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Poster
Bayesian Quadrature on Riemannian Data Manifolds
Christian Fröhlich · Alexandra Gessner · Philipp Hennig · Bernhard Schölkopf · Georgios Arvanitidis

Thu Jul 22 09:00 AM -- 11:00 AM (PDT) @
in Poster Session 5 »

Riemannian manifolds provide a principled way to model nonlinear geometric structure inherent in data. A Riemannian metric on said manifolds determines geometry-aware shortest paths and provides the means to define statistical models accordingly. However, these operations are typically computationally demanding. To ease this computational burden, we advocate probabilistic numerical methods for Riemannian statistics. In particular, we focus on Bayesian quadrature (BQ) to numerically compute integrals over normal laws on Riemannian manifolds learned from data. In this task, each function evaluation relies on the solution of an expensive initial value problem. We show that by leveraging both prior knowledge and an active exploration scheme, BQ significantly reduces the number of required evaluations and thus outperforms Monte Carlo methods on a wide range of integration problems. As a concrete application, we highlight the merits of adopting Riemannian geometry with our proposed framework on a nonlinear dataset from molecular dynamics.

Keywords: Convex Optimization · large scale learning · Algorithms · Gaussian Processes and Bayesian non-parametrics

Author Information

Christian Fröhlich (University of Tübingen)
Alexandra Gessner (University of Tuebingen)
Philipp Hennig (University of Tübingen)
Bernhard Schölkopf (MPI for Intelligent Systems Tübingen, Germany)

Bernhard Scholkopf received degrees in mathematics (London) and physics (Tubingen), and a doctorate in computer science from the Technical University Berlin. He has researched at AT&T Bell Labs, at GMD FIRST, Berlin, at the Australian National University, Canberra, and at Microsoft Research Cambridge (UK). In 2001, he was appointed scientific member of the Max Planck Society and director at the MPI for Biological Cybernetics; in 2010 he founded the Max Planck Institute for Intelligent Systems. For further information, see www.kyb.tuebingen.mpg.de/~bs.

Georgios Arvanitidis (MPI for Intelligent Systems, Tübingen)

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