Penalized Versus Constrained Generalized Eigenvalue Problems
研究了多元分析中广义特征值问题使用ℓ1惩罚与ℓ1约束的差异，发现ℓ1惩罚可能无法产生非常稀疏的解，而ℓ1约束能有效改进变量选择，并通过判别分析和主成分分析展示了约束的优势。
We investigate the difference between using an ℓ1 penalty versus an ℓ1 constraint in generalized eigenvalue problems arising in multivariate analysis. Our main finding is that the ℓ1 penalty may fail to provide very sparse solutions; a severe disadvantage for variable selection that can be remedied by using an ℓ1 constraint. Our claims are supported both by empirical evidence and theoretical analysis. Finally, we illustrate the advantages of the ℓ1 constraint in the context of discriminant analysis and principal component analysis. Supplementary materials for this article are available online.