  Extracting information from nonlinear measurements is a fundamental challenge
in data analysis. In this work, we consider separable inverse problems, where
the data are modeled as a linear combination of functions that depend
nonlinearly on certain parameters of interest. These parameters may represent
neuronal activity in a human brain, frequencies of electromagnetic waves,
fluorescent probes in a cell, or magnetic relaxation times of biological
tissues. Separable nonlinear inverse problems can be reformulated as
underdetermined sparse-recovery problems, and solved using convex programming.
This approach has had empirical success in a variety of domains, from
geophysics to medical imaging, but lacks a theoretical justification. In
particular, compressed-sensing theory does not apply, because the measurement
operators are deterministic and violate incoherence conditions such as the
restricted-isometry property. Our main contribution is a theory for sparse
recovery adapted to deterministic settings. We show that convex programming
succeeds in recovering the parameters of interest, as long as their values are
sufficiently distinct with respect to the correlation structure of the
measurement operator. The theoretical results are illustrated through numerical
experiments for two applications: heat-source localization and estimation of
brain activity from electroencephalography data.
