Cubic NK-SVD: An Algorithm for Designing Parametric Dictionary in Frequency Estimation

We propose a novel parametric dictionary learning algorithm for line spectral estimation, applicable in both single measurement vector (SMV) and multiple measurement vectors (MMV) scenarios. This algorithm, termed cubic Newtonized K-SVD (NK-SVD), extends the traditional K-SVD method by incorporating cubic regularization into Newton refinements. The proposed Gauss-Seidel scheme not only enhances the accuracy of frequency estimation over the continuum but also achieves better convergence by incorporating higher-order derivative information. A key contribution of this work is the rigorous convergence analysis of the proposed algorithm within the Block Coordinate Descent (BCD) framework. To the best of our knowledge, this is the first convergence analysis of BCD with a higher-order regularization scheme. Moreover, the convergence framework we develop is generalizable, providing a foundation for designing alternating minimization algorithms with higher-order regularization techniques. Extensive simulations demonstrate that cubic NK-SVD outperforms state-of-the-art methods in both SMV and MMV settings, particularly excelling in the challenging task of recovering closely-spaced frequencies. The code for our method is available at https://github.com/xzliu-opt/Cubic-NK-SVD.