A real-valued sequence encoding called Sinusoid matched an explicitly re-encoded signal after a 10-step shift, reaching a cosine similarity of 1.0 in a synthetic test. Its algebraic shift operation was also reported to be about five times faster than standard FPE at a vector dimensionality of 1,000. Both findings come from the paper's focused computational tests: one synthetic shift demonstration and one stated dimensionality.
The preprint examines whether real-valued position encodings can remain compatible with Hadamard-product binding while preserving the similarity structure and shift equivariance associated with Fractional Power Encoding. For a general reader, the practical question is whether a sequence can represent where something occurs in time and still be shifted through the representation itself. In this setting, shift equivariance means that moving a sequence by a temporal offset has a corresponding transformation in its encoded form.
What the preprint tests
It introduces three positional variants. Variant A uses the inverse-Fourier baseline. Variant B, Sinusoid RFF, adds an explicit shift operator to a real-valued representation. Variant C, Cosine-only RFF, uses cosine features without an algebraic shift operator.
Classification tests used MiniROCKET and MultiROCKET-HYDRA models on UCR time-series datasets. The researchers selected a parameter called beta through logarithmic grid search with cross-validation over 0, 0.41, 1.00, 1.83, 3.00, 4.66 and 7.00. They averaged results over 30 repetitions with different random seeds, using accuracy and error as the primary measures.
Accuracy stayed close
On the 38-dataset MiniROCKET comparison, the original model's mean accuracy was 80.79%. The three real-valued encodings were 82.23% for Standard FPE, 82.22% for Sinusoid FPE and 82.24% for Cosine FPE. Cosine had the highest displayed mean among those three, while the average relative accuracy changes were 1.44%, 1.43% and 1.45%, respectively.
The MultiROCKET-HYDRA comparison also covered 38 datasets. Mean accuracy was 82.72% for the original model, 83.80% for Standard FPE, 83.86% for Sinusoid FPE and 83.81% for Cosine FPE. Sinusoid was highest among the three encodings. Their average relative accuracy changes were 1.08%, 1.14% and 1.09%, respectively; average relative error changes were negative 6.0%, negative 7.0% and negative 6.0%.
A shift inside the representation
Sinusoid's central feature is an explicit shift operator. The operator takes the position representation at one temporal point and the representation of a shift, then returns the representation at the new point. In practical terms, a temporal move can be carried out inside the embedding space rather than by explicitly rebuilding the positional representation. The authors identify this exact shift-equivariant transformation as part of Sinusoid's favorable trade-off.
In the synthetic test, cosine similarity peaked at 1.0 at the true shift of 10 time steps, and the algebraically shifted representation matched explicit re-encoding. The reported shift operation scales linearly with vector dimensionality, compared with a standard FPE operation whose count scales with dimensionality times its logarithm. At a dimensionality of 1,000, the Sinusoid operation produced an approximately fivefold speed-up.
The trade-offs remain
The design also has costs. For the same number of sampled frequencies, Cosine-only has larger kernel-approximation variance. Sinusoid reduces that variance by encoding both phase components, but doing so doubles the vector dimensionality. The Sinusoid positional-vector norm remains 1 for any position, whereas the Cosine-only norm is generally not exactly constant at finite dimensionality; it is preserved in expectation, with fluctuations that decrease as dimensionality increases.
The classification figures need to be read as a selected comparison. The displayed table lists only datasets where the relative change in accuracy or error exceeds 5% in absolute value. The reported comparisons therefore cover 38-dataset MiniROCKET and MultiROCKET-HYDRA subsets, while the shift result comes from a separate synthetic test.
Within those boundaries, the authors identify Sinusoid as the most favorable balance between an efficient real-valued implementation and exact shift-equivariant transformations. The supplied document is an arXiv v1 preprint dated 28 Aug 2026.
Paper data and sources
Original title: Real-Valued Hyperdimensional Sequence Representations with Hadamard Product Binding and Shift Equivariance
Authors: Kenny Schlegel, Dmitri A. Rachkovskij, Denis Kleyko et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text