Robust Harmonic Retrieval via Block Successive Upper-Bound Minimization

Harmonic retrieval (HR) is a problem of significance with numerous applications. Many existing algorithms are explicitly or implicitly developed under Gaussian noise assumption, which, however, are not robust against non-Gaussian noise such as impulsive noise or outliers. In this paper, by employing the <inline-formula> <tex-math notation="LaTeX">$\ell _p$</tex-math></inline-formula>-fitting criterion and block successive upper-bound minimization (BSUM) technique, a variant of the classical RELAX algorithm named as BSUM-RELAX is devised for robust HR. It is revealed that the BSUM-RELAX successively performs alternating optimization along coordinate directions, i.e., it updates one harmonic by fixing the other <inline-formula><tex-math notation="LaTeX">$(K-1)$</tex-math> </inline-formula> components, such that the whole problem is split into <inline-formula><tex-math notation="LaTeX">$K$ </tex-math></inline-formula> single-tone HR problems, which are then solved by creating a surrogate function that majorizes the objective function of each subproblem. To further refine the frequency component, the Newton's method that takes linear complexity <inline-formula><tex-math notation="LaTeX">$\mathcal {O}(N)$</tex-math> </inline-formula> is derived for updating the frequency estimates. We prove that under the single-tone case, BSUM-RELAX converges to a Karush-Kuhn-Tucker point. Furthermore, the BSUM-RELAX is extended to the multidimensional HR case. Numerical results show that the proposed algorithm outperforms the state-of-the-art methods in heavy-tailed noise scenarios.