  In this paper we look at a particular problem related to under-determined
linear systems of equations with sparse solutions. $\ell_1$-minimization is a
fairly successful polynomial technique that can in certain statistical
scenarios find sparse enough solutions of such systems. Barriers of $\ell_1$
performance are typically referred to as its thresholds. Depending if one is
interested in a typical or worst case behavior one then distinguishes between
the \emph{weak} thresholds that relate to a typical behavior on one side and
the \emph{sectional} and \emph{strong} thresholds that relate to the worst case
behavior on the other side. Starting with seminal works
\cite{CRT,DonohoPol,DOnoho06CS} a substantial progress has been achieved in
theoretical characterization of $\ell_1$-minimization statistical thresholds.
More precisely, \cite{CRT,DOnoho06CS} presented for the first time linear lower
bounds on all of these thresholds. Donoho's work \cite{DonohoPol} (and our own
\cite{StojnicCSetam09,StojnicUpper10}) went a bit further and essentially
settled the $\ell_1$'s \emph{weak} thresholds. At the same time they also
provided fairly good lower bounds on the values on the \emph{sectional} and
\emph{strong} thresholds. In this paper, we revisit the \emph{sectional}
thresholds and present a simple mechanism that can be used to create solid
upper bounds as well. The method we present relies on a seemingly simple but
substantial progress we made in studying Hopfield models in
\cite{StojnicHopBnds10}.
