The accessible lasso models. Issue 4 (4th July 2017)

• Record Type: Journal Article
• Authors: Sepehri, Amir; Harris, Naftali
• Is Part Of: Statistics. Volume 51:Issue 4(2017)
• Page Start: 711
• Page End: 721
• Publication Date: 2017-07-04
• Subjects: Lasso geometry -- convex polytopes -- upper bound theorem -- model selection consistency; 62J07 -- 52B12; Mathematical statistics -- Periodicals; 519.505
• DOI: 10.1080/02331888.2016.1266632
• Languages: English
• ISSNs: 0233-1888

Abstract

ABSTRACT: A new line of research [Lee et al. Exact post-selection inference with the lasso. arXiv preprint arXiv:1311.6238 (2013)] on the lasso [Tibshirani. Regression shrinkage and selection via the lasso. J R Stat Soc Ser B (Methodological). 1996;267–288] exploits the beautiful geometric fact that the lasso fit is the residual from projecting the response vector y onto a certain convex polytope [Tibshirani et al. Degrees of freedom in lasso problems. Ann Stat. 2012;40:1198–1232]. This geometric picture also allows an exact geometric description of the set of accessible lasso models for a given design matrix, that is, which configurations of the signs of the coefficients it is possible to realize with some choice of y . In particular, the accessible lasso models are those that correspond to a face of the convex hull of all the feature vectors together with their negations. This convex hull representation then permits the enumeration and bounding of the number of accessible lasso models, which in turn provides a direct proof of model selection inconsistency when the size of the true model is greater than half the number of observations.