{(i)}$ denote the $i$-th 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 $(i/n - \pee_{(i)})/\sqrt{\pee_{(i)}(1-\pee_{(i)})}$. The ideal threshold optimizes the classification error. In \cite{PNAS} 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 {\it phase space}, a two-dimensional diagram with coordinates quantifying "rare" and "weak" in the RW model. 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 the exact same partition of the phase diagram. Other threshold methods, such as FDR threshold selection, are successful in a substantially smaller region of the phase space than either HCT or Ideal thresholding.