  This note extends an attribute of the LASSO procedure to a whole class of
related procedures, including square-root LASSO, square LASSO, LAD-LASSO, and
an instance of generalized LASSO. Namely, under the assumption that the input
matrix satisfies an $\ell_p$-restricted isometry property (which in some sense
is weaker than the standard $\ell_2$-restricted isometry property assumption),
it is shown that if the input vector comes from the exact measurement of a
sparse vector, then the minimizer of any such LASSO-type procedure has sparsity
comparable to the sparsity of the measured vector. The result remains valid in
the presence of moderate measurement error when the regularization parameter is
not too small.
