In various scenarios in the information sciences, one wishes to estimate a large number of parameters from highly incomplete / imperfect data samples. A growing body of recent work has demonstrated the effectiveness of convex relaxation --- in particular, semidefinite programming --- for solving many problems of this kind. However, the computational cost of such convex paradigms is often unsatisfactory, which limits applicability to large-dimensional data. This talk follows another route: by formulating the problems into nonconvex programs, we attempt to optimize the nonconvex objectives directly. To illustrate the power of this strategy, we present two concrete stories. The first involves solving a random quadratic system of equations, which spans many applications ranging from the century-old phase retrieval problem to various latent-variable models in machine learning. The second is about recovering a collection of discrete variables from noisy pairwise difference measurements, which arises when one wishes to jointly align multiple images or to retrieve the genome phases from paired sequencing reads. We propose novel nonconvex paradigms for solving these two problems. In both cases, the proposed solutions can be accomplished within linear time, while achieving a statistical accuracy that is nearly un-improvable.

Yuxin Chen is currently a postdoctoral scholar in the Department of Statistics at Stanford University, supervised by Prof. Emmanuel Candès. He received the Ph.D. degree in Electrical Engineering and M.S. in Statistics from Stanford University, M.S. in Electrical and Computer Engineering from the University of Texas at Austin, and B.E. in microelectronics from Tsinghua University. His research interests include high-dimensional structured estimation, convex and nonconvex optimization, statistical learning, and information theory.