Curved Algebraic Spaces Give AI a Sharper Grasp of Multimodal Knowledge

Knowledge graphs have quietly become one of the backbones of modern artificial intelligence. These vast networks of entities and relationships, drawn from sources such as Wikipedia, WordNet and curated databases like YAGO, power everything from question-answering systems to recommendation engines and drug discovery pipelines. But real-world knowledge rarely comes in a single, tidy format. Entities are described not only by symbolic triples linking them to one another, but also by images, free text, video and audio. Capturing that rich, multimodal character inside a single mathematical representation has remained one of the stubborn challenges of representation learning, and a new framework from researchers in Ho Chi Minh City offers an unusually elegant answer.

The framework, called CurvBi, short for Curvature-aware Bicomplex, is presented by Thanh Le, Vy Cao, Duong-Khang Au, Trong-Nghia Pham and Bac Le of the University of Science and Vietnam National University in Ho Chi Minh City. Writing in the journal Data Mining and Knowledge Discovery, the authors tackle the task known as multimodal knowledge graph completion: predicting missing links and entities in a graph by exploiting both its symbolic structure and the complementary information carried by textual and visual descriptions of its nodes. Their central insight is that the geometry used to fuse different modalities matters as much as the algebra used to model relationships between them.

To understand why, it helps to look at what existing approaches do. Most multimodal knowledge graph embedding methods treat the problem in two loosely connected stages. First, entity descriptions in each modality are projected into a shared Euclidean vector space, typically through simple concatenation, linear projection or a shallow fusion layer. Then, a relational model such as TransE, RotatE or a quaternion-based embedding scores candidate triples by comparing the vectors. This pipeline is convenient, but it flattens away the very structure that distinguishes modalities from one another. Text embeddings and image embeddings live on manifolds with different intrinsic curvature and different statistical character, and forcing them into a single flat space can blur or destroy the discriminative information they carry.

CurvBi departs from this convention on two fronts simultaneously. On the relational side, it adopts bicomplex embeddings, a representation built in a four-dimensional algebraic space formed by pairing two complex planes into a single number system. Bicomplex numbers are an extension of the complex numbers in which both the real and imaginary components are themselves complex, and their algebra is expressive enough to encode symmetric, asymmetric, inverse and cyclic relationships within one unified scoring function. This matters because real knowledge graphs contain all of these relational patterns at once: a relation like ‘is married to’ is symmetric, ‘is the parent of’ is asymmetric and invertible, and ‘is located in’ often participates in cyclic chains. Prior systems have typically needed separate mechanisms, or careful inductive-bias engineering, to cover this full catalogue of relation types.

The bicomplex formulation builds on a line of work the group has developed over several years, including quaternionic and bicomplex embeddings for temporal knowledge graphs and event prediction. In CurvBi, the four-dimensional algebraic structure plays a second, subtler role: it provides a natural home for the modality embeddings before they are fused. Rather than concatenating a text vector and an image vector end to end, CurvBi treats each modality-specific representation as a structured component within the bicomplex number, preserving the algebraic relationships between components while still allowing the model to learn how much weight each modality deserves for a given entity or relation.

The second front is geometric. Instead of projecting all modality embeddings into flat Euclidean space, CurvBi performs fusion on a Riemannian manifold, measuring the distance between modality-specific representations with the Fisher-Rao distance, a well-studied metric from information geometry that treats probability distributions as points on a curved manifold. Because the Fisher-Rao metric is curvature-aware, similarities between a text embedding and an image embedding are judged in a geometry that reflects their statistical structure rather than their raw coordinate differences. The result, according to the authors, is a fusion step that is geometry-sensitive in a way that Euclidean projection and shallow concatenation simply cannot be, aligning heterogeneous modalities while respecting the distinct manifold each of them inhabits.

This curvature-aware component connects CurvBi to a broader trend in machine learning. Hyperbolic and other non-Euclidean embedding spaces have been shown to represent hierarchies, cycles and scale-free structures far more efficiently than Euclidean space, and methods such as Poincaré embeddings and multicurvature adaptive schemes for temporal knowledge graphs have demonstrated measurable gains. What distinguishes CurvBi is the combination: the same framework unifies the algebraic machinery for relational reasoning and the differential-geometric machinery for modality alignment, so that neither stage of the pipeline is an afterthought bolted onto the other. The manifold structure informs how modalities are brought together, while the bicomplex algebra governs how the fused representation is used to score and complete triples.

The empirical case for the framework rests on four benchmark datasets for multimodal knowledge graph completion. Across all of them, CurvBi achieved consistently competitive results on standard evaluation metrics, including Hits at 1, Hits at 10 and mean reciprocal rank, which measure how often the correct missing entity appears at the top of the model’s ranked predictions. On certain benchmark settings the improvements were substantial: gains of up to 6 percent in Hits at 1 and 4.5 percent in mean reciprocal rank over state-of-the-art baselines. The authors also report that the framework scales well and generalizes across datasets with different sizes, relational vocabularies and modality configurations, an important practical consideration given that real multimodal knowledge graphs vary enormously in how much image and text data they attach to each entity.

The significance of the work extends beyond a leaderboard. Multimodal knowledge graphs are increasingly used in settings where text alone is ambiguous: in biomedical informatics, where molecular structures and microscopy images supplement gene and disease ontologies; in e-commerce, where product images carry information absent from catalog text; and in commonsense reasoning systems, where visual grounding helps language models resolve references. If the fusion geometry is wrong, the extra modalities can actively hurt, a phenomenon earlier studies have documented when visual context proved unhelpful for certain graph completion tasks. CurvBi’s results suggest that a principled geometric treatment of fusion, paired with an algebraically expressive relational model, is one way to make multimodal information reliably additive rather than subtractive.

The research, funded by the Vietnam National Foundation for Science and Technology Development under grant number 102.05-2025.75, also signals a maturing of geometric deep learning as applied to structured knowledge. Rather than treating curvature, complex numbers and manifold optimization as exotic mathematical decorations, the authors deploy them as load-bearing components of a system that demonstrably outperforms flatter alternatives. As knowledge graphs continue to absorb richer and more heterogeneous data streams, the lesson from CurvBi is likely to resonate: the shape of the space in which artificial intelligence reasons is not a technicality, but a first-class design decision that determines what that intelligence can and cannot see.

Subject of Research: Curvature-aware multimodal knowledge graph completion using bicomplex embeddings and Riemannian manifold fusion

Article Title: Learning curvature-aware multimodal representations with bicomplex embeddings

Article References: Le, T., Cao, V., Au, D.-K., Pham, T.-N., & Le, B. (2026). Learning curvature-aware multimodal representations with bicomplex embeddings. Data Mining and Knowledge Discovery, 40(5), Article 89. https://doi.org/10.1007/s10618-026-01257-0

Image Credits: AI Generated

DOI: 10.1007/s10618-026-01257-0

Keywords: knowledge graph completion, multimodal embeddings, bicomplex numbers, Riemannian geometry, Fisher-Rao distance, modality fusion, link prediction, representation learning, curvature-aware learning, quaternion algebra, data mining, artificial intelligence