Knowledge Manifold: A Riemannian Geometric Framework for Semantic Mapping and Geodesic Analysis of Scientific Literature
Authors: Tomonaga Okabe, Kazuhiko Komatsu
Organizations: Department of Aerospace Engineering, Tohoku University, 6-6-01, Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8579, Japan · Research Center for Green X-Tech, Tohoku University, 6-6-11, Aoba, Aramaki, Aoba-ku, Sendai, Miyagi 980-8579, Japan
Abstract
We present the knowledge manifold: a Riemannian geometric space in which a corpus of documents is arranged according to semantic positional relationships derived from character n-gram TF-IDF representations. The framework proceeds in five tightly coupled stages. First, each document is converted to a character-level n-gram TF-IDF vector (4-7 grams, up to 250,000 features, L2-normalized) and embedded in a two-dimensional knowledge map via constrained stress minimization with repulsion, variance, and centering regularizers. Second, knowledge at an arbitrary query point is estimated through Smoothed Particle Hydrodynamics (SPH) interpolation using a cubic-spline kernel, yielding an interpolated TF-IDF feature vector that can be linguistically characterized. Third, directional knowledge gradients at 0, 45, and 90 degrees are computed from the SPH interpolation map, and pairwise directional similarity is quantified via inner product and cosine similarity. Fourth, a Gaussian Process Regression (GPR) model, with a Constant x RBF + White kernel fitted on a 10-dimensional SVD projection, provides a Bayesian posterior mean, uncertainty estimate, and per-document contribution rate at the query point. Fifth, geodesics in the knowledge space are obtained by minimizing a discrete Riemannian path energy derived from the SPH-induced metric tensor, using L-BFGS-B with seven deterministic initial-path candidates. We apply the formulation to a corpus of 20 papers in fiber-reinforced composite materials and aerospace structural mechanics, showing that the semantic map recovers meaningful research clusters, geodesic paths reveal natural conceptual bridges between distant topics, and SPH/GPR interpolation enables the generation of virtual knowledge: hypothetical paper abstracts describing unstudied but geometrically predicted research directions.
Large language models extracting knowledge graphs from text capture only explicitly stated facts, often leaving semantically related entities disconnected across documents. We present an additive, engine-neutral second pass that discovers these latent ties without altering extracted facts. Each document is chunked and embedded once; top-k nearest- neighbor queries across existing chunks yield candidate node pairs via entity membership maps. Candidate pairs are scored using Shepard inverse-distance weighting with a rescaled chord distance metric, avoiding the threshold-collapsing flaw of affine cosine scoring behind a k-NN gate. Un-gated per-pair accumulators form a commutative monoid, ensuring the pipeline is strictly order-independent and scales incrementally without recomputing prior documents. Implemented across FalkorDB, Kinetica, ArangoDB, and Neo4j, our method shows that 768- and 240-dimensional embeddings retain 92% and 72% edge fidelity against a 3072-D baseline while achieving a 25x faster top-k formulation.
A Knowledge Graph (KG) represents facts as structured triples and is widely used to organize relational knowledge across diverse domains. Just as textual information ranges from words and sentences to complete documents, KG information can be interpreted at multiple levels, from entities, relations, and triples to subgraphs and entire KGs. However, existing KG embedding methods mainly focus on entities, relations, and triples, leaving graph-level semantics largely unaddressed. Conventional graph-level methods, which typically compare graphs based on structural patterns, are also insufficient because structural similarity alone cannot guarantee semantic similarity between KGs. To evaluate how well different methods capture such graph-level semantic information, we study graph-to-graph semantic similarity, which determines whether a pair of KGs represents semantically corresponding underlying information. To obtain reliable ground-truth correspondences, we construct a semantic matching dataset by modifying text documents, extracting KGs from both original and modified documents, and transferring their known correspondences to KG pairs. We compare text-based, structure-based, and KG embedding-based approaches on each dataset. For the KG embedding-based approach, we introduce two scoring functions: \textit{EmbPairSim}, which uses maximal pairwise entity similarity, and \textit{AvgEmbSim}, which uses a frequency-weighted centroid. Experiments on WikiText-2 and CC-News show that \textit{EmbPairSim} achieves up to 5.3 pp higher MRR than Sentence-BERT while using substantially fewer parameters. These results suggest that KGE representations can serve as compact and effective signals for graph-to-graph semantic similarity in KGs. Our code is available at https://github.com/SeungRyeolBaek/KG-to-KG-Semantic-Similarity.
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.
Brian Keith-Norambuena, Fausto German, Chris North