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2019 The Generalized Lasso Problem and Uniqueness
Alnur Ali, Ryan J. Tibshirani
Electron. J. Statist. 13(2): 2307-2347 (2019). DOI: 10.1214/19-EJS1569

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 [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.

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Alnur Ali. Ryan J. Tibshirani. "The Generalized Lasso Problem and Uniqueness." Electron. J. Statist. 13 (2) 2307 - 2347, 2019. https://doi.org/10.1214/19-EJS1569

Information

Received: 1 May 2018; Published: 2019
First available in Project Euclid: 9 July 2019

zbMATH: 07080059
MathSciNet: MR3980959
Digital Object Identifier: 10.1214/19-EJS1569

Subjects:
Primary: 62J07 , 62J07
Secondary: 90C46

Keywords: existence of solutions , generalized lasso , generalized linear models , high-dimensional , uniqueness of solutions

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Vol.13 • No. 2 • 2019
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