Weighted methods in sparse recovery

In the recent years we have successfully employed "weighted" algorithms to recover sparse signals from few linear, non-adaptive measurements. The general principle here is to use prior knowledge about the signal to be recovered, e.g., approximate locations of large-in-magnitude transform coefficients, if such information is available. An example for this is the use of weighted 1-norm minimization to improve wavefield reconstruction from randomized (sub)sampling. We will review these results and outline some new directions we have explored during the last year, such as weighted non-convex sparse recovery (see Ghadermarzy's talk), weighted analysis-based recovery (see Hargreaves's talk), and a weighted randomized Kaczmarz algorithm for solving large overdetermined systems of equations that are known to admit a (nearly) sparse solution. Various examples in seismic will be shown.