# Normalized Maximum Likelihood Code-Length on Riemannian Data Spaces

arXiv: https://arxiv.org/abs/2508.21466 · HTML: https://mlsanity.com/paper/2508.21466/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 55 references that are in the map.
- First posted on arXiv 29 Aug 2025.

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

- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [TorusE: Knowledge Graph Embedding on a Lie Group](https://mlsanity.com/paper/1711.05435/) (2017)
- [CO-SNE: Dimensionality Reduction and Visualization for Hyperbolic Data](https://mlsanity.com/paper/2111.15037/) (2021)
- [A Geometric Modeling of Occam's Razor in Deep Learning](https://mlsanity.com/paper/1905.11027/) (2019)
- [Detecting Signs of Model Change with Continuous Model Selection Based on Descriptive Dimensionality](https://mlsanity.com/paper/2302.12127/) (2023)
- [Foundation of Calculating Normalized Maximum Likelihood for Continuous Probability Models](https://mlsanity.com/paper/2409.08387/) (2024)

## Built on by (most cited first, 1 of 1)

- [The Normalized Maximum Likelihood for Regular Non-Smooth Models: Measure-Theoretic Foundations and Geometric Sampling](https://mlsanity.com/paper/2605.24477/) (2026)

## Abstract (arXiv)

In recent years, with the large-scale expansion of graph data, there has been an increased focus on Riemannian manifold data spaces other than Euclidean space. In particular, the development of hyperbolic spaces has been remarkable, and they have high expressive power for graph data with hierarchical structures. Normalized Maximum Likelihood (NML) is employed in regret minimization and model selection. However, existing formulations of NML have been developed primarily in Euclidean spaces and are inherently dependent on the choice of coordinate systems, making it non-trivial to extend NML to Riemannian manifolds. In this study, we define a new NML that reflects the geometric structure of Riemannian manifolds, called the Riemannian manifold NML (Rm-NML). This Rm-NML is invariant under coordinate transformations and coincides with the conventional NML under the natural parameterization in Euclidean space. We extend existing computational techniques for NML to the setting of Riemannian manifolds. Furthermore, we derive a method to simplify the computation of Rm-NML on Riemannian symmetric spaces, which encompass data spaces of growing interest such as hyperbolic spaces. To illustrate the practical application of our proposed method, we explicitly computed the Rm-NML for normal distributions on hyperbolic spaces.