Normalized Maximum Likelihood Code-Length on Riemannian Manifold Data Spaces

2025-08-29

AI Summary

Normalized Maximum Likelihood (NML) lacks geometric consistency and coordinate invariance on Riemannian manifolds, limiting its applicability to non-Euclidean data. Method: This paper introduces the first coordinate-invariant Riemannian manifold NML (Rm-NML), extending NML to Riemannian data spaces by integrating differential geometry and statistical manifold theory. It employs the intrinsic Riemannian metric to model data geometry and exploits symmetric space structure for computational tractability. Contribution/Results: Rm-NML reduces to classical NML under natural parameterization in Euclidean space, ensuring theoretical unification; on hyperbolic space, it yields an explicit closed-form coding length for the normal distribution, confirming both theoretical coherence and practical feasibility. As the first geometrically aware, coordinate-free Minimum Description Length principle for non-Euclidean graph data, Rm-NML establishes a foundational framework for principled modeling of manifold-structured data.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Data Mining & Knowledge Management: Other Foundations of Data Mining & Knowledge Management

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

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.

Problem

• Extending NML to Riemannian manifolds beyond Euclidean spaces
• Developing coordinate-invariant NML for geometric data structures
• Simplifying NML computation on symmetric spaces like hyperbolic geometry

Innovation

• Defined Rm-NML for Riemannian manifold geometric structure
• Extended NML computation techniques to Riemannian manifolds
• Simplified Rm-NML computation on Riemannian symmetric spaces