Correlation and high-dimensional consistency in pattern recognition

Classical discriminant analysis breaks down when the feature vectors are of extremely high dimension; for example, when the basic observation is a random function observed over a fine grid. Alternative methods have been developed assuming a simplified form for the covariance structure. We analyze the high-dimensional asymptotics of some of these methods, emphasizing the effects of correlations such as occur when the baseline is random. For instance, the Euclidean distance classifier, which has been proposed for generic use in high-dimensional classification problems, is dimensionally inconsistent under a simple repeated measurement model. We provide exponential bounds for the error rates of several classifiers. We develop new dimensionally consistent methods to deal with the effects of correlation in high-dimensional problems.