Functional Nonlinear Sparse Models
This addresses issues like grid mismatch and coherence in sparse signal recovery for applications such as imaging and classification, offering a more direct and potentially robust method, though it appears incremental in refining existing sparse modeling frameworks.
The paper tackles the challenges of infinite dimensionality and non-convexity in continuous nonlinear sparse signal processing problems, such as spectral estimation and robust classification, by proposing a functional optimization approach that avoids discretization and convex relaxations, proving no duality gap for efficient solution via convex methods.
Signal processing is rich in inherently continuous and often nonlinear applications, such as spectral estimation, optical imaging, and super-resolution microscopy, in which sparsity plays a key role in obtaining state-of-the-art results. Coping with the infinite dimensionality and non-convexity of these problems typically involves discretization and convex relaxations, e.g., using atomic norms. Nevertheless, grid mismatch and other coherence issues often lead to discretized versions of sparse signals that are not sparse. Even if they are, recovering sparse solutions using convex relaxations requires assumptions that may be hard to meet in practice. What is more, problems involving nonlinear measurements remain non-convex even after relaxing the sparsity objective. We address these issues by directly tackling the continuous, nonlinear problem cast as a sparse functional optimization program. We prove that when these problems are non-atomic, they have no duality gap and can therefore be solved efficiently using duality and~(stochastic) convex optimization methods. We illustrate the wide range of applications of this approach by formulating and solving problems from nonlinear spectral estimation and robust classification.