Influential Feature PCA for high dimensional clustering

We consider a clustering problem where we observe feature vectors $X_i \in R^p$, $i = 1, 2, \ldots, n$, from $K$ possible classes. The class labels are unknown and the main interest is to estimate them. We are primarily interested in the modern regime of $p \gg n$, where classical clustering methods face challenges. We propose Influential Features PCA (IF-PCA) as a new clustering procedure. In IF-PCA, we select a small fraction of features with the largest Kolmogorov-Smirnov (KS) scores, where the threshold is chosen by adapting the recent notion of Higher Criticism, obtain the first $(K-1)$ left singular vectors of the post-selection normalized data matrix, and then estimate the labels by applying the classical k-means to these singular vectors. It can be seen that IF-PCA is a tuning free clustering method. We apply IF-PCA to $10$ gene microarray data sets. The method has competitive performance in clustering. Especially, in three of the data sets, the error rates of IF-PCA are only $29\%$ or less of the error rates by other methods. We have also rediscovered a phenomenon on empirical null by \cite{Efron} on microarray data. With delicate analysis, especially post-selection eigen-analysis, we derive tight probability bounds on the Kolmogorov-Smirnov statistics and show that IF-PCA yields clustering consistency in a broad context. The clustering problem is connected to the problems of sparse PCA and low-rank matrix recovery, but it is different in important ways. We reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each.