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Sparse generalized eigenvalue problem: optimal statistical rates via truncated Rayleigh flow
Author(s) -
Tan Kean Ming,
Wang Zhaoran,
Liu Han,
Zhang Tong
Publication year - 2018
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12291
Subject(s) - mathematics , mathematical optimization , eigenvalues and eigenvectors , sparse approximation , rate of convergence , context (archaeology) , convex optimization , relaxation (psychology) , computer science , algorithm , regular polygon , key (lock) , biology , physics , geometry , quantum mechanics , psychology , paleontology , social psychology , computer security
Summary 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.