# Riemannian Convex Potential Maps

@inproceedings{Cohen2021RiemannianCP, title={Riemannian Convex Potential Maps}, author={Samuel Cohen and Brandon Amos and Yaron Lipman}, booktitle={ICML}, year={2021} }

Modeling distributions on Riemannian manifolds is a crucial component in understanding nonEuclidean data that arises, e.g., in physics and geology. The budding approaches in this space are limited by representational and computational tradeoffs. We propose and study a class of flows that uses convex potentials from Riemannian optimal transport. These are universal and can model distributions on any compact Riemannian manifold without requiring domain knowledge of the manifold to be integrated…

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## References

SHOWING 1-10 OF 54 REFERENCES

### Riemannian Continuous Normalizing Flows

- Mathematics, Computer ScienceNeurIPS
- 2020

Riemannian continuous normalizing flows is introduced, a model which admits the parametrization of flexible probability measures on smooth manifolds by defining flows as the solution to ordinary differential equations.

### Continuity of optimal transport maps and convexity of injectivity domains on small deformations of 𝕊2

- Mathematics
- 2009

Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to…

### A Jacobian Inequality for Gradient Maps on the Sphere and Its Application to Directional Statistics

- Mathematics
- 2009

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this article, the Jacobian determinant of a gradient map…

### Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)

- Mathematics
- 2008

Abstract The variant A3w of Ma, Trudinger and Wang's condition for regularity of optimal transportation maps is implied by the non-negativity of a pseudo-Riemannian curvature—which we call…

### Regularity of optimal transport maps on multiple products of spheres

- Mathematics
- 2010

This article addresses regularity of optimal transport maps for cost=“squared distance” on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and…

### Neural Manifold Ordinary Differential Equations

- Mathematics, Computer ScienceNeurIPS
- 2020

This paper introduces Neural Manifolds Ordinary Differential Equations, a manifold generalization of Neural ODEs, which enables the construction of Manifold Continuous Normalizing Flows (MCNFs), and finds that leveraging continuous manifold dynamics produces a marked improvement for both density estimation and downstream tasks.

### Normalizing Flows on Tori and Spheres

- Mathematics, Computer ScienceICML
- 2020

This paper proposes and compares expressive and numerically stable flows on spaces with more complex geometries, such as tori or spheres, and builds recursively on the dimension of the space, starting from flows on circles, closed intervals or spheres.

### Equivariant Hamiltonian Flows

- Computer Science, MathematicsArXiv
- 2019

This paper introduces equivariant hamiltonian flows, a method for learning expressive densities that are invariant with respect to a known Lie-algebra of local symmetry transformations while…

### On the regularity of solutions of optimal transportation problems

- Mathematics
- 2009

We give a necessary and sufficient condition on the cost function so that the map solution of Monge’s optimal transportation problem is continuous for arbitrary smooth positive data. This condition…

### Optimal transport mapping via input convex neural networks

- Computer ScienceICML
- 2020

This approach ensures that the transport mapping the authors find is optimal independent of how they initialize the neural networks, as gradient of a convex function naturally models a discontinuous transport mapping.