Vector Symbolic Architecture

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Vector Symbolic Architecture (VSA) is a kind of vector algebra used in an embedding space [en] to represent symbol manipulation [en] as vector operations.^{[1][2][3][4][5][6][7][8][9][10][11]}

It is also called hyperdimensional computing.

Its Turing completeness is not yet proven but a 2024 paper supports it since it has been found to be Cartesian closed [en].^[5]

HDC algebra reveals the logic of how and why systems makes decisions, unlike artificial neural networks. Physical world objects can be mapped to hypervectors, to be processed by the algebra.^[12]

Vector symbolic architectures (VSA) provided a systematic approach to high-dimensional symbol representations to support operations such as establishing relationships. Early examples include holographic reduced representations, binary spatter codes, and matrix binding of additive terms. HD computing advanced these models, particularly emphasizing hardware efficiency.^[13]

In 2015, Eric Weiss showed how to fully represent an image as a hypervector. A vector could contain information about all the objects in the image, including properties such as color, position, and size.^[14]

In 2023, Abbas Rahimi et al., used HDC with neural networks to solve Raven's progressive matrices.^[14]

In 2023, Mike Heddes et Al. under the supervision of Professors Givargis, Nicolau and Veidenbaum created a hyper-dimensional computing library^[15] that is built on top of PyTorch [en].

• Lambda calculus
• Turing machine
• JEPA
• Word embedding
• Support vector machine [en]

• Kleyko, Denis; Rachkovskij, Dmitri A.; Osipov, Evgeny; Rahimi, Abbas (2023-07-31). "A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations". ACM Computing Surveys. 55 (6): 1–40. arXiv:2111.06077. doi:10.1145/3538531. ISSN 0360-0300.
• Kleyko, Denis; Rachkovskij, Dmitri; Osipov, Evgeny; Rahimi, Abbas (2023-09-30). "A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part II: Applications, Cognitive Models, and Challenges". ACM Computing Surveys. 55 (9): 1–52. arXiv:2112.15424. doi:10.1145/3558000. ISSN 0360-0300.

• Stock, M.; Van Criekinge, W.; Boeckaerts, D.; Taelman, S.; Van Haeverbeke, M.; Dewulf, P.; De Baets, B. (2024), Dutt, V. (ed.), "Hyperdimensional computing: a fast, robust, and interpretable paradigm for biological data", PLOS Computational Biology, 20 (9) e1012426, Public Library of Science (PLOS), arXiv:2402.17572, doi:10.1371/journal.pcbi.1012426, PMC 11421772, PMID 39316621
• Cumbo, F.; Chicco, D. (2025), "Hyperdimensional computing in biomedical sciences: a brief review", PeerJ Computer Science, 11 (e2885) e2885, doi:10.7717/peerj-cs.2885, PMC 12192801, PMID 40567746
• Kanerva, Pentti (2009-06-01). "Hyperdimensional Computing: An Introduction to Computing in Distributed Representation with High-Dimensional Random Vectors". Cognitive Computation. 1 (2): 139–159. doi:10.1007/s12559-009-9009-8. ISSN 1866-9964. S2CID 733980.
• Neubert, Peer; Schubert, Stefan; Protzel, Peter (2019-12-01). "An Introduction to Hyperdimensional Computing for Robotics". KI – Künstliche Intelligenz. 33 (4): 319–330. doi:10.1007/s13218-019-00623-z. ISSN 1610-1987. S2CID 202642163.
• Neubert, Peer; Schubert, Stefan (2021-01-19). "Hyperdimensional computing as a framework for systematic aggregation of image descriptors". arXiv:2101.07720v1 [cs.CV].
• Stock, Michiel (2022-10-04). "Tutorial on Hyperdimensional Computing". Retrieved 2023-07-29.

• "HD/VSA". www.hd-computing.com. 2023-03-13. Retrieved 2023-04-15.
• Ananthaswamy, Anil (2023-04-13). "A New Approach to Computation Reimagines Artificial Intelligence". Quanta Magazine. Retrieved 2023-06-13.