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.
Vector Symbolic Architectures (VSAs) provide a well-defined algebraic framework for compositional representations in hyperdimensional spaces. We introduce HyperSpace, an open-source framework that decomposes VSA systems into modular operators for encoding, binding, bundling, similarity, cleanup, and regression. Using HyperSpace, we analyze and benchmark two representative VSA backends: Holographic Reduced Representations (HRR) and Fourier Holographic Reduced Representations (FHRR). Although FHRR provides lower theoretical complexity for individual operations, HyperSpaces modularity reveals that similarity and cleanup dominate runtime in spatial domains. As a result, HRR and FHRR exhibit comparable end-to-end performance. Differences in memory footprint introduce additional deployment trade-offs where HRR requires approximately half the memory of FHRR vectors. By enabling modular, system-level evaluation, HyperSpace reveals practical trade-offs in VSA pipelines that are not apparent from theoretical or operator-level comparisons alone.
Neurosymbolic AI combines neural networks with symbolic programs to create robust and explainable predictions. One such framework is NeurASP, which trains a neural network to predict concepts and reasons over them using rules written in answer set programming (ASP) to solve downstream tasks. Crucially, labels are only provided for the downstream prediction produced by the symbolic rules, not for the latent concepts themselves. Backpropagation through the non-differentiable ASP component requires expensive probability and gradient calculations, which has hindered scalability to more sophisticated tasks. In this paper, we address the current limitations of NeurASP by improving its computational performance through vectorization, batch processing and caching of intermediate computations during training. We compare computation speeds between the original and our new implementation of NeurASP and report speedups of multiple orders of magnitude for larger tasks. To this end, we propose a new dataset of difficult tasks involving playing cards, which we use to test the capabilities of NeurASP's enhanced learning function.
Neurosymbolic (NeSy) models integrate neural networks and symbolic reasoning for robust and interpretable AI. State-of-the-art NeSy models require that the symbolic component is expressed in a differentiable way, often complicating the use of approximate inference. We propose EM-NeSy which casts probabilistic NeSy learning as an instance of the Expectation-Maximization (EM) algorithm. In the expectation step, we compute the posterior over the neurally predicted symbols conditioned on the label via probabilistic inference. In the maximization step, we update the neural parameters based on this posterior using gradient descent only through the neural component. This formulation unlocks the full potential of the EM algorithm for NeSy learning. It allows NeSy to extend naturally to approximate reasoning without any additional modifications or differentiability requirements of the symbolic component. Furthermore, it recovers the standard end-to-end gradient-based NeSy setting under exact inference. Our experimental results demonstrate the scalability and computational efficiency of EM-NeSy.