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Degrees of freedom in lasso problems. (English) Zbl 1274.62469
Summary: We derive the degrees of freedom of the Lasso fit, placing no assumptions on the predictor matrix $$X$$. Like the well-known result of H. Zou, T. Hastie and R. Tibshirani [Ann. Stat. 35, No. 5, 2173–2192 (2007; Zbl 1126.62061)], which gives the degrees of freedom of the Lasso fit when $$X$$ has full column rank, we express our result in terms of the active set of a Lasso solution. We extend this result to cover the degrees of freedom of the generalized Lasso fit for an arbitrary predictor matrix $$X$$ (and an arbitrary penalty matrix $$D$$). Though our focus is degrees of freedom, we establish some intermediate results on the Lasso and generalized Lasso that may be interesting on their own.

##### MSC:
 62J07 Ridge regression; shrinkage estimators (Lasso) 90C46 Optimality conditions and duality in mathematical programming
##### Keywords:
Lasso; generalized Lasso; degrees of freedom; high-dimensional
PDCO
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##### References:
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