We review recent results for high-dimensional sparse linear regression in the
practical case of unknown variance. Different sparsity settings are covered,
including coordinate-sparsity, group-sparsity and variation-sparsity. The
emphasize is put on non-asymptotic analyses and feasible procedures. In
addition, a small numerical study compares the practical performance of three
schemes for tuning the Lasso esti- mator and some references are collected for
some more general models, including multivariate regression and nonparametric
regression.

We study the nonparametric covariance estimation of a stationary Gaussian
field X observed on a lattice. To tackle this issue, a neighborhood selection
procedure has been recently introduced. This procedure amounts to selecting a
neighborhood m by a penalization method and estimating the covariance of X in
the space of Gaussian Markov random fields (GMRFs) with neighborhood m. Such a
strategy is shown to satisfy oracle inequalities as well as minimax adaptive
properties. However, it suffers several drawbacks which make the method
difficult to apply in practice.

This is a technical appendix to "Adaptive estimation of stationary Gaussian
fields". We present several proofs that have been skipped in the main paper.