# Computing on Functions Using Randomized Vector Representations arXiv: https://arxiv.org/abs/2109.03429 ยท HTML: https://mlsanity.com/paper/2109.03429/ ## Key facts - Cited by 41 papers in the map (data through 14 Oct 2026). - Builds on 14 of its 141 references that are in the map. - First posted on arXiv 8 Sep 2021. ## Builds on (most cited first, 14 of 14) - [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018) - [On the Binding Problem in Artificial Neural Networks](https://mlsanity.com/paper/2012.05208/) (2020) - [Blessing of dimensionality: mathematical foundations of the statistical physics of data](https://mlsanity.com/paper/1801.03421/) (2018) - [Every Model Learned by Gradient Descent Is Approximately a Kernel Machine](https://mlsanity.com/paper/2012.00152/) (2020) - [Hyperdimensional computing as a framework for systematic aggregation of image descriptors](https://mlsanity.com/paper/2101.07720/) (2021) - [A theory of sequence indexing and working memory in recurrent neural networks](https://mlsanity.com/paper/1803.00412/) (2018) - [Variable Binding for Sparse Distributed Representations: Theory and Applications](https://mlsanity.com/paper/2009.06734/) (2020) - [Discrete-time signatures and randomness in reservoir computing](https://mlsanity.com/paper/2010.14615/) (2020) - [Robust computation with rhythmic spike patterns](https://mlsanity.com/paper/1901.07718/) (2019) - [Density Encoding Enables Resource-Efficient Randomly Connected Neural Networks](https://mlsanity.com/paper/1909.09153/) (2019) - [Autoscaling Bloom Filter: Controlling Trade-off Between True and False Positives](https://mlsanity.com/paper/1705.03934/) (2017) - [Neural Distributed Autoassociative Memories: A Survey](https://mlsanity.com/paper/1709.00848/) (2017) - [Integer Echo State Networks: Efficient Reservoir Computing for Digital Hardware](https://mlsanity.com/paper/1706.00280/) (2017) - [A short letter on the dot product between rotated Fourier transforms](https://mlsanity.com/paper/2007.13462/) (2020) ## Built on by (most cited first, 30 of 41) - [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations](https://mlsanity.com/paper/2111.06077/) (2021) - [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part II: Applications, Cognitive Models, and Challenges](https://mlsanity.com/paper/2112.15424/) (2021) - [A Neuro-vector-symbolic Architecture for Solving Raven's Progressive Matrices](https://mlsanity.com/paper/2203.04571/) (2022) - [Understanding Hyperdimensional Computing for Parallel Single-Pass Learning](https://mlsanity.com/paper/2202.04805/) (2022) - [Torchhd: An Open Source Python Library to Support Research on Hyperdimensional Computing and Vector Symbolic Architectures](https://mlsanity.com/paper/2205.09208/) (2022) - [Neuromorphic Visual Scene Understanding with Resonator Networks](https://mlsanity.com/paper/2208.12880/) (2022) - [Hyperseed: Unsupervised Learning with Vector Symbolic Architectures](https://mlsanity.com/paper/2110.08343/) (2021) - [Computing with Residue Numbers in High-Dimensional Representation](https://mlsanity.com/paper/2311.04872/) (2023) - [Towards Learning to Reason: Comparing LLMs with Neuro-Symbolic on Arithmetic Relations in Abstract Reasoning](https://mlsanity.com/paper/2412.05586/) (2024) - [A Linear Time and Space Local Point Cloud Geometry Encoder via Vectorized Kernel Mixture (VecKM)](https://mlsanity.com/paper/2404.01568/) (2024) - [Generalized Holographic Reduced Representations](https://mlsanity.com/paper/2405.09689/) (2024) - [Visual Odometry with Neuromorphic Resonator Networks](https://mlsanity.com/paper/2209.02000/) (2022) - [An Encoding Framework for Binarized Images using HyperDimensional Computing](https://mlsanity.com/paper/2312.00454/) (2023) - [Shift-Equivariant Similarity-Preserving Hypervector Representations of Sequences](https://mlsanity.com/paper/2112.15475/) (2021) - [Vector-Symbolic Architecture for Event-Based Optical Flow](https://mlsanity.com/paper/2405.08300/) (2024) - [The Discovery Engine: A Framework for AI-Driven Synthesis and Navigation of Scientific Knowledge Landscapes](https://mlsanity.com/paper/2505.17500/) (2025) - [HEAL: Brain-inspired Hyperdimensional Efficient Active Learning](https://mlsanity.com/paper/2402.11223/) (2024) - [Compositional Factorization of Visual Scenes with Convolutional Sparse Coding and Resonator Networks](https://mlsanity.com/paper/2404.19126/) (2024) - [Distributed Representations Enable Robust Multi-Timescale Symbolic Computation in Neuromorphic Hardware](https://mlsanity.com/paper/2405.01305/) (2024) - [Recursive Binding for Similarity-Preserving Hypervector Representations of Sequences](https://mlsanity.com/paper/2201.11691/) (2022) - [Efficient Hyperdimensional Computing](https://mlsanity.com/paper/2301.10902/) (2023) - [The Hyperdimensional Transform: a Holographic Representation of Functions](https://mlsanity.com/paper/2310.16065/) (2023) - [Decodable and Sample Invariant Continuous Object Encoder](https://mlsanity.com/paper/2311.00187/) (2023) - [Hyperdimensional computing: a fast, robust and interpretable paradigm for biological data](https://mlsanity.com/paper/2402.17572/) (2024) - [Self-Attention Based Semantic Decomposition in Vector Symbolic Architectures](https://mlsanity.com/paper/2403.13218/) (2024) - [PhasorFlow: A Python Library for Unit Circle Based Computing](https://mlsanity.com/paper/2603.15886/) (2026) - [Bridging the Gap Between Hyperdimensional Computing and Kernel Methods via the Nystr"om Method](https://mlsanity.com/paper/2608.06860/) (2026) - [Enhanced Detection of Transdermal Alcohol Levels Using Hyperdimensional Computing on Embedded Devices](https://mlsanity.com/paper/2403.12323/) (2024) - [Systematic Abductive Reasoning via Diverse Relation Representations in Vector-symbolic Architecture](https://mlsanity.com/paper/2501.11896/) (2025) - [On Design Choices in Similarity-Preserving Sparse Randomized Embeddings](https://mlsanity.com/paper/2501.14741/) (2024) ## Abstract (arXiv) Vector space models for symbolic processing that encode symbols by random vectors have been proposed in cognitive science and connectionist communities under the names Vector Symbolic Architecture (VSA), and, synonymously, Hyperdimensional (HD) computing. In this paper, we generalize VSAs to function spaces by mapping continuous-valued data into a vector space such that the inner product between the representations of any two data points represents a similarity kernel. By analogy to VSA, we call this new function encoding and computing framework Vector Function Architecture (VFA). In VFAs, vectors can represent individual data points as well as elements of a function space (a reproducing kernel Hilbert space). The algebraic vector operations, inherited from VSA, correspond to well-defined operations in function space. Furthermore, we study a previously proposed method for encoding continuous data, fractional power encoding (FPE), which uses exponentiation of a random base vector to produce randomized representations of data points and fulfills the kernel properties for inducing a VFA. We show that the distribution from which elements of the base vector are sampled determines the shape of the FPE kernel, which in turn induces a VFA for computing with band-limited functions. In particular, VFAs provide an algebraic framework for implementing large-scale kernel machines with random features, extending Rahimi and Recht, 2007. Finally, we demonstrate several applications of VFA models to problems in image recognition, density estimation and nonlinear regression. Our analyses and results suggest that VFAs constitute a powerful new framework for representing and manipulating functions in distributed neural systems, with myriad applications in artificial intelligence.