Electronic Journal of Statistics

The Generalized Lasso Problem and Uniqueness

Alnur Ali and Ryan J. Tibshirani

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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.

Article information

Electron. J. Statist., Volume 13, Number 2 (2019), 2307-2347.

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

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Digital Object Identifier

Primary: 62J07: Ridge regression; shrinkage estimators 62J07: Ridge regression; shrinkage estimators
Secondary: 90C46: Optimality conditions, duality [See also 49N15]

Generalized lasso high-dimensional uniqueness of solutions generalized linear models existence of solutions

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Ali, Alnur; Tibshirani, Ryan J. The Generalized Lasso Problem and Uniqueness. Electron. J. Statist. 13 (2019), no. 2, 2307--2347. doi:10.1214/19-EJS1569. https://projecteuclid.org/euclid.ejs/1562637626

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