OPTIMAL DISCRIMINANT ANALYSIS IN HIGH-DIMENSIONAL LATENT FACTOR MODELS

• 成果类型: Article
• 署名作者: Bing, Xin; Wegkamp, Marten
• 署名单位: University of Toronto; Cornell University; Cornell University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/23-AOS2289
• 发表日期: 2023
• 页码: 1232-1257
• 关键词: partial least-squares; tumor classification; principal-components; Adaptive estimation; SPARSE; CLASSIFIERS

摘要

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 lowdimensional 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.