  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.
