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$\ell_1$ optimization is a well known heuristic often employed for solvingvarious forms of sparse linear problems. In this paper we look at its a variantthat we refer to as the \emph{partial} $\ell_1$ and discuss its mathematicalproperties 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 avariety of probabilistic and geometric techniques that we have developed inrecent years we will first present general guidelines that conceptually fullycharacterize both, the PTs and the LDPs. After that we will put an emphasis onproviding a collection of explicit analytical solutions to all of theunderlying mathematical problems. As a nice bonus to the developed concepts,the forms of the analytical solutions will, in our view, turn out to be fairlyelegant as well.

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