Partial $\ell_1$ optimization in random linear systems -- phase transitions and large deviations

$\ell_1$ 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.