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Sparse learning models are popular in many application areas. Objective functions in sparse learning models are usually non-smooth, which makes it difficult to solve them numerically. We develop a fast and convergent two-step iteration scheme for solving a class of non-differentiable optimization models motivated from sparse learning. To overcome the difficulty of the non-differentiability of the models, we first present characterizations of their solutions as fixed-points of mappings involving the proximity operators of the functions appearing in the objective functions. We then introduce a two-step fixed-point algorithm to compute the solutions. We establish convergence results of the proposed two-step iteration scheme and compare it with the alternating direction method of multipliers (ADMM). In particular, we derive specific two-step iteration algorithms for three models in machine learning: l1-SVM classification, l1-SVM regression, and the SVM classification with the group LASSO regularizer. Numerical experiments with some synthetic datasets and some benchmark datasets show that the proposed algorithm outperforms ADMM and the linear programming method in computational time and memory storage costs. 

Abstract Seismic tomography has demonstrated that the shear‐wave velocity is relatively high over a 3,000‐km wide region in the lowermost mantle beneath southern and eastern Asia. This seismic anomaly demarcates the current position of slab remnants that may have subducted in the Cretaceous. To further characterize the seismic structure at smaller scales, we measure 929 residual travel time differences (δt) between the phasesScSandSusing recordings of eight earthquakes beneath the Indian Ocean at stations from the Chinese Digital Seismic Network. We interpret variations of δtup to 10 s as due to horizontal shear‐velocity variations in D″ beneath northern India, Nepal, and southwestern China. The shear velocity can vary by as much as 7% over distances shorter than 300 km. Our observations provide additional observational evidence that compositional heterogeneity and possibly melt contribute to the seismic structure of the lower mantle characterized by long‐term subduction and mantle downwelling.