🤖 AI Summary
This paper addresses classification under high-dimensional sparse settings. We propose a two-step discriminant method based on principal component analysis (PCA), grounded in an implicit low-rank factor model and featuring adaptive selection of the number of principal components. We establish, for the first time, a general risk analysis framework for high-dimensional two-step classifiers and rigorously derive the minimax-optimal convergence rate (up to logarithmic factors) for the PCA-based classifier—even when dimensionality far exceeds sample size. Theoretically, the excess risk achieves the optimal rate; simulations demonstrate robustness under model misspecification; and empirical evaluation on three real-world high-dimensional datasets shows significant improvement over state-of-the-art discriminant methods. Key contributions include: (i) a unified theoretical analysis paradigm for two-step classification, (ii) minimax-optimal rate guarantees, (iii) a data-driven, theoretically justified dimension-selection mechanism, and (iv) consistent empirical superiority across diverse high-dimensional benchmarks.
📝 Abstract
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. In this paper, 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 projections. We derive explicit rates of convergence of the excess risk of the proposed PC-based classifier. The obtained rates are further shown to be optimal up to logarithmic factors in the minimax sense. Our theory allows the lower-dimension to grow with the sample size and is also valid even when the feature dimension (greatly) 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.
