Newtonized Orthogonal Matching Pursuit: Frequency Estimation Over the Continuum

We propose a fast sequential algorithm for the fundamental problem of estimating frequencies and amplitudes of a noisy mixture of sinusoids. The algorithm is a natural generalization of Orthogonal Matching Pursuit (OMP) to the continuum using Newton refinements, and hence is termed Newtonized OMP (NOMP). Each iteration consists of two phases: detection of a new sinusoid, and sequential Newton refinements of the parameters of already detected sinusoids. The refinements play a critical role in two ways: 1) sidestepping the potential basis mismatch from discretizing a continuous parameter space and 2) providing feedback for locally refining parameters estimated in previous iterations. We characterize convergence and provide a constant false alarm rate (CFAR) based termination criterion. By benchmarking against the Cramér-Rao Bound, we show that NOMP achieves near-optimal performance under a variety of conditions. We compare the performance of NOMP with classical algorithms such as MUSIC and more recent Atomic norm Soft Thresholding (AST) and Lasso algorithms, both in terms of frequency estimation accuracy and run time.