Recent progress in Sign Language Translation (SLT) has focussed primarily on improving the representational capacity of large language models to incorporate Sign Language features. This work explores an alternative direction: enhancing the geometric properties of skeletal representations themselves. We propose GeoSign, a method that leverages the properties of hyperbolic geometry to model the hierarchical structure inherent in sign language kinematics. By projecting skeletal features derived from Spatio-Temporal Graph Convolutional Networks (ST-GCNs) into the Poincaré ball model, we aim to create more discriminative embeddings, particularly for fine-grained motions like finger articulations. We introduce a hyperbolic projection layer, a weighted Fréchet mean aggregation scheme, and a geometric contrastive loss operating directly in hyperbolic space. These components are integrated into an end-to-end translation framework as a regularisation function, to enhance the representations within the language model. This work demonstrates the potential of hyperbolic geometry to improve skeletal representations for Sign Language Translation, improving on SOTARGB methods while preserving privacy and improving computational efficiency.

Large language models (LLMs) have demonstrated remarkable performance across various tasks. However, it remains an open question whether the default Euclidean space is the most suitable choice for LLMs. In this study, we investigate the geometric characteristics of LLMs, focusing specifically on tokens and their embeddings. Our findings reveal that token frequency follows a power-law distribution, where high-frequency tokens (e.g., "the," "that") constitute the minority, while low-frequency tokens (e.g., "apple," "dog") constitute the majority. Furthermore, high-frequency tokens cluster near the origin, whereas low-frequency tokens are positioned farther away in the embedding space.

Many real datasets are hierarchically structured.

In this paper, we study the gyrovector space structure (gyro-structure) of matrix manifolds. Our work is motivated by the success of hyperbolic neural networks (HNNs) that have demonstrated impressive performance in a variety of applications. At the heart of HNNs is the theory of gyrovector spaces that provides a powerful tool for studying hyperbolic geometry. Here we focus on two matrix manifolds, i.e., Symmetric Positive Definite (SPD) and Grassmann manifolds, and consider connecting the Riemannian geometry of these manifolds with the basic operations, i.e., the binary operation and scalar multiplication on gyrovector spaces. Our work reveals some interesting facts about SPD and Grassmann manifolds. First, SPD matrices with an Affine-Invariant (AI) or a Log-Euclidean (LE) geometry have rich structure with strong connection to hyperbolic geometry. Second, linear subspaces, when equipped with our proposed basic operations, form what we call gyrocommutative and gyrononreductive gyrogroups. Furthermore, they share remarkable analogies with gyrovector spaces. We demonstrate the applicability of our approach for human activity understanding and question answering.

Large language models (LLMs) have achieved remarkable success and demonstrated superior performance across various tasks, including natural language processing (NLP), weather forecasting, biological protein folding, text generation, and solving mathematical problems. However, many real-world data exhibit highly non-Euclidean latent hierarchical anatomy, such as protein networks, transportation networks, financial networks, brain networks, and linguistic structures or syntactic trees in natural languages. Effectively learning intrinsic semantic entailment and hierarchical relationships from these raw, unstructured input data using LLMs remains an underexplored area. Due to its effectiveness in modeling tree-like hierarchical structures, hyperbolic geometry -- a non-Euclidean space -- has rapidly gained popularity as an expressive latent representation space for complex data modeling across domains such as graphs, images, languages, and multi-modal data. Here, we provide a comprehensive and contextual exposition of recent advancements in LLMs that leverage hyperbolic geometry as a representation space to enhance semantic representation learning and multi-scale reasoning. Specifically, the paper presents a taxonomy of the principal techniques of Hyperbolic LLMs (HypLLMs) in terms of four main categories: (1) hyperbolic LLMs through exp/log maps; (2) hyperbolic fine-tuned models; (3) fully hyperbolic LLMs, and (4) hyperbolic state-space models. We also explore crucial potential applications and outline future research directions. A repository of key papers, models, datasets, and code implementations is available at https://github.com/sarangp2402/Hyperbolic-LLM-Models.

Selective state-space models excel at long-sequence modeling, but their capacity for language representation -- in complex hierarchical reasoning -- remains underexplored. Most large language models rely on \textit{flat} Euclidean embeddings, limiting their ability to capture latent hierarchies. To address this, we propose {\it Hierarchical Mamba (HiM)}, integrating efficient Mamba2 with hyperbolic geometry to learn hierarchy-aware language embeddings for deeper linguistic understanding. Mamba2-processed sequences are projected to the Poincaré ball or Lorentzian manifold with ``learnable'' curvature, optimized with a hyperbolic loss. Our HiM model facilitates the capture of relational distances across varying hierarchical levels, enabling effective long-range reasoning for tasks like mixed-hop prediction and multi-hop inference in hierarchical classification. Experimental results show both HiM variants effectively capture hierarchical relationships across four linguistic and medical datasets, surpassing Euclidean baselines, with HiM-Poincaré providing fine-grained distinctions with higher h-norms, while HiM-Lorentz offers more stable, compact, and hierarchy-preserving embeddings-favoring robustness. The source code is publicly available at https://github.com/BerryByte/HiM.