Carlos Fernandez-Granda,
CIMS

Title: Separable nonlinear inverse problems in theory and
practice

Abstract:

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.