Complexity and Applications of the Homotopy Principle for Uniformly Constrained Sparse Minimization

Abstract

In this paper, we investigate the homotopy path related to $$\ell _{1}$$ -norm minimization problems with $$\ell _{\infty }$$ -norm constraints. We establish an enhanced upper bound on the number of linear segments in the path and provide an example showing that the number of segments is exponential in the number of variables in the worst case. We also use the homotopy framework to develop grid independent (cross-)validation schemes for sparse linear discriminant analysis and classification that make use of the entire path. Several numerical and statistical examples illustrate the applicability of the framework.