Iterative mix thresholding algorithm with continuation technique for mix sparse optimization and application

Mix sparse structure is inherited in a wide class of practical applications, namely, the sparse structure appears as the inter-group and intra-group manners simultaneously. In this paper, we propose an iterative mix thresholding algorithm with continuation technique (IMTC) to solve the $$\ell _0$$ ℓ 0 regularized mix sparse optimization. The significant advantage of the IMTC is that it has a closed-form expression and low storage requirement, and it is able to promote the mix sparse structure of the solution. We prove the convergence property and the linear convergence rate of the ITMC to a local minimum; moreover, we show that the ITMC approaches an approximate true mix sparse solution within a tolerance relevant to the noise level under an assumption of restricted isometry property. We also apply the mix sparse optimization to model the differential optical absorption spectroscopy analysis with the wavelength misalignment, and numerical results indicate that the IMTC can exactly and quantitatively predict the existing materials and the factual wavelength misalignment simultaneously within 0.1 s, which meets the demand of improvement of the automatic analysis software.