The dual and degrees of freedom of linearly constrained generalized lasso

Hu, Qinqin; Zeng, Peng; Lin, Lu

doi:10.1016/j.csda.2014.12.010

Cited by 21 publications

(29 citation statements)

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“…When there is no constraint, Theorem degenerates to the result in Tibshirani & Taylor (). When X has full column rank,

\dim (col (X P_{null})) = \dim (null (G_{- scriptA, scriptB}))

, which is exactly Theorem in Hu et al ().…”

Section: Degrees Of Freedommentioning

confidence: 56%

“…Hence, we can calculate degrees of freedom as

df (\hat{μ}) = E [(▿ \cdot g) (y)],

where

▿ \cdot g = \sum_{i = 1}^{n} \partial g_{i} / \partial y_{i}

is the divergence of y . See Meyer & Woodroofe (), Efron (), Zou et al (), Tibshirani & Taylor (), Hu et al () and references therein.…”

Section: Degrees Of Freedommentioning

confidence: 99%

“…The derived formula for degrees of freedom makes it possible to apply an information criterion (IC), which greatly saves the need of computing. It is necessary to point it out that this paper is a non‐trivial extension of Hu et al () from a full‐rank case to a rank‐deficiency case. When X does not have full column rank, X T X is not invertible, and hence, generalized inverse has to be introduced.…”

Section: Introductionmentioning

confidence: 96%

“…In these applications, prior knowledge is formulated in terms of linear equality or linear inequality constraints on the parameters. The estimation algorithm and properties of lcg‐lasso solution have been studied by He (), James et al (), Hu et al () and others. However, there is no discussion on the consequence of rank deficiency on the estimate and fit.…”

Section: Introductionmentioning

confidence: 99%

“…A formula for degrees of freedom exists for lasso (Zou et al, ; Dossal et al, ) and generalized lasso (Tibshirani & Taylor, ; ). Recently, Hu et al () derived a formula for degrees of freedom for lcg‐lasso fit when X has full column rank. However, there is no existing result for lcg‐lasso fit with rank deficiency yet.…”

Section: Introductionmentioning

confidence: 99%

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Geometry and Degrees of Freedom of Linearly Constrained Generalized Lasso

Zeng

2017

Scandinavian J Statistics

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The least squares fit in a linear regression is always unique even when the design matrix has rank deficiency. In this paper, we extend this classic result to linearly constrained generalized lasso. It is shown that under a mild condition, the fit can be represented as a projection onto a polytope and, hence, is unique no matter whether design matrix X has full column rank or not. Furthermore, a formula for the degrees of freedom is derived to characterize the effective number of parameters. It directly yields an unbiased estimate of degrees of freedom, which can be incorporated in an information criterion for model selection.

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“…When there is no constraint, Theorem degenerates to the result in Tibshirani & Taylor (). When X has full column rank,