Geometric classifiers for high-dimensional noisy data

Abstract

We consider the quadratic classification for high-dimensional data under the strongly spiked eigenvalue (SSE) model. High-dimensional data contain much information, however, it also contains huge amount of noise. We detect the high-dimensional noise as a spiked eigenstructure of high-dimensional covariance matrices. In order to find the difference between two populations, we utilize a geometric feature of high-dimensional data. The classification analysis based on the geometric feature of high-dimensional data is called geometrical quadratic discriminant analysis (GQDA). We create new GQDA on the basis of the high-dimensional spiked eigenstructures. We precisely study the influence of the spiked eigenstructure on GQDA using several examples. In order to remove the spiked noise, we use a data transformation technique. We show that our proposed classifier has a consistency property with respect to the error rate of misclassifying an individual. By using computer simulation, we discuss the performance of the proposed classifier. Finally, we give several demonstrations of data analysis using a microarray data set.