A $ q$-Gaussian Maximum Correntropy Adaptive Filtering Algorithm for Robust Spare Recovery in Impulsive Noise

This letter proposes a robust formulation for sparse signal reconstruction from compressed measurements corrupted by impulsive noise, which exploits the <inline-formula><tex-math notation="LaTeX">$\boldsymbol {q}$</tex-math> </inline-formula>-Gaussian generalized correntropy <inline-formula><tex-math notation="LaTeX">$\boldsymbol {(1< q< 3)}$</tex-math></inline-formula> as the loss function for the residual error and utilizes a <inline-formula> <tex-math notation="LaTeX">$\ell _{0}$</tex-math></inline-formula>-norm penalty term for sparsity inducing. To solve this formulation efficiently, we develop a gradient-based adaptive filtering algorithm which incorporates a zero-attracting regularization term into the framework of adaptive filtering. This new proposed algorithm blending the advantages of adaptive filtering and <inline-formula><tex-math notation="LaTeX">$\boldsymbol {q}$</tex-math> </inline-formula>-Gaussian generalized correntropy can obtain accurate reconstruction and satisfactory robustness with a proper shape parameter <inline-formula><tex-math notation="LaTeX">$\boldsymbol {q}$</tex-math></inline-formula>. Numerical experiments on both synthetic sparse signals and natural images are conducted to illustrate the superior recovery performance of the proposed algorithm to the state-of-the-art robust sparse signal reconstruction algorithms.