Translator Disclaimer
December 2016 Influential features PCA for high dimensional clustering
Jiashun Jin, Wanjie Wang
Ann. Statist. 44(6): 2323-2359 (December 2016). DOI: 10.1214/15-AOS1423

Abstract

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, 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 procedure to these singular vectors. In this procedure, the only tuning parameter is the threshold in the feature selection step. We set the threshold in a data-driven fashion by adapting the recent notion of Higher Criticism. As a result, 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 Efron [J. Amer. Statist. Assoc. 99 (2004) 96–104] 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.

Citation

Download Citation

Jiashun Jin. Wanjie Wang. "Influential features PCA for high dimensional clustering." Ann. Statist. 44 (6) 2323 - 2359, December 2016. https://doi.org/10.1214/15-AOS1423

Information

Received: 1 July 2014; Revised: 1 December 2015; Published: December 2016
First available in Project Euclid: 23 November 2016

zbMATH: 1359.62249
MathSciNet: MR3576543
Digital Object Identifier: 10.1214/15-AOS1423

Subjects:
Primary: 62G32, 62H30
Secondary: 62E20

Rights: Copyright © 2016 Institute of Mathematical Statistics

JOURNAL ARTICLE
37 PAGES


SHARE
Vol.44 • No. 6 • December 2016
Back to Top