Inverse Problems, Imaging and Tensor Decomposition
Perspectives from computational algebra and non-convex optimization are brought to bear on a scientific application and a data science application. In the first part of the talk, I will discuss cryo-electron microscopy (cryo-EM), an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D projections, recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM presents a particularly rich inverse problem, with unknown orientations, extreme noise, big data and conformational heterogeneity. In particular, this motivates a general framework for statistical estimation under compact group actions, connecting information theory and group invariant theory. In the second part of the talk, I will discuss tensor rank decomposition, a higher-order variant of PCA broadly applicable in data science. A fast algorithm is introduced and analyzed, combining ideas of Sylvester and the power method.
Joe Kileel is currently a Simons Postdoctoral Research Associate in the Program in Applied and Computational Mathematics, Princeton University, working with Amit Singer’s group. In 2017, he received a Mathematics PhD from UC Berkeley under the supervision of Bernd Sturmfels, where his thesis was awarded the Bernard Friedman Memorial Prize for best in applied mathematics. Joe’s research interests center on mathematical data science, with focuses on imaging science, tensor methods, computational statistics and inverse problems. He is especially interested in the development of scalable and robust nonlinear algebraic techniques for scientific computing and data