A minimax dictionary expansion for sparse continuous reconstruction

The concept of dictionary expansion has been applied in inverse problems as a means to overcome a problem known as off-grid deviation. Within this framework and under the assumption that the off-grid deviations obey an uniform distribution, we propose a minimax error criterion to build expanded dictionaries. To this end, we formulate the problem as a polynomial regression and cast it as a second-order cone program. A robust method for the recovery of continuous time shifts and amplitudes from reconstructed expanded coefficients is also presented. Empirical results with a greedy algorithm and a convex optimization algorithm, both conceived to work with expanded dictionaries, show that the proposed expanded basis provides accurate reconstruction of continuous-time located events in the presence of noise.