Fast and Robust Shortest Paths on Manifolds Learned from Data

Georgios Arvanitidis, Søren Hauberg, Philipp Hennig, Michael Schober

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Abstract

We propose a fast, simple and robust algorithm for computing shortest paths and distances on Riemannian manifolds learned from data. This amounts to solving a system of ordinary differential equations (ODEs) subject to boundary conditions. Here standard solvers perform poorly because they require well-behaved Jacobians of the ODE, and usually, manifolds learned from data imply unstable and ill-conditioned Jacobians. Instead, we propose a fixed-point iteration scheme for solving the ODE that avoids Jacobians. This enhances the stability of the solver, while reduces the computational cost. In experiments involving both Riemannian metric learning and deep generative models we demonstrate significant improvements in speed and stability over both general-purpose state-of-the-art solvers as well as over specialized solvers.
Original languageEnglish
Title of host publicationProceedings of the 22nd International Conference on Artificial Intelligence and Statistics
Volume89
Publication date2019
Pages1506-1515
Publication statusPublished - 2019
Event22nd International Conference on Artificial Intelligence and Statistics - LOISIR Hotel, Naha, Japan
Duration: 16 Apr 201918 Apr 2019

Conference

Conference22nd International Conference on Artificial Intelligence and Statistics
LocationLOISIR Hotel
CountryJapan
CityNaha
Period16/04/201918/04/2019

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