  We study a set of regularization methods for high-dimensional linear
regression models. These penalized estimators have the square root of the
residual sum of squared errors as loss function, and any weakly decomposable
norm as penalty function. This fit measure is chosen because of its property
that the estimator does not depend on the unknown standard deviation of the
noise. On the other hand, a generalized weakly decomposable norm penalty is
very useful in being able to deal with different underlying sparsity
structures. We can choose a different sparsity inducing norm depending on how
we want to interpret the unknown parameter vector $\beta$. Structured sparsity
norms, as defined in Micchelli et al. [18], are special cases of weakly
decomposable norms, therefore we also include the square root LASSO (Belloni et
al. [3]), the group square root LASSO (Bunea et al. [10]) and a new method
called the square root SLOPE (in a similar fashion to the SLOPE from Bogdan et
al. [6]). For this collection of estimators our results provide sharp oracle
inequalities with the Karush-Kuhn-Tucker conditions. We discuss some examples
of estimators. Based on a simulation we illustrate some advantages of the
square root SLOPE.
