# Gram-Space: Structure-Preserving Codebook Compression for Memory-Efficient Neuro-Symbolic AI

arXiv: https://arxiv.org/abs/2608.01528 · HTML: https://mlsanity.com/paper/2608.01528/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 17 references that are in the map.
- First posted on arXiv 2 Aug 2026.

## Builds on (most cited first, 10 of 10)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [RAVEN: A Dataset for Relational and Analogical Visual rEasoNing](https://mlsanity.com/paper/1903.02741/) (2019)
- [A Neuro-vector-symbolic Architecture for Solving Raven's Progressive Matrices](https://mlsanity.com/paper/2203.04571/) (2022)
- [Classification using Hyperdimensional Computing: A Review](https://mlsanity.com/paper/2004.11204/) (2020)
- [Learning with Holographic Reduced Representations](https://mlsanity.com/paper/2109.02157/) (2021)
- [Probabilistic Abduction for Visual Abstract Reasoning via Learning Rules in Vector-symbolic Architectures](https://mlsanity.com/paper/2401.16024/) (2024)
- [Towards Learning Abductive Reasoning using VSA Distributed Representations](https://mlsanity.com/paper/2406.19121/) (2024)
- [Generalized Holographic Reduced Representations](https://mlsanity.com/paper/2405.09689/) (2024)
- [A-I-RAVEN and I-RAVEN-Mesh: Two New Benchmarks for Abstract Visual Reasoning](https://mlsanity.com/paper/2406.11061/) (2024)
- [DPQ-HD: Post-Training Compression for Ultra-Low Power Hyperdimensional Computing](https://mlsanity.com/paper/2505.05413/) (2025)

## Abstract (arXiv)

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.