# Implicit Riemannian Concave Potential Maps

arXiv: https://arxiv.org/abs/2110.01288 · HTML: https://mlsanity.com/paper/2110.01288/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 32 references that are in the map.
- First posted on arXiv 4 Oct 2021.

## Builds on (most cited first, 20 of 20)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [OptNet: Differentiable Optimization as a Layer in Neural Networks](https://mlsanity.com/paper/1703.00443/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimization](https://mlsanity.com/paper/2012.05942/) (2020)
- [Sampling using SU(N) gauge equivariant flows](https://mlsanity.com/paper/2008.05456/) (2020)
- [Semi-Amortized Variational Autoencoders](https://mlsanity.com/paper/1802.02550/) (2018)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Equivariant Hamiltonian Flows](https://mlsanity.com/paper/1909.13739/) (2019)
- [Scalable Log Determinants for Gaussian Process Kernel Learning](https://mlsanity.com/paper/1711.03481/) (2017)
- [Equivariant Manifold Flows](https://mlsanity.com/paper/2107.08596/) (2021)
- [Riemannian Convex Potential Maps](https://mlsanity.com/paper/2106.10272/) (2021)
- [Implicitly Defined Layers in Neural Networks](https://mlsanity.com/paper/2003.01822/) (2020)

## Built on by (most cited first, 10 of 10)

- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [The Monge Gap: A Regularizer to Learn All Transport Maps](https://mlsanity.com/paper/2302.04953/) (2023)
- [Spherical Sliced-Wasserstein](https://mlsanity.com/paper/2206.08780/) (2022)
- [On amortizing convex conjugates for optimal transport](https://mlsanity.com/paper/2210.12153/) (2022)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [Symmetry-Based Representations for Artificial and Biological General Intelligence](https://mlsanity.com/paper/2203.09250/) (2022)
- [Differentiation Through Black-Box Quadratic Programming Solvers](https://mlsanity.com/paper/2410.06324/) (2024)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)
- [HOTS3D: Hyper-Spherical Optimal Transport for Semantic Alignment of Text-to-3D Generation](https://mlsanity.com/paper/2407.14419/) (2024)
- [Entropic Riemannian Neural Optimal Transport](https://mlsanity.com/paper/2605.04255/) (2026)

## Abstract (arXiv)

We are interested in the challenging problem of modelling densities on Riemannian manifolds with a known symmetry group using normalising flows. This has many potential applications in physical sciences such as molecular dynamics and quantum simulations. In this work we combine ideas from implicit neural layers and optimal transport theory to propose a generalisation of existing work on exponential map flows, Implicit Riemannian Concave Potential Maps, IRCPMs. IRCPMs have some nice properties such as simplicity of incorporating symmetries and are less expensive than ODE-flows. We provide an initial theoretical analysis of its properties and layout sufficient conditions for stable optimisation. Finally, we illustrate the properties of IRCPMs with density estimation experiments on tori and spheres.