Relations between β and δ for QP and LP in Compressed Sensing Computations
Abstract
In many compressed sensing applications, linear programming (LP) has been used to reconstruct a sparse signal. When observation is noisy, the LP formulation is extended to allow an inequality constraint and the solution is dependent on a parameter δ, related to the observation noise level. Recently, some researchers also considered quadratic programming (QP) for compressed sensing signal reconstruction and the solution in this case is dependent on a Lagrange multiplier β. In this work, we investigated the relation between δ and β and derived an upper and a lower bound on β in terms of δ. For a given δ, these bounds can be used to approximate β. Since δ is a physically related quantity and easy to determine for an application while there is no easy way in general to determine β, our results can be used to set β when the QP is used for compressed sensing. Our results and experimental verification also provide some insight into the solutions generated by compressed sensing.