  $\ell_1$ optimization is a well known heuristic often employed for solving
various forms of sparse linear problems. In this paper we look at its a variant
that we refer to as the \emph{partial} $\ell_1$ and discuss its mathematical
properties when used for solving linear under-determined systems of equations.
We will focus on large random systems and discuss the phase transition (PT)
phenomena and how they connect to the large deviation principles (LDP). Using a
variety of probabilistic and geometric techniques that we have developed in
recent years we will first present general guidelines that conceptually fully
characterize both, the PTs and the LDPs. After that we will put an emphasis on
providing a collection of explicit analytical solutions to all of the
underlying mathematical problems. As a nice bonus to the developed concepts,
the forms of the analytical solutions will, in our view, turn out to be fairly
elegant as well.
