# Real-Valued Hyperdimensional Sequence Representations with Hadamard Product Binding and Shift Equivariance

arXiv: https://arxiv.org/abs/2608.28334 · HTML: https://mlsanity.com/paper/2608.28334/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 28 references that are in the map.
- First posted on arXiv 28 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [MINIROCKET: A Very Fast (Almost) Deterministic Transform for Time Series Classification](https://mlsanity.com/paper/2012.08791/) (2020)
- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part I: Models and Data Transformations](https://mlsanity.com/paper/2111.06077/) (2021)
- [MultiRocket: Multiple pooling operators and transformations for fast and effective time series classification](https://mlsanity.com/paper/2102.00457/) (2021)
- [A Survey on Hyperdimensional Computing aka Vector Symbolic Architectures, Part II: Applications, Cognitive Models, and Challenges](https://mlsanity.com/paper/2112.15424/) (2021)
- [HYDRA: Competing convolutional kernels for fast and accurate time series classification](https://mlsanity.com/paper/2203.13652/) (2022)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)
- [Variable Binding for Sparse Distributed Representations: Theory and Applications](https://mlsanity.com/paper/2009.06734/) (2020)
- [HDC-MiniROCKET: Explicit Time Encoding in Time Series Classification with Hyperdimensional Computing](https://mlsanity.com/paper/2202.08055/) (2022)
- [Computing with Residue Numbers in High-Dimensional Representation](https://mlsanity.com/paper/2311.04872/) (2023)
- [Shift-Equivariant Similarity-Preserving Hypervector Representations of Sequences](https://mlsanity.com/paper/2112.15475/) (2021)
- [Compositional Factorization of Visual Scenes with Convolutional Sparse Coding and Resonator Networks](https://mlsanity.com/paper/2404.19126/) (2024)
- [Recursive Binding for Similarity-Preserving Hypervector Representations of Sequences](https://mlsanity.com/paper/2201.11691/) (2022)

## Abstract (arXiv)

Encoding temporal order is a fundamental requirement for sequence representations in Hyperdimensional Computing. Fractional Power Encoding provides similarity-preserving position vectors whose inner products approximate shift-invariant kernels, and it supports shift-equivariant transformations of encoded sequence representations. However, standard formulations of Fractional Power Encoding are primarily designed for binding operations such as circular convolution or complex-valued multiplication, which limits their compatibility with Hadamard product binding of real-valued vectors. This paper develops real-valued position encodings motivated by Random Fourier Features, aiming to retain the desirable properties of Fractional Power Encoding while supporting Hadamard-based operations. We propose three real-valued position-encoding variants: a real-valued baseline based on the inverse Fourier transform, and Sinusoid and Cosine-only representations derived from Random Fourier Features. Among them, the Sinusoid variant provides an explicit algebraic shift operator, allowing temporal shifts to be applied directly to the vector-encoded sequence representation without re-encoding the shifted sequence. Experiments on time-series classification datasets show that the proposed real-valued representations achieve performance comparable to standard Fractional Power Encoding while enabling computationally efficient Hadamard product binding. The Sinusoid variant offers the most favorable trade-off, combining efficient real-valued implementation with exact shift-equivariant transformations.