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

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

Problem Statement
Graph-based narrative extraction relies on a coherence function to score transitions between events (documents). While operational metrics exist—ranging from word-influence random walks to KL divergence with temporal decay—they lack a principled mathematical foundation. Specifically, current metrics are not derived from an underlying geometry of the document space, do not necessarily satisfy proper metric properties, and lack an axiomatic derivation. This paper addresses the specific composite metric $C = \sqrt{A \cdot T}$ introduced by Keith and Mitra [2], where $A$ is angular similarity and $T$ is topic similarity derived via Jensen-Shannon distance. The goal is to provide an information-geometric justification for this specific combination and to characterize the geometric mean as the natural combinator under specific conditions.

Methodology and Theoretical Framework
The paper constructs a framework based on Information Geometry and Product Riemannian Manifolds.

1. Product Manifold Structure: The authors model the document space as a product manifold $S^{d-1} \times \Delta^{K-1}_+$.

   • The angular component ($A$) resides on the unit sphere $S^{d-1}$, measuring geodesic distance between document embeddings.
   • The topic component ($T$) resides on the probability simplex $\Delta^{K-1}_+$, measuring proximity via the Jensen-Shannon distance ($d_{JS}$) of soft cluster membership distributions.
   • Proposition 1 demonstrates that the negative log-coherence decomposes additively: $-\log C = \frac{1}{2}[-\log(1 - d_{ang}/\pi)] + \frac{1}{2}[-\log(1 - d_{JS})]$. This treats the angular and topic costs as additive components on the product manifold.

2. Local Chentsov Compatibility:

   • The paper invokes Chentsov's Theorem, which states that the Fisher-Rao metric is the unique Riemannian metric on the simplex invariant under sufficient statistics.
   • Theorem 1 and Corollary 1 establish that the Riemannian metric tensor induced by the Jensen-Shannon distance ($d_{JS}$) is exactly proportional to the Fisher information matrix. Thus, the topic component is locally consistent with the Fisher-Rao geometry singled out by Chentsov's theorem.

3. Axiomatic Characterization:

   • The authors define a "compensability spectrum" of power-mean combinators (Min, HM, GM, AM, Max).
   • Theorem 3 provides an axiomatic characterization showing that the Geometric Mean (GM) is the unique combinator satisfying four natural axioms:
     • Boundary/Veto: $F(0, t) = 0$ (a failure in one channel yields zero coherence).
     • Symmetry: $F(a, t) = F(t, a)$.
     • Log-additivity: $-\log F(a, t)$ decomposes into a sum of functions of $-\log a$ and $-\log t$.
     • Normalization: $F(a, a) = a$.
   • The paper proves that dropping any single axiom admits non-geometric-mean solutions.

4. Information-Theoretic Connections:

   • The angular cost is identified as the Shannon surprisal of a random-hyperplane (SimHash) collision.
   • The topic cost is identified as a strictly increasing function of the mutual information between a sample and a latent label (Proposition 10).
   • The framework relies on Scale Complementarity: the two channels capture information at different scales (fine-grained semantic direction vs. coarse-grained thematic membership) such that they are non-redundant despite structural coupling.

Experimental Validation
The framework was validated across four corpora (Cuba, COVID, VisPub, AMiner) ranging from 40 to 6,000 documents, using various embedding models (GPT-4, MPNet, MiniLM, SPECTER2) and topic models (LDA, soft k-means, GMM).

• Metric Properties: The Fisher identity ($d_{JS} \approx d_{FR}/\sqrt{2 \ln 2}$) held with Pearson correlation $R \ge 0.99$. The geometric mean tracked the proper product metric $d_\times$ with rank correlation $\rho = 0.999$.
• Triangle Inequality: While $1 - C_{GM}$ is not a metric in general, no violations were observed in empirical sampling (except in degenerate cases with very few clusters, $K=2$).
• Downstream Consistency: Using an "LLM-as-judge" protocol, the geometric mean was not empirically dominated by any alternative combinator (AM, HM, Quad, Min, Max) or single-channel baseline. Different combinators produced structurally different storylines but statistically indistinguishable aggregate quality scores.
• Bottleneck-Gap Profile: Sweeping the compensability exponent $\alpha$, the symmetric component of the separation between coherent and incoherent storylines was maximized exactly at the geometric mean ($\alpha = 0$) across all corpora, including a human-navigation corpus (Wikispeedia) and a cross-modal image narrative.

Key Contributions

1. Geometric Reading: The paper provides the first information-geometric interpretation of the composite coherence metric, mapping it to a product manifold where negative log-coherence decomposes additively.
2. Local Fisher-Rao Consistency: It demonstrates that the Jensen-Shannon distance used in the topic component induces a metric tensor proportional to the Fisher information matrix, aligning the metric with the unique geometry characterized by Chentsov's theorem.
3. Axiomatic Uniqueness: It proves that the geometric mean is the unique combinator satisfying a specific set of four axioms (boundary, symmetry, log-additivity, normalization), distinguishing it from other power means.
4. Empirical Justification: It validates that the geometric mean is not empirically dominated and that the design premise (incoherent transitions are more channel-imbalanced) holds, maximizing the symmetric separation of storylines at the geometric mean.

Significance and Claims
The paper claims to provide an information-geometric justification for the composite coherence metric $C = \sqrt{A \cdot T}$, which has been used operationally in systems like Narrative Maps and Narrative Trails. It does not claim that the geometric mean is universally optimal for all possible narrative extraction tasks or that it produces "better" stories in a human-perceptible sense compared to other combinators (as downstream quality was statistically indistinguishable).

Instead, the significance lies in:

• Theoretical Grounding: Moving the metric from an ad-hoc operational definition to one derived from the geometry of the document space and axiomatic principles.
• Modeling Decision: Framing the choice of combinator not as a hyperparameter to be tuned for quality, but as a modeling decision about which narrative structures to foreground (e.g., enforcing a veto vs. allowing compensation).
• Diagnostic Utility: Highlighting that the geometric mean's log-additive structure allows for the diagnostic attribution of path costs to specific channels (angular vs. topic), a feature lost in other combinators like the product metric squared ($d_\times^2$).

The authors conclude that the geometric mean is the "natural choice" under the stated operational requirements (veto, symmetry, log-additivity) and the design premise of scale complementarity, providing a rigorous foundation for its continued use in event-based narrative extraction.