Fast and Accurate Algorithms for Re-Weighted $\ell _{1}$-Norm Minimization

To recover a sparse signal from an underdetermined system, we often solve a constrained
$\ell _{1}$
-norm minimization problem. In many cases, the signal sparsity and recovery performance can be further improved by replacing the
$\ell _{1}$
norm with a “weighted”
$\ell _{1}$
norm. Without prior information about the signal's nonzero elements, the procedure for selecting weights is iterative in nature. Common approaches update the weights at every iteration using the solution of a weighted
$\ell _{1}$
problem from the previous iteration. This paper presents two homotopy-based algorithms that efficiently solve reweighted
$\ell _{1}$
problems. First, we present an algorithm that quickly updates the solution of a weighted
$\ell _{1}$
problem as the weights change. Since the solution changes only slightly with small changes in weights, we develop a homotopy algorithm that replaces old weights with new ones in a small number of computationally inexpensive steps. Second, we propose an algorithm that solves a weighted
$\ell _{1}$
problem by adaptively selecting weights while estimating the signal. This algorithm integrates the reweighting into every step along the homotopy path by changing the weights according to changes in the solution and its support, allowing us to achieve a high quality signal reconstruction by solving a single homotopy problem. We compare both algorithms' performance, in terms of reconstruction accuracy and computational complexity, against state-of-the-art solvers and show that our methods have smaller computational cost. We also show that the adaptive selection of the weights inside the homotopy often yields reconstructions of higher quality.