Sparse generalized eigenvalue problem: optimal statistical rates via truncated Rayleigh flow

• 成果类型: Article
• 署名作者: Tan, Kean Ming; Wang, Zhaoran; Liu, Han; Zhang, Tong
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; Northwestern University
• 刊物名称: JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY
• ISSN/ISSBN: 1369-7412
• DOI: 10.1111/rssb.12291
• 发表日期: 2018
• 页码: 1057-1086
• 关键词: sliced inverse regression; Dimension Reduction; DISCRIMINANT-ANALYSIS; class prediction; gene-expression; PCA; CLASSIFICATION; components; selection; cancer

摘要

The sparse generalized eigenvalue problem (GEP) plays a pivotal role in a large family of high dimensional statistical models, including sparse Fisher's discriminant analysis, canonical correlation analysis and sufficient dimension reduction. The sparse GEP involves solving a non-convex optimization problem. Most existing methods and theory in the context of specific statistical models that are special cases of the sparse GEP require restrictive structural assumptions on the input matrices. We propose a two-stage computational framework to solve the sparse GEP. At the first stage, we solve a convex relaxation of the sparse GEP. Taking the solution as an initial value, we then exploit a non-convex optimization perspective and propose the truncated Rayleigh flow method (which we call rifle') to estimate the leading generalized eigenvector. We show that rifle converges linearly to a solution with the optimal statistical rate of convergence. Theoretically, our method significantly improves on the existing literature by eliminating structural assumptions on the input matrices. To achieve this, our analysis involves two key ingredients: a new analysis of the gradient-based method on non-convex objective functions, and a fine-grained characterization of the evolution of sparsity patterns along the solution path. Thorough numerical studies are provided to validate the theoretical results.