# Lagrangian Manifold Monte Carlo on Monge Patches

arXiv: https://arxiv.org/abs/2202.00755 · HTML: https://mlsanity.com/paper/2202.00755/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 35 references that are in the map.
- First posted on arXiv 1 Feb 2022.

## Builds on (most cited first, 7 of 7)

- [A Conceptual Introduction to Hamiltonian Monte Carlo](https://mlsanity.com/paper/1701.02434/) (2017)
- [Latent Space Oddity: on the Curvature of Deep Generative Models](https://mlsanity.com/paper/1710.11379/) (2017)
- [The Tamed Unadjusted Langevin Algorithm](https://mlsanity.com/paper/1710.05559/) (2017)
- [Langevin diffusions and the Metropolis-adjusted Langevin algorithm](https://mlsanity.com/paper/1309.2983/) (2013)
- [Fisher Information and Natural Gradient Learning of Random Deep Networks](https://mlsanity.com/paper/1808.07172/) (2018)
- [Introducing an Explicit Symplectic Integration Scheme for Riemannian Manifold Hamiltonian Monte Carlo](https://mlsanity.com/paper/1910.06243/) (2019)
- [An Efficient Implementation of Riemannian Manifold Hamiltonian Monte Carlo for Gaussian Process Models](https://mlsanity.com/paper/1810.11893/) (2018)

## Built on by (most cited first, 11 of 11)

- [Riemannian Laplace approximations for Bayesian neural networks](https://mlsanity.com/paper/2306.07158/) (2023)
- [Coin Sampling: Gradient-Based Bayesian Inference without Learning Rates](https://mlsanity.com/paper/2301.11294/) (2023)
- [Scalable Stochastic Gradient Riemannian Langevin Dynamics in Non-Diagonal Metrics](https://mlsanity.com/paper/2303.05101/) (2023)
- [Riemannian Laplace Approximation with the Fisher Metric](https://mlsanity.com/paper/2311.02766/) (2023)
- [What's Inside Your Diffusion Model? A Score-Based Riemannian Metric to Explore the Data Manifold](https://mlsanity.com/paper/2505.11128/) (2025)
- [Log-density gradient covariance and automatic metric tensors for Riemann manifold Monte Carlo methods](https://mlsanity.com/paper/2211.01746/) (2022)
- [Warped geometric information on the optimisation of Euclidean functions](https://mlsanity.com/paper/2308.08305/) (2023)
- [Geodesic Slice Sampler for Multimodal Distributions with Strong Curvature](https://mlsanity.com/paper/2502.21190/) (2025)
- [Reducing Memorisation in Generative Models via Riemannian Bayesian Inference](https://mlsanity.com/paper/2602.00199/) (2026)
- [A Likely Geometry of Generative Models](https://mlsanity.com/paper/2510.26266/) (2025)
- [Disentanglement Beyond Generative Models with Riemannian ICA](https://mlsanity.com/paper/2605.22531/) (2026)

## Abstract (arXiv)

The efficiency of Markov Chain Monte Carlo (MCMC) depends on how the underlying geometry of the problem is taken into account. For distributions with strongly varying curvature, Riemannian metrics help in efficient exploration of the target distribution. Unfortunately, they have significant computational overhead due to e.g. repeated inversion of the metric tensor, and current geometric MCMC methods using the Fisher information matrix to induce the manifold are in practice slow. We propose a new alternative Riemannian metric for MCMC, by embedding the target distribution into a higher-dimensional Euclidean space as a Monge patch and using the induced metric determined by direct geometric reasoning. Our metric only requires first-order gradient information and has fast inverse and determinants, and allows reducing the computational complexity of individual iterations from cubic to quadratic in the problem dimensionality. We demonstrate how Lagrangian Monte Carlo in this metric efficiently explores the target distributions.