Optimization on manifolds: methods and applications

Pierre-Antoine Absil (University of Louvain, Belgium)

Abstract

Optimization on manifolds is now a well established research field. The first developments can be traced back to the 1970s, and the area has been very active for the past few years. Optimization on manifolds is concerned with problems that can be formulated as finding an optimum of a real-valued cost function defined on a smooth nonlinear search space. Oftentimes, the search space is a "matrix manifold" (or more generally a "tensor manifold"), in the sense that its points admit natural representations in the form of matrices. In most cases, the matrix manifold structure is due either to the presence of certain nonlinear constraints (such as orthogonality or rank constraints), or to invariance properties in the cost function that need to be factored out in order to obtain a nondegenerate optimization problem. Manifolds that come up in practical applications include the rotation group SO(3) (generation of rigid body motions from sample points), the set of fixed-rank matrices (low-rank models, e.g., in collaborative filtering), the set of 3x3 symmetric positive-definite matrices (interpolation of diffusion tensors), and the shape manifold (morphing). The practical importance of optimization problems on manifolds has stimulated the development of geometric optimization algorithms that exploit the differential structure of the manifold search space. This course will give an overview of geometric optimization algorithms and their applications, with an emphasis on the underlying geometric concepts and on the numerical efficiency of the algorithm implementations. The course will end with recent developments related to curve fitting and nonsmooth optimization on manifolds.

The second day will include computer exercises. Computers with Matlab are available in the seminar room, or you can bring your own laptop with Matlab installed.

Registration

No registration required.

Date and Place

• Monday, 24.04.2017 in Seminar Room 02.06.011
  • 10:15 - 11:45 and 16:15 - 17:45

• Tuesday, 25.04.2017: in Seminar Room 03.04.011
  • 10:15 - 11:45 and 16:15 - 17:45

-- DianeClaytonWinter - 21 Feb 2017