Bio

Elina Robeva is a fourth-year graduate student in mathematics at UC Berkeley advised by Bernd Sturmfels. Originally from Bulgaria, Elina’s career as a mathematician started in middle school when she took part in many competitions in mathematics and computer science. After winning two silver medals from the international mathematical olympiad in high-school, she started her undergraduate degree at Stanford University in 2007. There she pursued her interests in mathematics and wrote two combinatorics papers with Professor Sam Payne. She received the Dean’s award, the Sterling award, the undergraduate research award, and an honorable mention for the Morgan prize. Elina completed software engineering internships at Facebook and Google and decided to pursue a PhD where she could apply her mathematical skills to problems in computer science and other applied disciplines. She commenced her PhD at Harvard University in 2011 and transferred to UC Berkeley in 2012 to work with Professor Bernd Sturmfels. Her papers are focused on the interplay between algebraic geometry statistics and optimization. They include work on mixture models and the EM algorithm, orthogonal tensor decomposition, factorizations through the cone of positive semidefinite matrices, and super-resolution imaging.

Super-resolution without Separation.

Super-resolution without Separation

This is joint work with Benjamin Recht and Geoffrey Schiebinger at UC Berkeley.
We provide a theoretical analysis of diffraction-limited super-resolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Given a lo-resolution blurred signal of M point sources of light, super-resolution imaging aims to recover the correct locations of the point sources and the intensity of light at each of them. Caused by the imaging device (telescope, microscope, camera, or others), every point source of light is blurred by a given point spread function. We assume that the incoming signal is a linear combination of M shifted copies (centered at each of the M point sources) of a known point spread function with unknown shifts (the locations of the point sources) and intensities, and one only observes a finite collection of evaluations of this signal.

To recover the locations and intensities, practitioners solve a convex program, which is a weighted version of basis pursuit over a continuous dictionary. Despite the recent success in many empirical disciplines, the theory of super-resolution imaging remains limited. More precisely, our aim is to show that the true point source locations and intensities are the unique optimal solution to the above mentioned convex program. Much of the existing proofs to date rely heavily on the assumption that the point sources are separated by more than some minimum amount. Building on polynomial interpolation techniques and tools from compressed sensing, we show that under some reasonable conditions on the point spread function, arbitrarily close point sources can be resolved by the above convex program from 2M+1 observations. Moreover, we show that the Gaussian point spread function satisfies these conditions.