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arxiv: 2508.21466 · v2 · submitted 2025-08-29 · 💻 cs.LG · cs.IT· math.IT

Normalized Maximum Likelihood Code-Length on Riemannian Data Spaces

Kota Fukuzawa , Atsushi Suzuki , Kenji Yamanishi This is my paper

Pith reviewed 2026-05-18 20:49 UTC · model grok-4.3

classification 💻 cs.LG cs.ITmath.IT
keywords Normalized Maximum LikelihoodRiemannian ManifoldsHyperbolic SpaceCoordinate InvarianceModel SelectionVolume FormSymmetric SpacesRegret Minimization
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The pith

A coordinate-invariant normalized maximum likelihood can be defined on Riemannian manifolds by using the volume form induced by the metric.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a new version of normalized maximum likelihood, called Rm-NML, that respects the geometry of any Riemannian manifold instead of being tied to Euclidean coordinates. The central move is to replace the ordinary Lebesgue measure in the normalizing integral with the Riemannian volume element, which automatically makes the resulting code length unchanged when the coordinate system is altered. This new quantity reduces exactly to the classical NML when the manifold is flat Euclidean space equipped with its standard coordinates. The authors also extend existing approximation methods and obtain simplified expressions on symmetric spaces, with an explicit formula worked out for normal distributions on hyperbolic space.

Core claim

We define the Riemannian manifold NML (Rm-NML) by replacing the Lebesgue measure in the conventional NML normalizing constant with the Riemannian volume form. This ensures invariance under coordinate transformations and agreement with standard NML in Euclidean space. We extend computational techniques and derive simplifications for Riemannian symmetric spaces, including explicit computation for normal distributions on hyperbolic spaces.

What carries the argument

Rm-NML, formed by integrating the maximum likelihood density against the Riemannian volume measure rather than Lebesgue measure, which carries the invariance property.

If this is right

  • Model selection and regret minimization can now be performed directly on manifold-valued data without coordinate artifacts.
  • On hyperbolic spaces the method supplies a concrete code length for hierarchical graph models.
  • Existing NML approximation algorithms extend to the manifold setting once the volume form is substituted.
  • The same construction applies to any Riemannian symmetric space arising in data analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach suggests trying Rm-NML on sphere-valued or Grassmannian data to see whether it improves compression over Euclidean approximations.
  • One could compare regret achieved by Rm-NML versus Euclidean NML when data are known to lie on a curved manifold.
  • Closed-form expressions might be derived for other common distributions on hyperbolic space beyond Gaussians.

Load-bearing premise

A Riemannian metric and its associated volume measure must exist on the data space so the normalizing integral can be written in a coordinate-free manner.

What would settle it

Evaluate the Rm-NML value for the same distribution on the sphere or hyperbolic plane in two different coordinate systems and check whether the numerical results match exactly.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper defines a Riemannian manifold version of Normalized Maximum Likelihood (Rm-NML) by replacing the Euclidean Lebesgue measure in the normalizing integral with the Riemannian volume form induced by a chosen metric on the data manifold. The resulting quantity is claimed to be invariant under coordinate reparameterizations, to reduce exactly to ordinary NML on Euclidean space with the standard metric, and to admit simplified evaluation on Riemannian symmetric spaces. An explicit closed-form or computable expression is derived for the normal distribution on hyperbolic space.

Significance. If the finiteness of the new normalizing integral and the invariance property are rigorously established, the construction supplies a geometrically intrinsic model-selection criterion for data lying on manifolds that arise in graph embedding and hierarchical modeling. The reduction to the Euclidean case and the explicit hyperbolic-normal example are direct consequences of standard differential-geometric identities and therefore constitute a clean extension of existing NML techniques.

major comments (2)
  1. [§3] §3, Definition of Rm-NML: the normalizing integral is asserted to be finite once the Riemannian volume form is substituted, yet no explicit integrability argument or decay estimate is supplied for the non-compact hyperbolic case; without such a bound the central claim that Rm-NML is well-defined remains unsupported.
  2. [§5] §5, hyperbolic-normal example: the derivation of the explicit Rm-NML expression relies on the volume element of the hyperboloid model, but the manuscript does not verify that the resulting integral converges or that the maximum-likelihood density is properly normalized with respect to this measure.
minor comments (2)
  1. [Notation] The notation for the Riemannian metric g and its induced volume form dV_g is introduced without a short reminder of the coordinate-change rule for the volume element; a one-sentence recall would improve readability for readers outside differential geometry.
  2. [Introduction] A reference to Rissanen’s original NML papers or to the standard Euclidean regret bounds would help situate the new definition relative to the literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and for highlighting the importance of rigorously establishing the well-definedness of Rm-NML. We address each major comment below and will incorporate the necessary additions in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3, Definition of Rm-NML: the normalizing integral is asserted to be finite once the Riemannian volume form is substituted, yet no explicit integrability argument or decay estimate is supplied for the non-compact hyperbolic case; without such a bound the central claim that Rm-NML is well-defined remains unsupported.

    Authors: We agree that an explicit proof of finiteness is essential for the non-compact case. In the revised manuscript, we will include a dedicated subsection providing a decay estimate for the integrand. Specifically, we will show that the maximum likelihood density for the Riemannian normal distribution on hyperbolic space decays exponentially with the hyperbolic distance, which, when integrated against the exponentially growing volume element, yields a convergent integral. This argument relies on standard estimates for hyperbolic geometry and ensures Rm-NML is well-defined. revision: yes

  2. Referee: [§5] §5, hyperbolic-normal example: the derivation of the explicit Rm-NML expression relies on the volume element of the hyperboloid model, but the manuscript does not verify that the resulting integral converges or that the maximum-likelihood density is properly normalized with respect to this measure.

    Authors: The maximum-likelihood density is normalized by construction with respect to the Riemannian volume measure, as it is the density of the probability distribution on the manifold. For the Rm-NML integral, we acknowledge the lack of explicit verification in the original text. We will add a verification step in the revised version, either by direct computation for the specific parameters or by providing bounds that confirm convergence using the explicit volume form in the hyperboloid model. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in Rm-NML definition

full rationale

The central construction replaces the Lebesgue measure with the Riemannian volume form inside the normalizing integral of the maximum-likelihood density. This is a direct, coordinate-free redefinition that inherits invariance from the intrinsic properties of the volume element and reduces to ordinary NML on Euclidean space by specialization of the metric; neither step relies on fitted parameters renamed as predictions nor on self-citations whose content is presupposed. Extensions to symmetric spaces and the hyperbolic-normal example invoke only standard differential-geometric identities that are independent of the present paper's results. The derivation chain is therefore self-contained against external geometric facts and introduces no load-bearing circular reductions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a Riemannian volume measure that replaces the Euclidean Lebesgue measure and on the technical assumption that the resulting integral can be normalized and computed on symmetric spaces.

axioms (1)
  • domain assumption Data space admits a Riemannian metric whose volume form can be used to define an invariant normalizing constant for the maximum-likelihood integral.
    Invoked when the authors replace coordinate-dependent Lebesgue measure with the intrinsic volume element to achieve invariance.

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Reference graph

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