Abstract

 
 
 Three relevant facts about the least absolute shrinkage and selection operator (Lasso) are studied: The estimatives follows piecewise linear curves in relation to tuning parameter, the number of nonzero selected covariates is an unbiased estimator of its degrees of freedom and when the number of covariates p is greater than the numbers of observations n at most n covariates are selected. These results are well known and described in the literature, but with no simple demonstrations. We present, based on a geometrical approach, simple and intuitive heuristics proofs for these results.
 
 
Highlights
Suppose the usual regression situation: data xi,yi, i = 1, . . . , n, where xi = is a vector of predictors variables and yi is the corresponding n response.Consider as usual that the observations are independent and 1 n xij = i=1 n1 n x2ij = 1. Tibshirani (1996) defines the least absolute shrinkage and selection operator (Lasso) estimative as the solution of i=1 the quadratic convex optimization problem: Rev
To obtain the Lasso estimative we have to find the point in Kp closest to the data vector y
We will be concerned only with the tangent point closest to the origin. It is a typical problem in Mathematical analysis to show that tangents points between these two families defines a smooth curve, which is called by definition β(lasso) (t)
Summary
To obtain the Lasso estimative we have to find the point in Kp closest to the data vector y. To do this we project y orthogonally into the subspace image of X (yP∗ = PIm(X)y) and find in Kp the point yp closest to yP∗. This is equivalent to find in K the point closest, in the Mahalanobis distance, β1,β2 m = β1 X Xβ2, to the ordinary least squared estimative βols. This can be done by constructing several hyperboloids on βols, until one of these reach a tangent point on K.
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