Adaptive efficient sparse estimator achieving oracle properties

Compressed Sensing is the new trend in the signal processing context which aims to sample a compressible signal with a rate less than the Nyquist lower bound sampling rate. The main challenge arises due to the non-convex optimisation problem to be solved in the reconstruction stage. This paper introduces a suitable objective function in order to simultaneously recover the true support of the underlying sparse signal while achieving an acceptable estimation error. Inspired by the well-known Lasso objective function, we have developed an objective function based on a new penalty denoted by the Linearised Exponentially Decaying (LED) penalty. The comprehensive analysis of the LED based objective function shows that the new approach satisfies the oracle properties, as opposed to the conventional Lasso objective function. Furthermore, we have developed a Sequential Adaptive Coordinate-wise (SAC) solution for the proposed objective function. The simulation results for the proposed LED-SAC reconstruction algorithm are given and compared with other state of the art methods. It is shown that LED-SAC approaches the least mean squared error criterion. Moreover, compared to the other methods, LED-SAC has much more adaptation rate in terms of tracking the variations in the support of the underlying sparse signal.