# Expected path length on random manifolds

arXiv: https://arxiv.org/abs/1908.07377 · HTML: https://mlsanity.com/paper/1908.07377/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2019.

## Built on by (most cited first, 9 of 9)

- [Learning Riemannian Manifolds for Geodesic Motion Skills](https://mlsanity.com/paper/2106.04315/) (2021)
- [Pulling back information geometry](https://mlsanity.com/paper/2106.05367/) (2021)
- [A prior-based approximate latent Riemannian metric](https://mlsanity.com/paper/2103.05290/) (2021)
- [Reactive Motion Generation on Learned Riemannian Manifolds](https://mlsanity.com/paper/2203.07761/) (2022)
- [Adversarial robustness of VAEs through the lens of local geometry](https://mlsanity.com/paper/2208.03923/) (2022)
- [Identifying latent distances with Finslerian geometry](https://mlsanity.com/paper/2212.10010/) (2022)
- [Geometry and Generalization: Eigenvalues as predictors of where a network will fail to generalize](https://mlsanity.com/paper/2107.06386/) (2021)
- [Geometric instability of out of distribution data across autoencoder architecture](https://mlsanity.com/paper/2201.11902/) (2022)
- [Extended Neural Contractive Dynamical Systems: On Multiple Tasks and Riemannian Safety Regions](https://mlsanity.com/paper/2411.11405/) (2024)

## Abstract (arXiv)

Manifold learning seeks a low dimensional representation that faithfully captures the essence of data. Current methods can successfully learn such representations, but do not provide a meaningful set of operations that are associated with the representation. Working towards operational representation learning, we endow the latent space of a large class of generative models with a random Riemannian metric, which provides us with elementary operators. As computational tools are unavailable for random Riemannian manifolds, we study deterministic approximations and derive tight error bounds on expected distances.