The generalized lasso problem and uniqueness

Abstract

We study uniqueness in the generalized lasso problem, where the penalty is the ℓ1 norm of a matrix D times the coefficient vector. We derive a broad result on uniqueness that places weak assumptions on the predictor matrix X and penalty matrix D; the implication is that, if D is fixed and its null space is not too large (the dimension of its null space is at most the number of samples), and X and response vector y jointly follow an absolutely continuous distribution, then the generalized lasso problem has a unique solution almost surely, regardless of the number of predictors relative to the number of samples. This effectively generalizes previous uniqueness results for the lasso problem [32] (which corresponds to the special case D = I). Further, we extend our study to the case in which the loss is given by the negative log-likelihood from a generalized linear model. In addition to uniqueness results, we derive results on the local stability of generalized lasso solutions that might be of interest in their own right.