Exact reconstruction conditions and error bounds for regularized Modified Basis Pursuit (Reg-modified-BP)

We study the problem of reconstructing a sparse signal from a limited number of linear measurements, when a part of its support and the signal estimate on it are known. The support and signal estimate can be obtained from prior knowledge, e.g., in a real-time dynamic MRI application, they could be the support and signal estimate from the previous time instant. We propose regularized Modified Basis Pursuit (Reg-mod-BP). We also provide the exact reconstruction conditions and we argue that they can be weaker than modified-CS. We then bound its reconstruction error when exact reconstruction can not happen and we show that the bound is much smaller than modified-CS when the available measurements are few. We also use Monte Carlo to verify that reg-mod-BP has better exact reconstruction conditions than other methods with very few measurements. We also compare the average errors when exact reconstruction can not be achieved and show that the errors are smaller than other methods.