Information Terra: A Narrative-Anchored Semantic-First Projection of Document Embeddings

We introduce Information Terra, a narrative-anchored semantic-first projection that places a document corpus on an Earth-like globe whose poles are two user-chosen endpoint documents and whose prime meridian is the great-circle geodesic between them on the embedding hypersphere -- so latitude encodes narrative progress and longitude thematic deviation. Land features are recovered from document density via kernel density estimation and labeled by theme. A narrative trail built from the underlying narrative coherence graph, and constrained to be monotone in geodesic progress, provides a readable storyline. The projection's axes are semantically grounded in the user's chosen narrative endpoints, and the globe metaphor affords rotation and antipodal reading. We demonstrate the method on a 540-article Cuban Protests corpus, showing a storyline from Obama's 2016 visit to the 2021 International Aid during the protests.

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

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 $\mathbb{S}^{d-1}\timesΔ^{K-1}$, 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 ($ρ=0.999$), 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.

On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants

We provide a unified theoretical analysis of Linear Discriminant Analysis with simultaneous multilabel scatter matrix formulations and Stiefel orthogonality constraints. Our contributions span both algebraic structure and statistical guarantees. On the algebraic side, we characterize the rank of the multilabel between-class scatter matrix, showing that the effective discriminant dimensionality can strictly exceed the classical single-label bound of $C-1$; we establish a multilabel partition of variance and prove that all four Fisher objectives are equivalent under the $W^\top S_t^{ML} W = I_r$ constraint while characterizing their divergence under the Stiefel constraint; and we prove a two-sided label-distance preservation bound relating projected distances to Hamming distances in label space. On the statistical side, we establish a finite-sample $O(k_{\max}\sqrt{d\log d/n}/gap_r)$ bound on the subspace estimation error under sub-Gaussian noise with a matching $Ω(σ^2 d/(n\,gap_r))$ minimax lower bound, establishing a near-minimax-optimal rate (matching up to logarithmic and $k_{\max}$ factors) for multilabel discriminant subspace estimation. We further provide high-probability distance concentration, robustness guarantees under label interactions, and a regularization analysis preserving the spectral structure when $d \gg n$. All results are verified numerically on synthetic data generated from the linear label-effect model, covering both the algebraic identities and the multilabel-specific quantities ($k_{\max}$, $κ(S_t^{ML})$, $\|Γ/n\|_2$, $Δ_r$) that govern the statistical bounds. The numerical experiments are designed as a sanity check for the theorems rather than as an empirical benchmark; evaluation on real multilabel datasets is left to future work targeting application-oriented venues.

Toward a Machine Bertin: Why Visualization Needs Design Principles for Machine Cognition

Visualization's design knowledge-effectiveness rankings, encoding guidelines, color models, preattentive processing rules -- derives from six decades of psychophysical studies of human vision. Yet vision-language models (VLMs) increasingly consume chart images in automated analysis pipelines, and a growing body of benchmark evidence indicates that this human-centered knowledge base does not straightforwardly transfer to machine audiences. Machines exhibit different encoding performance patterns, process images through patch-based tokenization rather than holistic perception, and fail on design patterns that pose no difficulty for humans-while occasionally succeeding where humans struggle. Current approaches address this gap primarily by bypassing vision entirely, converting charts to data tables or structured text. We argue that this response forecloses a more fundamental question: what visual representations would actually serve machine cognition well? This paper makes the case that the visualization field needs to investigate machine-oriented visual design as a distinct research problem. We synthesize evidence from VLM benchmarks, visual reasoning research, and visualization literacy studies to show that the human-machine perceptual divergence is qualitative, not merely quantitative, and critically examine the prevailing bypassing approach. We propose a conceptual distinction between human-oriented and machine-oriented visualization-not as an engineering architecture but as a recognition that different audiences may require fundamentally different design foundations-and outline a research agenda for developing the empirical foundations the field currently lacks: the beginnings of a "machine Bertin" to complement the human-centered knowledge the field already possesses.