Solution-recovery in `1-norm for non-square linear systems: deterministic conditions and open questions

Consider an over-determined linear system Ax ≈ b and an under-determined linear system By = c. Given b = A x̂+h, under what conditions x̂ will minimize the residual ATx− b in `1-norm (i.e., ‖h‖1 = minx ‖ATx− b‖1)? On the other hand, given c = Bh, under what conditions h will be the minimum `1-norm solution of By = c? These two “solution-recovery” problems have been the focus of a number of recent works. Moreover, these two problems are equivalent under appropriate conditions on the data sets (A, b) and (B, c). In this paper, we give deterministic conditions for these solutionrecoverary problems and raise a few open questuions. Some of the results in this paper are already known or partially known, but our derivations are different and thus may provide different perspectives.