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Semi-Riemannian Manifold Optimization
We introduce in this paper a manifold optimization framework that utiliz...
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Fat Triangulations and Differential Geometry
We study the differential geometric consequences of our previous result ...
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Fast and Robust Shortest Paths on Manifolds Learned from Data
We propose a fast, simple and robust algorithm for computing shortest pa...
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Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope
We present an entirely new geometric and probabilistic approach to synch...
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Probabilistic Solutions to Differential Equations and their Application to Riemannian Statistics
We study a probabilistic numerical method for the solution of both bound...
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A Generalization of the Pearson Correlation to Riemannian Manifolds
The increasing application of deep-learning is accompanied by a shift to...
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Statistical models and probabilistic methods on Riemannian manifolds
This entry contains the core material of my habilitation thesis, soon to...
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Bayesian Quadrature on Riemannian Data Manifolds
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.
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