mlsanityData to 7 Oct 2026

Learning with Holographic Reduced Representations

arXiv 2109.02157 ↗First posted 5 Sep 2021cs.AI cs.LG cs.NEcited by 25

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Abstract (arXiv)

Holographic Reduced Representations (HRR) are a method for performing symbolic AI on top of real-valued vectors by associating each vector with an abstract concept, and providing mathematical operations to manipulate vectors as if they were classic symbolic objects. This method has seen little use outside of older symbolic AI work and cognitive science. Our goal is to revisit this approach to understand if it is viable for enabling a hybrid neural-symbolic approach to learning as a differentiable component of a deep learning architecture. HRRs today are not effective in a differentiable solution due to numerical instability, a problem we solve by introducing a projection step that forces the vectors to exist in a well behaved point in space. In doing so we improve the concept retrieval efficacy of HRRs by over 100×. Using multi-label classification we demonstrate how to leverage the symbolic HRR properties to develop an output layer and loss function that is able to learn effectively, and allows us to investigate some of the pros and cons of an HRR neuro-symbolic learning approach. Our code can be found at https://github.com/NeuromorphicComputationResearchProgram/Learning-with-Holographic-Reduced-Representations

Key facts. Cited by 25 papers in the map (data through 7 Oct 2026). Builds on 22 of its 60 references that are in the map. First posted on arXiv 5 Sep 2021.

Builds on

22 of 60 references.

References · 22

  1. 2021Linear Transformers Are Secretly Fast Weight Programmerscited by 471
  2. 2019The Neuro-Symbolic Concept Learner: Interpreting Scenes, Words, and Sentences From Natural Supervisioncited by 369
  3. 2020A Metric Learning Reality Checkcited by 317
  4. 2020On the Binding Problem in Artificial Neural Networkscited by 230
  5. 2020Logic Tensor Networkscited by 197
  6. 2021Accounting for Variance in Machine Learning Benchmarkscited by 195
  7. 2015Holographic Embeddings of Knowledge Graphscited by 181
  8. 2019Are We Really Making Much Progress? A Worrying Analysis of Recent Neural Recommendation Approachescited by 181
  9. 2018AttentionXML: Label Tree-based Attention-Aware Deep Model for High-Performance Extreme Multi-Label Text Classificationcited by 101
  10. 2016DiSMEC - Distributed Sparse Machines for Extreme Multi-label Classificationcited by 70
  11. 2020A comparison of Vector Symbolic Architecturescited by 63
  12. 2021LightXML: Transformer with Dynamic Negative Sampling for High-Performance Extreme Multi-label Text Classificationcited by 61
10 more
  1. 2013Large-scale Multi-label Learning with Missing Labelscited by 58
  2. 2018A no-regret generalization of hierarchical softmax to extreme multi-label classificationcited by 52
  3. 2016Associative Long Short-Term Memorycited by 37
  4. 2019Extreme Classification in Log Memory using Count-Min Sketch: A Case Study of Amazon Search with 50M Productscited by 36
  5. 2021DECAF: Deep Extreme Classification with Label Featurescited by 33
  6. 2018A theory of sequence indexing and working memory in recurrent neural networkscited by 29
  7. 2018Learning to Reason with Third-Order Tensor Productscited by 27
  8. 2019CircConv: A Structured Convolution with Low Complexitycited by 8
  9. 2019Accelerating Extreme Classification via Adaptive Feature Agglomerationcited by 7
  10. 2016SSH (Sketch, Shingle, & Hash) for Indexing Massive-Scale Time Seriescited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 6; 2023: 3; 2024: 4; 2025: 4; 2026: 6.

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  1. 2021A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformationscited by 98
  2. 2021A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part II: Applications, Cognitive Models, and Challengescited by 68
  3. 2022Torchhd: An Open Source Python Library to Support Research on Hyperdimensional Computing and Vector Symbolic Architecturescited by 17
  4. 2023Recasting Self-Attention with Holographic Reduced Representationscited by 14
  5. 2022A Brain-Inspired Low-Dimensional Computing Classifier for Inference on Tiny Devicescited by 11
  6. 2024What Do Machine Learning Researchers Mean by "Reproducible"?cited by 8
  7. 2022Deploying Convolutional Networks on Untrusted Platforms Using 2D Holographic Reduced Representationscited by 7
  8. 2023Factorizers for Distributed Sparse Block Codescited by 4
  9. 2023Towards Generalization in Subitizing with Neuro-Symbolic Loss using Holographic Reduced Representationscited by 4
  10. 2024Vector-Symbolic Architecture for Event-Based Optical Flowcited by 4
  11. 2024A Walsh Hadamard Derived Linear Vector Symbolic Architecturecited by 4
  12. 2022Lempel-Ziv Networkscited by 3
13 more
  1. 2022Recursive Binding for Similarity-Preserving Hypervector Representations of Sequencescited by 2
  2. 2025Linearithmic Clean-up for Vector-Symbolic Key-Value Memory with Kroneker Rotation Productscited by 2
  3. 2022Optimal quadratic binding for relational reasoning in vector symbolic neural architecturescited by 1
  4. 2024Multi-label Learning with Random Circular Vectorscited by 1
  5. 2025FoGE: Fock Space inspired encoding for graph promptingcited by 0
  6. 2025Efficient Vector Symbolic Architectures from Histogram Recoverycited by 0
  7. 2025A Vector Symbolic Approach to Multiple Instance Learningcited by 0
  8. 2026Reinforcement Learning Enhancement Using Vector Semantic Representation and Symbolic Reasoning for Human-Centered Autonomous Emergency Brakingcited by 0
  9. 2026Disentanglement with Holographic Reduced Representationscited by 0
  10. 2026Recursive Binding on a Budget: Subspace Carving in Order-p Tensor Memoriescited by 0
  11. 2026Gram-Space: Structure-Preserving Codebook Compression for Memory-Efficient Neuro-Symbolic AIcited by 0
  12. 2026Learning a Vector-Symbolic Model for Socio-Cultural Taskscited by 0
  13. 2026Learning Holographic Reduced Representations with Clifford Variational Autoencoderscited by 0