Compressed Sensing with Prior Information: Optimal Strategies, Geometry, and Bounds

We address the problem of compressed sensing (CS) with prior information: reconstruct a target CS signal with the aid of a similar signal that is known beforehand, our prior information. We integrate the additional knowledge of the similar signal into CS via l1-l1 and l1-l2 minimization. We then establish bounds on the number of measurements required by these problems to successfully reconstruct the original signal. Our bounds and geometrical interpretations reveal that if the prior information has good enough quality, l1-l1 minimization improves the performance of CS dramatically. In contrast, l1-l2 minimization has a performance very similar to classical CS and brings no significant benefits. All our findings are illustrated with experimental results.