2012/01/18 by Dheeraj Singaraju, Singaraju, Dheeraj, Ehsan Elhamifar +7
Computer Science · Engineering · #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Machine Learning and Algorithms #Microwave Imaging and Scattering Analysis #Sparse and Compressive Sensing Techniques
paper · pdf · doi:10.48550/arxiv.1201.3674
openalex publication_date 2012/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recent results in Compressive Sensing have shown that, under certain conditions, the solution to an underdetermined system of linear equations with sparsity-based regularization can be accurately recovered by solving convex relaxations of the original problem. In this work, we present a novel primal-dual analysis on a class of sparsity minimization problems. We show that the Lagrangian bidual (i.e., the Lagrangian dual of the Lagrangian dual) of the sparsity minimization problems can be used to derive interesting convex relaxations: the bidual of the ℓ0-minimization problem is the ℓ1-minimization problem; and the bidual of the ℓ0,1-minimization problem for enforcing group sparsity on structured data is the ℓ1,∞-minimization problem. The analysis provides a means to compute per-instance non-trivial lower bounds on the (group) sparsity of the desired solutions. In a real-world application, the bidual relaxation improves the performance of a sparsity-based classification framework applied to robust face recognition.