Skip to main content
padlock icon - secure page this page is secure

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

Buy Article:

$43.00 + tax (Refund Policy)

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.
No References
No Citations
No Supplementary Data
No Article Media
No Metrics

Keywords: Convex relaxation; Non‐convex optimization; Sparse Fisher's discriminant analysis; Sparse canonical correlation analysis; Sparse sufficient dimension reduction

Document Type: Research Article

Publication date: 01 November 2018

  • Access Key
  • Free content
  • Partial Free content
  • New content
  • Open access content
  • Partial Open access content
  • Subscribed content
  • Partial Subscribed content
  • Free trial content
Cookie Policy
X
Cookie Policy
Ingenta Connect website makes use of cookies so as to keep track of data that you have filled in. I am Happy with this Find out more