Lasso and Dantzig Selectors for Non-parametric and High-dimensional Regression

where we wish to estimate the function g given the data under the assumption that the Wi are independent indentically distributed Gaussian error terms and g lies inside of some function space F . One reasonable method of doing this is to select a finite dictionary of functions {f1, . . . , fM} such that any function in F can be well approximated by sparse span of the dictionary. In this setting it is generally assumed that M > n, and we have thus reduced the non-parametric regression in equation 1, to the high-dimensional regression problem: