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arxiv: 2606.29118 · v1 · pith:TWMDGVNXnew · submitted 2026-06-28 · 💻 cs.IT · cs.CL· cs.IR· cs.LG· math.IT

An Information-Geometric Justification for Composite Coherence in Event-Based Narrative Extraction

Brian Keith-Norambuena This is my paper

Pith reviewed 2026-06-30 02:59 UTC · model grok-4.3

classification 💻 cs.IT cs.CLcs.IRcs.LGmath.IT
keywords narrative extractioncoherence metricinformation geometrygeometric meanJensen-Shannon distanceproduct manifoldFisher informationevent graphs
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The pith

The geometric mean is the unique combinator satisfying four axioms for combining angular and topic similarities in narrative coherence.

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

The paper provides an information-geometric foundation for the composite coherence function C equals square root of A times T that scores transitions between events in graph-based narrative extraction. It demonstrates that the negative log of this function decomposes additively into separate costs on the product manifold formed by the sphere of document embeddings and the simplex of topic memberships. The geometric mean is shown to be the only combination rule that obeys a boundary condition, symmetry, log-additivity, and normalization. The topic component aligns locally with the Fisher-Rao metric because the Jensen-Shannon distance induces a Riemannian tensor proportional to the Fisher information matrix. Experiments across corpora, embeddings, and topic models confirm the decomposition and show the geometric mean tracks the induced product metric while remaining competitive with alternatives.

Core claim

On the product manifold S to the d-1 times Delta to the K-1, the negative log-coherence decomposes additively into an angular cost from embedding similarity and a topic cost from the Jensen-Shannon distance. The topic cost is locally consistent with the Fisher-Rao metric because the Riemannian metric tensor induced by the Jensen-Shannon distance is proportional to the Fisher information matrix. Within the compensability spectrum of combinators, the geometric mean is the unique rule satisfying the four axioms of a boundary or veto condition, symmetry, log-additivity, and normalization, and this choice induces a proper product metric d sub times on the manifold.

What carries the argument

The geometric-mean combinator that forms sqrt of A times T and is the sole function on the compensability spectrum obeying the boundary, symmetry, log-additivity, and normalization axioms.

Load-bearing premise

The Riemannian metric tensor induced by the Jensen-Shannon distance on the simplex is proportional to the Fisher information matrix.

What would settle it

A calculation on the same topic models showing that the correlation between the Jensen-Shannon induced distances and the Fisher information matrix drops below 0.99, or an LLM-as-judge evaluation in which some other combinator or single-channel baseline outperforms the geometric mean on the extracted storylines.

Figures

Figures reproduced from arXiv: 2606.29118 by Brian Keith-Norambuena.

Figure 1
Figure 1. Figure 1: The product-manifold reading of the composite coherence metric. The angular com [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The compensability spectrum of power-mean combinators [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Geometry of the triangle-inequality counterexample for [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Approximation quality on Cuba, both panels colored by the minimum base-2 Shannon [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Joint distribution of angular similarity [PITH_FULL_IMAGE:figures/full_fig_p029_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Cross-corpus validation across four corpora: point estimates with [PITH_FULL_IMAGE:figures/full_fig_p030_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The bottleneck-gap profile g(α) (Experiment 7, Proposition 5). (a) Wikispeedia, hu￾man versus random navigation: the gap g(α) (blue, star at its peak α = +0.10) and its even part E(α) (red), maximized exactly at the geometric mean α = 0. (b) The four narrative corpora, narrative-trail versus random chronological sequence, each gap normalized by its value g(0) at the geometric mean; every profile peaks at α… view at source ↗
read the original abstract

Graph-based narrative extraction relies on a coherence function to score transitions between events, but the coherence metrics in current use are defined operationally and lack an information-theoretic foundation. We study the composite metric $C=\sqrt{A\cdot T}$, where $A$ is the angular similarity of document embeddings and $T=1-d_{\mathrm{JS}}$ is a topic proximity from the Jensen-Shannon distance of soft memberships, and give it an information-geometric reading together with an axiomatic characterization of the geometric-mean combinator. On the product manifold $\mathbb{S}^{d-1}\times\Delta^{K-1}$, the negative log-coherence decomposes additively into an angular and a topic cost. Because the Riemannian metric tensor induced by the Jensen-Shannon distance on the simplex is proportional to the Fisher information matrix, the topic component is locally consistent with the Fisher-Rao metric singled out by Chentsov's theorem. Within the compensability spectrum of combinators, the geometric mean is the unique one consistent with four natural axioms (a boundary/veto condition, symmetry, log-additivity, normalization), and the construction motivates a proper product metric $d_\times$. Experiments on four corpora, three embedding families, and three topic models are consistent with the framework: the Fisher identity holds ($R\ge0.99$), the geometric mean tracks $d_\times$ closely ($\rho=0.999$), and a downstream LLM-as-judge check finds it is not dominated by any alternative combinator or single-channel baseline. Sweeping the spectrum, the bottleneck-coherence gap between extracted and random storylines splits into a symmetric component, maximized at the geometric mean across five corpora, and a displacement term; a cross-modal image-narrative case study reproduces the effect. These results justify the composite coherence metric and articulate when the geometric mean is the natural choice.

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

0 major / 2 minor

Summary. The paper claims to provide an information-geometric justification for the composite coherence metric C=√(A·T) in event-based narrative extraction. It axiomatizes the geometric mean as the unique combinator satisfying four axioms (boundary/veto condition, symmetry, log-additivity, normalization) within the compensability spectrum, shows that the JS distance induces a Riemannian metric proportional to the Fisher information matrix on the simplex (consistent with Chentsov's theorem), motivates a product metric d_× on the product manifold, and reports empirical consistency with R≥0.99 for the Fisher identity and ρ=0.999 for tracking d_× across experiments on four corpora, three embeddings, and three topic models, along with LLM-as-judge and cross-modal validations.

Significance. If the central claims hold, the work supplies a principled axiomatic and geometric foundation for composite coherence, uniquely characterizing the geometric mean without free parameters and linking it to the Fisher-Rao metric. The use of standard results like Chentsov's theorem and the explicit four-axiom derivation are strengths that provide a parameter-free justification. The empirical checks, while supportive, require careful verification for the reported correlations.

minor comments (2)
  1. [Experiments] Experiments section: the reported values R≥0.99 and ρ=0.999 for the Fisher identity and geometric mean tracking lack accompanying details on sample sizes, data splits, exclusion criteria, or confidence intervals, which would strengthen the verification of the local consistency with the Fisher-Rao metric.
  2. [Abstract / product manifold section] The proportionality between the JS-induced metric tensor and the Fisher information matrix is asserted in the abstract and product manifold discussion but the explicit constant of proportionality or its derivation (via the Hessian of JS) could be stated more clearly for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly identifies the core contributions: the axiomatic uniqueness of the geometric mean under the four stated axioms and the link to the Fisher-Rao metric via Chentsov's theorem on the product manifold. No major comments requiring point-by-point rebuttal were listed in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation rests on four explicitly listed axioms for the geometric-mean combinator and on the external Chentsov uniqueness theorem; the claimed proportionality between the JS-induced Riemannian tensor and the Fisher information matrix is a standard property of the Jensen-Shannon divergence, not obtained by fitting inside the paper. The additive decomposition on the product manifold follows directly from the definitions of angular and topic costs. No equation reduces by construction to a fitted parameter, no load-bearing premise is justified solely by self-citation, and no ansatz is smuggled via prior work by the same authors. Experiments supply consistency checks rather than the central justification.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The paper rests on four stated axioms for the combinator and on the proportionality between the Jensen-Shannon-induced metric and the Fisher information matrix; no free parameters or new entities are introduced.

axioms (5)
  • domain assumption boundary/veto condition
    If either input similarity is zero then the composite coherence is zero.
  • standard math symmetry
    The combinator treats its two arguments symmetrically.
  • domain assumption log-additivity
    The logarithm of the combinator is additive in the two channels.
  • domain assumption normalization
    The combinator satisfies a normalization condition that fixes its scale.
  • domain assumption Jensen-Shannon metric tensor proportional to Fisher information
    The Riemannian metric induced by Jensen-Shannon distance on the simplex is proportional to the Fisher information matrix (invoked for consistency with Chentsov's theorem).

pith-pipeline@v0.9.1-grok · 5875 in / 1612 out tokens · 61754 ms · 2026-06-30T02:59:58.510986+00:00 · methodology

discussion (0)

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