Carlos Fernandez-Granda, CIMS
Title: Separable nonlinear inverse problems in theory and practice 

In separable inverse problems the data are modeled as a linear combination of functions that depend nonlinearly on certain parameters of interest. The parameters may represent locations of neural activity, frequencies of electromagnetic waves, or fluorescent probes in a cell. Parameter estimation can be reformulated as an underdetermined sparse-recovery problem, and solved using convex programming. The approach has had empirical success in a variety of domains, but lacks a theoretical justification. In this talk we present a theory of sparse recovery tailored to this setting, which establishes exact recovery of the parameters of interest, as long as their values are sufficiently distinct with respect to the correlation structure of the measurements. In addition, we describe an application in magnetic-resonance imaging, where the parameters are magnetic relaxation times of biological tissues. Finally, we illustrate the potential of learning-based methodology for these inverse problems by describing a novel data-driven technique to perform super-resolution of line spectra.