A Direct Estimation Approach to Sparse Linear Discriminant AnalysisTony Cai and Weidong Liu
A Direct Estimation Approach to Sparse Linear Discriminant AnalysisTony Cai and Weidong Liu
- Abstract: This paper considers sparse linear discriminant analysis of high-dimensional data. In contrast to the existing methods which are based on separate estimation of the precision matrix Ω and the difference δ of the mean vectors, we introduce a simple and effective classifier by estimating the product Ωδ directly through constrained l1 minimization. The estimator can be implemented efficiently using linear programming and the resulting classifier is called the linear programming discriminant (LPD) rule. The LPD rule is shown to have desirable theoretical and numerical properties. It exploits the approximate sparsity of Ωδ and as a consequence allows cases where it can still perform well even when Ω and/or δ cannot be estimated consistently. Asymptotic properties of the LPD rule are investigated and consistency and rate of convergence results are given. The LPD classifier has superior finite sample performance and significant computational advantages over the existing methods that require separate estimation of Ω and δ. The LPD rule is also applied to analyze real datasets from lung cancer and leukemia studies. The classifier performs favorably in comparison to existing methods.
- Paper: pdf file.
- Other related papers:
- Cai, T. & Zhou, H. (2010).
Optimal rates of convergence for sparse covariance matrix estimation.
Technical report.
- Cai, T., Zhang, C.-H. & Zhou, H. (2010).
Optimal rates of convergence for covariance matrix estimation
The Annals of Statistics 38, 2118-2144.
- Cai, T. & Liu, W. (2011).
Adaptive thresholding for sparse covariance matrix estimation.
J. American Statistical Association 106, 672-684.
- Cai, T., Liu, W. & Luo, X. (2011).
A constrained l1 minimization approach to sparse precision matrix estimation.
J. American Statistical Association 106, 594-607.