# Neural FIM for learning Fisher Information Metrics from point cloud data

arXiv: https://arxiv.org/abs/2306.06062 · HTML: https://mlsanity.com/paper/2306.06062/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 1 Jun 2023.

## Builds on (most cited first, 13 of 13)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Manifold Interpolating Optimal-Transport Flows for Trajectory Inference](https://mlsanity.com/paper/2206.14928/) (2022)
- [The Shape of Data: Intrinsic Distance for Data Distributions](https://mlsanity.com/paper/1905.11141/) (2019)
- [Wasserstein Proximal of GANs](https://mlsanity.com/paper/2102.06862/) (2021)
- [Diffusion Curvature for Estimating Local Curvature in High Dimensional Data](https://mlsanity.com/paper/2206.03977/) (2022)
- [Out-of-Sample Extension for Dimensionality Reduction of Noisy Time Series](https://mlsanity.com/paper/1606.08282/) (2016)
- [On the Diffusion Geometry of Graph Laplacians and Applications](https://mlsanity.com/paper/1611.03033/) (2016)
- [Diffusion geometry unravels the emergence of functional clusters in collective phenomena](https://mlsanity.com/paper/1704.07068/) (2017)
- [On a prior based on the Wasserstein information matrix](https://mlsanity.com/paper/2202.03217/) (2022)
- [Error Metrics for Learning Reliable Manifolds from Streaming Data](https://mlsanity.com/paper/1611.04067/) (2016)

## Built on by (most cited first, 3 of 3)

- [Geometry-Aware Generative Autoencoders for Warped Riemannian Metric Learning and Generative Modeling on Data Manifolds](https://mlsanity.com/paper/2410.12779/) (2024)
- [Enforcing Latent Euclidean Geometry in Single-Cell VAEs for Manifold Interpolation](https://mlsanity.com/paper/2507.11789/) (2025)
- [From Layers to Networks: Comparing Neural Representations via Diffusion Geometry](https://mlsanity.com/paper/2605.15901/) (2026)

## Abstract (arXiv)

Although data diffusion embeddings are ubiquitous in unsupervised learning and have proven to be a viable technique for uncovering the underlying intrinsic geometry of data, diffusion embeddings are inherently limited due to their discrete nature. To this end, we propose neural FIM, a method for computing the Fisher information metric (FIM) from point cloud data - allowing for a continuous manifold model for the data. Neural FIM creates an extensible metric space from discrete point cloud data such that information from the metric can inform us of manifold characteristics such as volume and geodesics. We demonstrate Neural FIM's utility in selecting parameters for the PHATE visualization method as well as its ability to obtain information pertaining to local volume illuminating branching points and cluster centers embeddings of a toy dataset and two single-cell datasets of IPSC reprogramming and PBMCs (immune cells).