Unified-Mathematical Form of Successfully Non-Convex Regularization for Enhancing Sparse Signals

Many signal processing algorithms, including sampling, channel coding, sparse dictionary representation, array processing and channel estimation, are designed to work with the sparse signal — the signal in which most elements are zero. The quality of the sparse signal directly impacts the efficiency of these algorithms. Therefore, we propose the novel non-convex regularization and its shrinkage function to enhance the sparse signal. Our regularization is based on the generalized inverse tangent function that can exhibit the small-bias property between the observed and enhanced signals. Many successful regularizers are the subset of the proposed regularization by setting its parameter. Moreover, our shrinkage function has the multivariate (non-separable) property that leads to better efficiency than functions with the univariate (separable) property. We present our regularization and its shrinkage function in closed (analytic) forms. Our method is compared with the state-of-the-art methods in many generated random and real multipath (many power values in the wireless communication) signals, and demonstrates superior performance in terms of the root mean square error (RMSE) and visualization.