Functional Nonlinear Sparse Models
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Signal processing in inherently continuous and often nonlinear applications, such as radar, magnetic resonance 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 estimation problems typically involves discretization and convex relaxations, e.g., using atomic norms. Although successful, this approaches are not without issues. Discretization often leads to high dimensional, potentially ill-conditioned optimization problems. Moreover, due to grid mismatch and other coherence issues, a sparse signal in the continuous domain may no longer be sparse when discretized. Finally, nonlinear problems remain non-convex even after relaxing the sparsity objective. And even in the linear case, performance guarantees for atomic norm relaxations hold under assumptions that may be hard to meet in practice. We propose to address these issues by directly tackling the continuous, nonlinear problem cast as a sparse functional optimization program. We prove that these problems have no duality gap and show that they can be solved efficiently using duality and a (stochastic) subgradient ascent-type algorithm. We illustrate the wide range of applications for this new approach by formulating and solving sparse problems in super-resolution (nonlinear line spectral estimation) and vector field estimation (spectrum cartography).
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