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.

There are two types of ordinary differential equations (ODEs): initial value problems (IVPs) and boundary value problems (BVPs). While many probabilistic numerical methods for the solution of IVPs have been presented to-date, there exists no efficient probabilistic general-purpose solver for nonlinear BVPs. Our method based on iterated Gaussian process (GP) regression returns a GP posterior over the solution of nonlinear ODEs, which provides a meaningful error estimation via its predictive posterior standard deviation. Our solver is fast (typically of quadratic convergence rate) and the theory of convergence can be transferred from prior non-probabilistic work. Our method performs on par with standard codes for an established benchmark of test problems.

Time-of-flight (TOF) depth cameras provide robust depth inference at low power requirements in a wide variety of consumer and industrial applications. These cameras reconstruct a single depth frame from a given set of infrared (IR) frames captured over a very short exposure period. Operating in this mode the camera essentially forgets all information previously captured - and performs depth inference from scratch for every frame. We challenge this practice and propose using previously captured information when inferring depth. An inherent problem we have to address is camera motion over this longer period of collecting observations. We derive a probabilistic framework combining a simple but robust model of camera and object motion, together with an observation model. This combination allows us to integrate information over multiple frames while remaining robust to rapid changes. Operating the camera in this manner has implications in terms of both computational efficiency and how information should be captured. We address these two issues and demonstrate a realtime TOF system with robust temporal integration that improves depth accuracy over strong baseline methods including adaptive spatio-temporal filters.