Lasso and equivalent quadratic penalized models

The least absolute shrinkage and selection operator (lasso) and ridge regression produce usually different estimates although input, loss function and parameterization of the penalty are identical. In this paper we look for ridge and lasso models with identical solution set. It turns out, that the lasso model with shrink vector λ and a quadratic penalized model with shrink matrix as outer product of λ with itself are equivalent, in the sense that they have equal solutions. To achieve this, we have to restrict the estimates to be positive. This doesn’t limit the area of application since we can decompose every estimate in a positive and negative part. The resulting problem can be solved with a non negative least square algorithm and may benefit from algorithms with high numerically accuracy. This model can also deal with mixtures of ridge and lasso penalties like the elastic net, leading to a continuous solution path as a function of the mixture proportions. Beside this quadratic penalized model, an augmented regression model with positive bounded estimates is developed which is also equivalent to the lasso model, but is probably faster to solve.