PJ-RoPE: A Fourier-Jet-Affine Position Space for Relative Attention
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Computer Science > Machine Learning
Title:PJ-RoPE: A Fourier-Jet-Affine Position Space for Relative Attention
Abstract:We unify RoPE's Fourier phase, Jordan-RoPE's finite jets, and ALiBi's affine recency into a single learnable relative-position space, and study which regions of this space are selected by different tasks. PJ-RoPE is a Fourier-Jet-Affine formulation for relative attention, with an optional Poincare-type reading as the affine completion of a homogeneous Fourier-jet positional representation. Algebraically, the same primitives form a finite constant-coefficient difference module: simple roots of the lag-shift operator give Fourier/RoPE characters, repeated nonzero roots give Jordan/Fourier jets, and the repeated unit root gives ALiBi-like affine recency.
The framework separates scalar PJ-bias kernels from exact PJ-rotary feature transforms, introduces adaptive sector diagnostics, and uses LC/rapidity coordinates to stabilize high-order jets. Controlled probes verify sector containment and selection; small language runs expose an affine/recency boundary; music-token streams provide the clearest case where LC/affine variants remain strong while carrying measurable high-order corrections; and LC diagnostics show a scale-stability gain coupled to phase-resolution loss.
| Comments: | 26 pages, 6 figures, 10 tables. Code available at this https URL |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2606.05345 [cs.LG] |
| (or arXiv:2606.05345v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2606.05345
arXiv-issued DOI via DataCite (pending registration)
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