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Sparse generalized eigenvalue problem (GEP) plays a pivotal role in a largefamily of high-dimensional statistical models, including sparse Fisher'sdiscriminant analysis, canonical correlation analysis, and sufficient dimensionreduction. Sparse GEP involves solving a non-convex optimization problem. Mostexisting methods and theory in the context of specific statistical models thatare special cases of the sparse GEP require restrictive structural assumptionson the input matrices. In this paper, we propose a two-stage computationalframework to solve the sparse GEP. At the first stage, we solve a convexrelaxation of the sparse GEP. Taking the solution as an initial value, we thenexploit a nonconvex optimization perspective and propose the truncated Rayleighflow method (Rifle) to estimate the leading generalized eigenvector. We showthat Rifle converges linearly to a solution with the optimal statistical rateof convergence for many statistical models. Theoretically, our methodsignificantly improves upon the existing literature by eliminating structuralassumptions on the input matrices for both stages. To achieve this, ouranalysis involves two key ingredients: (i) a new analysis of the gradient basedmethod on nonconvex objective functions, and (ii) a fine-grainedcharacterization of the evolution of sparsity patterns along the solution path.Thorough numerical studies are provided to validate the theoretical results.

Sparse Generalized Eigenvalue Problem: Optimal Statistical Rates via Truncated Rayleigh Flow