Relations between $\beta$ and $\delta$ for QP and LP in Compressed Sensing Computations

TL;DR

This paper explores the relationship between two parameters,  and , in compressed sensing methods using LP and QP, providing bounds to help set  based on , with experimental validation.

Contribution

It derives bounds linking  and , enabling better parameter setting in quadratic programming for compressed sensing.

Findings

Derived upper and lower bounds on  in terms of 

Provided a method to approximate  from  for practical applications

Experimental results support the theoretical bounds

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 $\delta$, 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 $\beta$. In this work, we investigated the relation between $\delta$ and $\beta$ and derived an upper and a lower bound on $\beta$ in terms of $\delta$. For a given $\delta$, these bounds can be used to approximate $\beta$. Since $\delta$ is a physically related quantity and easy to determine for an application while there is no easy way in general to determine $\beta$, our results can be used to set $\beta$ when the QP is used for compressed sensing. Our results and experimental verification also provide some insight into the solutions generated by compressed sensing.