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

Year: 2026

Type: preprint

Abstract: 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.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2608.28334

Authors: Schlegel, Kenny, Rachkovskij, Dmitri A., Kleyko, Denis, Loutfi, Amy, Streif, Stefan, Osipov, Evgeny

Institution: Chemnitz University of Technology

Topic: Time Series Analysis and Forecasting

Subfield: Signal Processing

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: Stiftelsen för Strategisk Forskning, Knut och Alice Wallenbergs Stiftelse, Vetenskapsrådet

Awards: Securing technological leadership with efficient AGI, Vector-symbolic architectures as an algebraic programming language for neuromorphic computers, G3676281905, Harnessing Structure, Recurrence, and Randomness for Efficient Neural Networks, G4632909464, G539454170, Beräkning i Superposition, NAISS

OpenAlex ID: W7204767590

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