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The Generalized Lasso Problem and Uniqueness

20 May 2018
Alnur Ali
Robert Tibshirani
ArXiv (abs)PDFHTML
Abstract

We study uniqueness in the generalized lasso problem, where the penalty is the ℓ1\ell_1ℓ1​ norm of a matrix DDD times the coefficient vector. We derive a broad result on uniqueness that places weak assumptions on the predictor matrix XXX and penalty matrix DDD; the implication is that, if DDD is fixed and its null space is not too large (the dimension of its null space is at most the number of samples), and XXX and response vector yyy 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=ID=ID=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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