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.
Funding Statement
Wegkamp is supported in part by the National Science Foundation grants DMS 2015195 and DMS 2210557. Bing is partially supported by a discovery grant from the Natural Sciences and Engineering Research Council of Canada.
Acknowledgments
The authors would like to thank the Editor, Associate Editor and two referees for their careful reading and very constructive suggestions.
Citation
Xin Bing. Marten Wegkamp. "Optimal discriminant analysis in high-dimensional latent factor models." Ann. Statist. 51 (3) 1232 - 1257, June 2023. https://doi.org/10.1214/23-AOS2289
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