Geometry and Degrees of Freedom of Linearly Constrained Generalized Lasso

Abstract

The least squares fit in a linear regression is always unique even when the design matrix has rank deficiency. In this paper, we extend this classic result to linearly constrained generalized lasso. It is shown that under a mild condition, the fit can be represented as a projection onto a polytope and, hence, is unique no matter whether design matrix X has full column rank or not. Furthermore, a formula for the degrees of freedom is derived to characterize the effective number of parameters. It directly yields an unbiased estimate of degrees of freedom, which can be incorporated in an information criterion for model selection.

Algorithms for Estimating Trends in Global Temperature Volatility

“…However, such analysis needs either the computation of the degrees of freedom or requires cross validation. Previous work has investigated the degrees of freedom in generalized lasso problems with Gaussian likelihood [15,29,34], but, results for non-Gaussian likelihood remains an open problem, and cross validation is too expensive. In this paper, therefore, we use a heuristic method for choosing λ t and λ s : we compute the solutions for a range of values of and choose those which minimize L(λ t , λ s ) = −l(y|h) + Dh .…”