HyperSpace: A Generalized Framework for Spatial Encoding in Hyperdimensional Representations

Vector symbolic architectures (VSA) are widely used for reasoning in neuro-symbolic (NeSy) AI, yet high-dimensional codebooks often create severe memory bottlenecks that limit scalability and deployment. In this paper, we propose Gram-Space, a compression framework that applies Gram-Schmidt orthogonalization to represent codebook vectors in a compact orthonormal coordinate system. Gram-Space preserves the dot-product structure required by matrix-based VSA operators, which supports numerically equivalent execution of matrix similarity, probability vectorization, and attention score computations. We provide a correctness analysis showing that inner products are preserved under the orthonormal basis representation. Using modern GPU hardware, we benchmark the Gram-Space framework on standard neuro-symbolic reasoning datasets. Experimental evaluations across state-of-the-art VSA models show that Gram-Space reduces model-level GPU memory usage by up to 15.75x and improves inference latency by up to 3.62x. Profiling results further indicate that Gram-Space reduces allocation-heavy overhead in codebook-associated stages and improves hardware utilization for NeSy workloads.

Vision-Language Models (VLMs) struggle with compositional reasoning that requires understanding inter-object relationships. A natural remedy is to inject explicit scene graph triplets $\langle s, p, o \rangle$ from an off-the-shelf scene graph generator (SGG), but we show this backfires: discrete text labels collide with the continuous visual modality, degrading GQA accuracy from 60.38% to 58.86%. We propose \textbf{HyperVis}, which bypasses the SGG semantic bottleneck entirely. From $N$ class-agnostic region proposals, we compute a dense $O(N^2)$ visual relation tensor via spatially-biased cross-attention, project it onto a Lorentz hyperboloid, and enforce hierarchy through spatial physics, namely IoA-driven entailment cones and exterior-angle repulsion. We discover that HyperVis contributes in two complementary ways: (1) as a \emph{training-time regularizer}, the hyperbolic relational losses shape LoRA representations that improve generative VQA (GQA 61.03% vs.\ 57.21% for LoRA fine-tuning without relational losses, recovering and surpassing the baseline); and (2) as an \emph{inference-time relational encoder}, hyperbolic prefix tokens boost discriminative compositional scoring (SugarCrepe 79.94%, $+$6.25pp over baseline). The learned curvature stabilises at $κ{=}4.0$, an order of magnitude above prior hyperbolic VLMs where $κ$ typically collapses toward zero, indicating that continuous visual features genuinely require the exponential volume of strongly curved space. A controlled Euclidean ablation confirms this decomposition: the relational pipeline regularises LoRA comparably in flat space (GQA 60.81%), but the compositionality gain is specifically hyperbolic (SugarCrepe $+$4.58pp over Euclidean), with entailment loss ${\sim}6{\times}$ higher in Euclidean training. Codes are available at TBA.

In this work, we aim to discretize the high-dimensional visual representations to bridge the gap with language models - a non-trivial challenge, as existing quantization methods suffer from codebook collapse, failing to scale while preserving semantic coherence. We identify the root cause as metric mismatch: standard Euclidean codebook objectives are fundamentally misaligned with the anisotropic geometry of representation space, leading to codebook embeddings with high-variance magnitude scales and uneven angular distributions that hinder scalability. To address this, we propose Hyper-Spherical Quantization (HSQ), which decouples semantic content from feature magnitude via angular routing, preventing code assignment from being dominated by scale rather than meaning. The resulting discrete Representation Autoencoder (dRAE) achieves high-fidelity reconstruction while preserving semantic integrity and supporting scalable codebook budget. Extensive experiments demonstrate consistent performance gains as the vocabulary size scales to 131{,}072, along with 100% codebook utilization, simplified training pipeline, and strong performance across understanding and generation tasks.