A half-quadratic block-coordinate descent method for spectral estimation

In short-time spectral estimation, Sacchi et al. (IEEE Trans. Signal Process. 46(1) (1998) 31) and Ciuciu et al. (IEEE Trans. Signal Process. 49 (2001) 2202) derived new nonlinear spectral estimators defined as minimizers of penalized criteria. The first contributors have introduced separable penalizations for line spectra (LS) recovering, whereas the latter have proposed circular Gibbs–Markov functions for smooth spectra (SS) restoration, and combined both contributions for estimation of “mixed” spectra (MS), i.e., frequency peaks superimposed on a homogeneous background (Ciuciu et al., 2001). Sacchi et al. resorted to the iteratively reweighted least squares (IRLS) algorithm for the minimization stage. Here, we show that IRLS is a block-coordinate descent (BCD) method performing the minimization of a half-quadratic (HQ) energy. The latter, derived from the Geman and Reynolds construction, has the same minimizer as the initial criterion but depends on more variables. After proving that such a construction is not available for Gibbs–Markov penalizations, we extend the pioneering work of Geman and Yang (IEEE Trans. Image Process. 4(7) (1995) 932) that leads to a suitable HQ energy for any kind of penalization encountered in Ciuciu et al. (2001). The BCD algorithm used for minimizing such HQ criteria is actually an original residual steepest descent (RSD) procedure (IEEE Trans. Acoust. Speech Signal Process. ASSP-33(1) (1985) 174) and thus converges in any convex case. A comparison between RSD, IRLS when available, and a pseudo-conjugate gradient algorithm is addressed in any case. [Copyright &y& Elsevier]