Vector Symbolic Architectures

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Sep 23, 2026cs.LG

Learning Holographic Reduced Representations with Clifford Variational Autoencoders

Vector Symbolic Algebras project data structures into a hyperdimensional vector space through the application of their vector algebras to randomly generated atomic vector symbols and fractional power encodings of real-valued data. Embedding unstructured data remains an open question. We present \textit{Clifford-VAE}, a variational autoencoder that learns to project data onto a Clifford torus in arbitrary dimensions. Experiments using the MNIST, FashionMNIST, and CIFAR-10 datasets demonstrate that Clifford-VAE produces representations that are competitive with those produced by Gaussian and Hyperspherical VAEs for semi-supervised classification tasks while outperforming Gaussian and Hyperspherical counterparts in the VSA benchmark tests of self-binding and unbinding, role-filler recovery, and bundle capacity. Clifford-VAE provides a principled technique for grounding perceptual data into a symbolic reasoning framework, providing a new approach to a long-standing problem in the VSA literature.

Mohamed Malek Abid, P. Michael Furlong

Institute of Neuroinformatics, University of Zurich & ETH Zurich · National Research Council Canada & Centre for Theoretical Neuroscience, Systems Design Engineering, University of Waterloo, Waterloo, Canada

Variational AutoencodersVisual Representation LearningHyperdimensional ComputingVector Symbolic Architectures
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Aug 3, 2026cs.CL

Learning a Vector-Symbolic Model for Socio-Cultural Tasks

How can we better represent the impact of sociocultural structures on decision making in computational cognitive models? Modeling this impact requires traversing multiple levels of semantic representation, however it is not immediately clear to a modeler which levels of representation are most salient to a given situation. Though large language models and cognitively grounded corpus models can represent broad semantic associations through co-occurences, the role of self representations in memory should be accounted for to determine how cultural associations shape decision making. We propose a declarative memory system to be used in the ACT-R cognitive architecture that represents semantic associations at multiple levels via a vector-symbolic autoencoder. We use a simple HRR operation to encode episodic memories differently from semantic memory vectors extracted from text to produce a final chunk activation for a memory request. We use ACT-R cognitive models of a racially contextualized implicit association test (IAT) to test this new declarative memory system.

Meera Ray, Swapnika Dulam, Christopher L. Dancy

Department of Computer Science and Engineering, The Pennsylvania State University W209 Westgate Building, University Park, PA 16802. USA · Department of Industrial and Manufacturing Engineering, The Pennsylvania State University M310 Leonhard Building, University Park, PA 16802. USA

Computational Cognitive ModelingVector Symbolic Architectures
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Aug 2, 2026cs.LG

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

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.

Weilun Wang, Wantong Li

University of California, Riverside

Memory-Efficient InferenceVector Symbolic ArchitecturesNeuro-Symbolic ReasoningNeural Network Compression
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Jul 18, 2026cs.NE

How to Build Marcus's Algebraic Mind: From Minsky's Emotion-Machine Viewpoint

This paper reports a step that had already been taken. Marcus's three components of an adequate cognitive architecture -- variables, recursive structure, individuals vs kinds -- left the neural substrate open; a companion paper answers it with VaCoAl, built on XOR-and-shift over GF(2). Those pillars are horizontal: how a mind represents the world. Minsky's Emotion Machine supplies the orthogonal, vertical dimension -- a mind reasoning about its own reasoning. The vertical step needed no new machinery. A representation exactly recoverable becomes introspection when what is recovered is the reasoner's own deliberative trace: Minsky's Reflective layer. Pillar 2 says nothing about whose structure is recovered; point the same unbind inward and Reflection follows. Only the argument changes. Capability: exact reversible binding at O(N), kept exact within a Frontier Size, yields a trace readable constituent by constituent: a self-report faithful by construction, not verbalization. Only exact-algebraic traces, not probabilistic ones, make the reflexive read faithful. Necessity: audit and valuation are one circuit at opposite settings of one parameter. Repair every collision and the trace is exactly auditable but no path outranks another; tolerate them and the CR2 decay that credits the direct route makes the read approximate. Position: we claim neither to have built the Reflective layer nor to surpass large language models; it supplies the auditable trace probabilistic deliberation cannot leave. The same substrate gives panalogy (content-addressable retrieval scored by CR) and credit assignment as counterfactual unbind-rebind, meeting Pearl's causal axis. Not claimed: Reflection's sufficient conditions (meta-control loop, context-tagging) are future work; Reflection is introspection, not evaluation; its reachability is argued, not shown; speed and power unbenchmarked.

Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

Institute of Innovation Research, Hitotsubashi University (Professor Emeritus) · Meisei University (Professor Emeritus) · Shuhari System

Computational Cognitive ModelingVector Symbolic ArchitecturesReasoning Trace Analysis
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Jul 18, 2026cs.NE

How to Build Marcus's Algebraic Mind: From Thagard's Brain--Mind Viewpoint

This paper reports a convergence neither program was looking for. Marcus's The Algebraic Mind named three things any architecture needs -- operations over variables, recursive structure, and individuals distinct from kinds -- showed multilayer perceptrons have none, and left a register-and-treelet implementation as conjecture. Thagard's Brain-Mind ran it from the other end, making binding the mechanism from which the whole of mind is assembled, and circular convolution load-bearing. Marcus leaves his register algebra open; Thagard's is lossy, degrading under the very recursion his own account demands. VaCoAl is a hyperdimensional computing architecture built end-to-end on XOR-and-shift over GF(2) via primitive-polynomial LFSRs; PyVaCoAl is its extended software realization (all results here); the silicon substrate exists as CASRAM. Bind(R,F) = R XOR Shift(F) is exactly reversible and non-commutative: it supplies all three pillars at fixed dimension and removes convolution's depth degradation. Capability: exact reversible binding at O(N) yields compositional generalization with post-hoc auditability, which no lossy or learned substrate offers. Necessity: two independent architecture programs and the dentate gyrus-CA3 circuit require the same algebra -- convergence, not biomimicry. New here: discrimination and failure-tolerance are one. Repair every collision (RR = 1) and the system is bit-identical to a hash dictionary: candidates become indistinguishable and the confidence path-integral collapses; a memory that never fails keeps no record of which routes were hard. Position: we do not surpass large language models but supply the auditable, multi-hop relational reasoning embeddings lack. No consciousness is implemented and no cognitive experiment reported; bit-exactness holds in silicon, approximately under biological noise; speed and power remain unbenchmarked.

Hiroyuki Chuma, Kanji Otsuka, Yoichi Sato

Institute of Innovation Research, Hitotsubashi University (Professor Emeritus) · Meisei University (Professor Emeritus) · Shuhari System

Computational Cognitive ModelingHyperdimensional ComputingVector Symbolic ArchitecturesAlgorithmic Reasoning
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Jun 30, 2026cs.AI

Creating Intelligence: A Computational Foundation for AGI

This work introduces a new computational theory of mind grounded in set theory and hyperdimensional computing. Whereas traditional neural networks rely on continuous weights and matrix multiplication, this framework works with sparse binary data. It represents information as discrete sets, directly modeling biological neural population codes. I demonstrate that associative memory emerges naturally from network topologies featuring a combinatorially expanded hidden layer. Learning is driven by topological plasticity rather than scalar weight adjustments. This architecture unifies auto-associative and hetero-associative learning under a single core algorithm: information retrieval via subset pattern matching and exact nearest-neighbor search. Operating with constant-time complexity, these mechanisms bridge perceptual data (sparse distributed representations) and symbols (sparse holographic representations) without continuous bottlenecks. Mapping this framework to neuroanatomy, I propose that both the cerebellum and the neocortex implement variants of this algorithm, making subset pattern matching the fundamental engine of cognition. Because it relies on discrete logic rather than matrix arithmetic, this algorithm translates directly into in-memory hardware. This opens a new route toward synthetic intelligence with human-level energy efficiency.

Peter Overmann

Computational Cognitive ModelingHyperdimensional ComputingVector Symbolic ArchitecturesAssociative Memory
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Jun 9, 2026cs.LG

Recursive Binding on a Budget: Subspace Carving in Order-p Tensor Memories

Tensor Product Representations provide the structural fidelity required for symbolic reasoning in models but suffer from exponential dimensionality growth when encoding deep recursive structures. Conversely, Vector Symbolic Architectures maintain constant dimensionality but sacrifice capacity and fidelity due to noisy compression via superposition. In this work, we propose Orthogonal Subspace Carving (OSC), a memory architecture that binds fillers to roles by projecting onto the null space of the role basis before aggregating into a fixed order-p tensor. OSC uses projections to enforce geometric orthogonality between bound structures within a static memory trace. We show that this mechanism decouples the tensor order from the structural depth, enabling deep recursive binding within a constant memory footprint. By performing retrieval via recognition, this construction allows for component vectors that are orders of magnitude smaller than the memory tensor, giving superior memory efficiency in settings involving high superposition. We also show that TPR is a special case of binding in Clifford algebra, and give a Clifford formulation of OSC.

Travis Pence, Daisuke Yamada, Vikas Singh

Department of Computer Science, University of Wisconsin-Madison, United States · Department of Biostatistics and Med. Info., University of Wisconsin-Madison, United States

Memory-Efficient InferenceVector Symbolic ArchitecturesAssociative Memory
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Jun 8, 2026cs.LG

Disentanglement with Holographic Reduced Representations

Disentanglement, the separation of factors of variation in data using neural networks, remains a long-standing challenge in machine learning. Prior work has addressed this problem with variational autoencoders and generative adversarial networks that incorporate ideas from variational inference and information-theoretic constraints. In contrast to methods that rely on continuous representations, we propose a design that treats disentangled representations as symbolic structures, motivated by the compositional relationships among the concepts that make up samples from a distribution. However, learning discrete symbolic structures with neural networks while maintaining differentiability is difficult and often requires complex architectures. To address this, we introduce an unsupervised learning algorithm that uses holographic reduced representations (HRR) for neural disentanglement. We show that the HRR unbinding operation provides an inductive bias for separating factors and yields competitive results against baselines, as measured by latent traversals and disentanglement metrics. We complement these empirical findings with an information-theoretic analysis of the HRR unbinding channel. We prove that unbinding induces approximately independent symbol-value pairs and derive a per-slot capacity bound that quantifies how many distinct symbolic concepts can be reliably encoded, giving a quantitative account of the inductive bias toward disentanglement. The resulting representations differ from standard autoencoder-based models, in that their latent units are vectors that are summed together, rather than scalar dimensions of a low-dimensional latent vector. We show that this HRR representation is more robust to noise than other disentangled representations and maintains reconstruction quality across a range of SNRs.

Jhonny J. Velasquez Olivera, Christo K. Thomas, Walid Saad

Virginia Tech · Worcester Polytechnic Institute

Disentangled Representation LearningUnsupervised Representation LearningVector Symbolic Architectures
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May 29, 2026cs.AI

Geodesic Flow Matching for Denoising High-Dimensional Structured Representations

Vector Symbolic Algebras (VSAs) enable robust neurosymbolic reasoning by encoding symbolic information into high-dimensional distributed representations. For continuous domains, Spatial Semantic Pointers (SSPs) extend this framework by mapping variables onto continuous toroidal manifolds. However, standard approaches like Flow Matching assume a flat Euclidean geometry, which fails to account for the geometric constraints imposed on valid SSP states. We demonstrate that this assumption fails for SSPs: Euclidean linear interpolants ``cut through" the manifold's interior, destroying the phase and magnitude structure required for accurate decoding. To resolve this, we employ Geodesic Flow Matching, adapting Riemannian transport dynamics to strictly restrict the denoising flow to the SSP toroidal manifold. We validate this approach in a Spiking Neural SLAM system, showing that manifold-aware cleanup stabilizes path integration against drift. The method achieves a 72% reduction in tracking error and enables a 40% increase in neural efficiency compared to competitive baselines. Code is available at https://github.com/kremHabashy/CleanupSSP .

Karim Habashy, Chris Eliasmith

Department of Systems Design Engineering, University of Waterloo, Waterloo, Canada · Center For Theoretical Neuroscience, University of Waterloo, Waterloo, Canada · Department of Philosophy, University of Waterloo, Waterloo, Canada

Flow MatchingRepresentation LearningRiemannian Flow MatchingVector Symbolic Architectures
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May 20, 2026cs.NE

How to Build Marcus's Algebraic Mind: Algebro-Deterministic Substrate over Galois Fields

In The Algebraic Mind, Gary Marcus identified three components essential for any adequate cognitive architecture: operations over variables, recursively structured representations, and a distinction between mental representations of individuals and kinds. He argued that standard multilayer perceptrons supported none of these, acknowledging that a neural implementation using registers and treelets, constructed via developmental programs rather than gradient descent, remained a programmatic conjecture. Twenty-five years later, the required substrate is now available. Our newly developed PyVaCoAl/VaCoAl is a hyperdimensional computing architecture organized end-to-end around a single algebraic primitive: XOR-and-shift over GF(2), implemented by primitive-polynomial linear-feedback shift registers. The architecture supports reversible variable binding via Bind(R,F) = R XOR shift(F), non-commutative compositional bundling that distinguishes "the dog bites the man" from "the man bites the dog," and address-space individual/kind separation under the same algebra. A companion perspective argues that the dentate gyrus-CA3 circuit is a biological homologue of this same engine, with developmentally specified mossy-fiber targeting supplying the innate microcircuitry Marcus anticipated. In this paper, we map the correspondence between Marcus's three pillars and the operational commitments of PyVaCoAl/VaCoAl. We reinterpret the treelet as an algebraic register set indexed by a primitive generator polynomial, arguing that this architecture provides a functional neural substrate meeting Marcus's specifications far more closely than the tensor products, circular convolution, or temporal synchrony available in 2001. We also demonstrate how this substrate naturally extends to Pearl's rung-3 counterfactual reasoning, a capability the original treelet program did not directly target.

Hiroyuki Chuma, Kanji Otsuk, Yoichi Sato

Institute of Innovation Research, Hitotsubashi University, Kunitachi, Tokyo 186-8603, Japan (Professor Emeritus) · Meisei University, Hino, Tokyo, Japan (Professor Emeritus) · Shuhari System, Tokyo, Japan

Computational Cognitive ModelingHyperdimensional ComputingVector Symbolic Architectures
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May 20, 2026cs.LG

Sutra: Tensor-Op RNNs as a Compilation Target for Vector Symbolic Architectures

Sutra is a typed, purely functional programming language whose compiled forward pass is a PyTorch neural network. The compiler beta-reduces the whole program -- primitives, control flow, string I/O -- to one fused tensor-op graph over a frozen embedding substrate. Rotation binding, unbind, bundle, polynomial Kleene three-valued logic, and tail-recursive loops all lower to tensor operations; the Kleene connectives are Lagrange-interpolated polynomials exact on the {-1, 0, +1} truth grid. Validation is one fact tested two ways. (1) The same program runs on four frozen embeddings spanning two modalities -- three text encoders (nomic-embed-text, all-minilm, mxbai-embed-large) and one protein language model (ESM-2) -- and decodes bundles at 100% accuracy through width k=8 on every substrate, where the textbook Hadamard product has already collapsed (2.5% on mxbai-embed-large, 7.5% on all-minilm). (2) PyTorch autograd flows through the actually compiled graph: a fuzzy-rule classifier written in .su trains from random init (18.7 +/- 9.5%; chance = 20%, five classes) to 100.0 +/- 0.0% (three seeds) by backpropagating through the emitted graph, the symbolic source unmodified. A weighted variant additionally trains a scalar cosine gain and writes it back into the .su source as a numeric literal; recompiling reproduces the trained behaviour to ~2e-7 per logit, so the trained model is itself legible, recompilable code. The same artifact is therefore both a logic program and a trainable neural network.

Emma Leonhart

Logical ReasoningDifferentiable ProgrammingVector Symbolic ArchitecturesNeuro-Symbolic Reasoning
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Apr 21, 2026cs.AI

Overcoming the Impedance Mismatch: A Theoretical Roadmap for Fusing Foundation Models and Knowledge Graphs

Modern artificial intelligence remains fundamentally divided between the continuous, probabilistic spaces of Foundation Models and the discrete, deterministic structures of Knowledge Graphs. While Retrieval-Augmented Generation (RAG) attempts to connect them by serializing graph data into text, we argue this lexical bridging is merely a superficial patch. In this paper, we formalize the underlying structural and geometric friction as the \textit{Impedance Mismatch}. By categorizing current neuro-symbolic integration strategies into a three-tiered hierarchy, we demonstrate that neither surface-level prompt injection nor continuous representation alignment can preserve the strict logical motifs required for reliable multi-hop reasoning. We define the specific mathematical limits, such as the Lexical Bottleneck and Topological Collapse, that show current architectures will eventually hallucinate or conflate semantic nodes. To achieve true semantic fusion, we propose a rigorous theoretical roadmap. We advocate for natively internalizing discrete symbolic structures through Structured Residual Streams, utilizing Vector Symbolic Architectures for latent sub-graph injection, and performing model updates via Orthogonal Subspace Editing. This actionable framework paves the way for models that seamlessly fuse the precision of symbolic logic with the expressivity of parametric memory.

Sahil Rajesh Dhayalkar

Arizona State University

Vector Symbolic ArchitecturesNeuro-Symbolic Reasoning
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Apr 16, 2026cs.AI

SRMU: Relevance-Gated Updates for Streaming Hyperdimensional Memories

Sequential associative memories (SAMs) are difficult to build and maintain in real-world streaming environments, where observations arrive incrementally over time, have imbalanced sampling, and non-stationary temporal dynamics. Vector Symbolic Architectures (VSAs) provide a biologically-inspired framework for building SAMs. Entities and attributes are encoded as quasi-orthogonal hyperdimensional vectors and processed with well defined algebraic operations. Despite this rich framework, most VSA systems rely on simple additive updates, where repeated observations reinforce existing information even when no new information is introduced. In non-stationary environments, this leads to the persistence of stale information after the underlying system changes. In this work, we introduce the Sequential Relevance Memory Unit (SRMU), a domain- and cleanup-agnostic update rule for VSA-based SAMs. The SRMU combines temporal decay with a relevance gating mechanism. Unlike prior approaches that solely rely on cleanup, the SRMU regulates memory formation by filtering redundant, conflicting, and stale information before storage. We evaluate the SRMU on streaming state-tracking tasks that isolate non-uniform sampling and non-stationary temporal dynamics. Our results show that the SRMU increases memory similarity by 12.6%12.6\% and reduces cumulative memory magnitude by 53.5%53.5\%. This shows that the SRMU produces more stable memory growth and stronger alignment with the ground-truth state.

Shay Snyder, Andrew Capodieci, David Gorsich +1

George Mason University · George Mason University, USA · Neya Robotics +3

Hyperdimensional ComputingVector Symbolic ArchitecturesState TrackingAssociative Memory
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Apr 16, 2026cs.AI

HyperSpace: A Generalized Framework for Spatial Encoding in Hyperdimensional Representations

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.

Shay Snyder, Andrew Capodieci, David Gorsich +1

George Mason University · George Mason University, Fairfax, Virginia, USA · Neya Robotics +3

Hyperdimensional ComputingVector Symbolic Architectures
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