The Generalized Lasso Problem and Uniqueness

TL;DR

This paper establishes broad conditions under which the generalized lasso problem has a unique solution, extending previous results and including cases with generalized linear models, with implications for stability and predictor configurations.

Contribution

It provides a general uniqueness theorem for the generalized lasso with minimal assumptions on predictor and penalty matrices, extending prior work on the standard lasso.

Findings

Uniqueness holds almost surely under broad conditions.

Results extend to generalized linear models.

Analysis includes local stability of solutions.

Abstract

We study uniqueness in the generalized lasso problem, where the penalty is the $\ell_1$ 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 (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.