Relations between $β$ and $δ$ for QP and LP in Compressed Sensing Computations

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.

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