Generic chaining and the ℓ1-penalty

We address the choice of the tuning parameter λ in ℓ 1 - penalized M-estimation. Our main concern is models which are highly non-linear, such as the Gaussian mixture model. The number of parameters p is moreover large, possibly larger than the number of observations n. The generic chaining technique of Talagrand (2005) is tailored for this problem. It leads to the choice λ ≈ log p / n , as in the standard Lasso procedure (which concerns the linear model and least squares loss).