EE Systems Seminar
What is the role of curvature in complexity of optimization on manifolds?
ABSTRACT The talk is about solving optimization problems of the form: min f(x), where x lives on a (known) smooth manifold. For example, the manifold could be a sphere, a set of orthonormal matrices, complex phases, fixed-rank matrices or tensors, rigid motions, or a more abstract quotient space owing to symmetry. This comes up in signal and image processing, computer vision, machine learning, inverse problems etc.
After a brief description of how Riemannian geometry enables efficient algorithms, I'll discuss which properties of the cost function and of the manifold affect the worst-case complexity of computing approximate stationary points (both first and second order). In particular, I'll share some thoughts about the role of Riemannian curvature on that front.
BIO Nicolas Boumal is an assistant professor in the mathematics department at Princeton University, where he was also an instructor with the Program in Applied and Computational Mathematics 2016-2018. He studies non-convex optimization, numerical analysis and statistical estimation, exploiting mathematical structures such as smooth geometry, convex geometry and low rank. He is the author of a popular Riemannian optimization toolbox called Manopt.
He obtained his PhD in mathematical engineering from the Université catholique de Louvain in Belgium in 2014, and was a postdoc with the computer science department of the Ecole Normale Supérieure de Paris in 2015. His research is partially supported by a grant of the National Science Foundation.
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