  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.
