  We consider the framework of penalized estimation where the penalty term is
given by a real-valued polyhedral gauge, which encompasses methods such as
LASSO, generalized LASSO, SLOPE, OSCAR, PACS and others. Each of these
estimators can uncover a different structure or ``pattern'' of the unknown
parameter vector. We define a novel and general notion of patterns based on
subdifferentials and formalize an approach to measure pattern complexity. For
pattern recovery, we provide a minimal condition for a particular pattern to be
detected by the procedure with positive probability, the so-called
accessibility condition. Using our approach, we also introduce the stronger
noiseless recovery condition. For the LASSO, it is well known that the
irrepresentability condition is necessary for pattern recovery with probability
larger than $1/2$ and we show that the noiseless recovery plays exactly the
same role in our general framework, thereby unifying and extending the
irrepresentability condition to a broad class of penalized estimators. We also
show that the noiseless recovery condition can be relaxed when turning to
so-called thresholded penalized estimators: we prove that the accessibility
condition is already sufficient (and necessary) for sure pattern recovery by
thresholded penalized estimation provided that the signal of the pattern is
large enough. Throughout the article, we demonstrate how our findings can be
interpreted through a geometrical lens.
