  In this paper we look at a well known linear inverse problem that is one of
the mathematical cornerstones of the compressed sensing field. In seminal works
\cite{CRT,DOnoho06CS} $\ell_1$ optimization and its success when used for
recovering sparse solutions of linear inverse problems was considered.
Moreover, \cite{CRT,DOnoho06CS} established for the first time in a statistical
context that an unknown vector of linear sparsity can be recovered as a known
existing solution of an under-determined linear system through $\ell_1$
optimization. In \cite{DonohoPol,DonohoUnsigned} (and later in
\cite{StojnicCSetam09,StojnicUpper10}) the precise values of the linear
proportionality were established as well. While the typical $\ell_1$
optimization behavior has been essentially settled through the work of
\cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}, we in this
paper look at possible upgrades of $\ell_1$ optimization. Namely, we look at a
couple of algorithms that turn out to be capable of recovering a substantially
higher sparsity than the $\ell_1$. However, these algorithms assume a bit of
"feedback" to be able to work at full strength. This in turn then translates
the original problem of improving upon $\ell_1$ to designing algorithms that
would be able to provide output needed to feed the $\ell_1$ upgrades considered
in this papers.
