# Normalized Maximum Likelihood Code-Length on Riemannian Data Spaces

## TL;DR

This paper introduces a coordinate-invariant Normalized Maximum Likelihood (NML) tailored for Riemannian manifolds, enabling model selection and regret minimization on non-Euclidean data spaces like hyperbolic spaces.

## Contribution

It defines a new Riemannian NML (Rm-NML) that respects geometric structures, extends computational methods, and simplifies calculations on symmetric spaces.

## Key findings

- Derived Rm-NML for hyperbolic spaces
- Extended NML computation techniques to Riemannian manifolds
- Explicitly computed Rm-NML for normal distributions on hyperbolic spaces

## Abstract

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.