  In compressed sensing one uses known structures of otherwise unknown signals
to recover them from as few linear observations as possible. The structure
comes in form of some compressibility including different notions of sparsity
and low rankness. In many cases convex relaxations allow to efficiently solve
the inverse problems using standard convex solvers at almost-optimal sampling
rates. A standard practice to account for multiple simultaneous structures in
convex optimization is to add further regularizers or constraints. From the
compressed sensing perspective there is then the hope to also improve the
sampling rate. Unfortunately, when taking simple combinations of regularizers,
this seems not to be automatically the case as it has been shown for several
examples in recent works. Here, we give an overview over ideas of combining
multiple structures in convex programs by taking weighted sums and weighted
maximums. We discuss explicitly cases where optimal weights are used reflecting
an optimal tuning of the reconstruction. In particular, we extend known lower
bounds on the number of required measurements to the optimally weighted maximum
by using geometric arguments. As examples, we discuss simultaneously low rank
and sparse matrices and notions of matrix norms (in the "square deal" sense)
for regularizing for tensor products. We state an SDP formulation for
numerically estimating the statistical dimensions and find a tensor case where
the lower bound is roughly met up to a factor of two.
