Abstract:
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In high-dimensional classification problems, a commonly used approach is to first project the high-dimensional features into a lower dimensional space, and base the classification on the resulting lower dimensional projections. We formulate a latent-variable model with a hidden low-dimensional structure to justify this two-step procedure and to guide which projection to choose. We propose a computationally efficient classifier that takes certain principal components (PCs) of the observed features as projections, with the number of retained PCs selected in a data-driven way. A general theory is established for analyzing such two-step classifiers based on any low-dimensional projections. We derive explicit rates of convergence of the excess risk of the proposed PC-based classifier, and prove that these rates are minimax optimal. Our theory allows, but does not require, the lower-dimension to grow with the sample size and is also valid even when the feature dimension exceeds the sample size. Extensive simulations corroborate our theoretical findings. The proposed method also performs favorably relative to other existing discriminant methods on three real data examples.
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