FRAME: Learning the Adaptation Domain with a Mixture of Fractional-Fourier Experts
Parameter-efficient fine-tuning (PEFT) reparameterizes weight updates in a fixed basis: low-rank adapters operate in the spatial domain, while spectral methods operate in a fixed Fourier domain. This paper introduces Fractional-Fourier Mixture of Experts, where each expert has a learnable fractional-Fourier order that interpolates between spatial and Fourier domains. Routing tokens through experts of different orders allows low-rank updates to be placed in their most compact domain, and the experts are naturally decorrelated, reducing interference and improving multi-task composition. The method adds negligible cost and outperforms strong baselines on LLaMA-3.1-8B and Qwen2.5-7B across various benchmarks.
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[Submitted on 30 Jun 2026]
Title:FRAME: Learning the Adaptation Domain with a Mixture of Fractional-Fourier Experts
View a PDF of the paper titled FRAME: Learning the Adaptation Domain with a Mixture of Fractional-Fourier Experts, by Tom Saliencro and 4 other authors
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Abstract:Parameter-efficient fine-tuning (PEFT) reparameterizes weight updates in a fixed basis: low-rank adapters operate in the spatial domain, while a recent line of spectral methods operates in a fixed Fourier domain. We argue that the choice of domain is itself a design degree of freedom that should be learned, and that no single basis is optimal across tasks, layers, or tokens. We introduce Fractional-Fourier Mixture of Experts, a mixture-of-experts adapter in which every expert carries a learnable fractional-Fourier order that continuously interpolates between the spatial domain (recovering vanilla LoRA) and the Fourier domain (recovering a spectral adapter). Routing tokens through experts that occupy different points on this spatial-spectral continuum lets the model place each low-rank update in the domain where it is most compact, and -- because fractional-Fourier operators of different orders are mutually incoherent -- makes the experts naturally decorrelated, which reduces interference and improves multi-task composition. The order is a single scalar per expert, trained with a separate optimizer, and the transform is computed with an $\mathcal{O}(d\log d)$ chirp--FFT surrogate, so Fractional-Fourier Mixture of Experts adds negligible cost over standard MoE-LoRA. Across commonsense, mathematical, code, and knowledge benchmarks on LLaMA-3.1-8B and Qwen2.5-7B, Fractional-Fourier Mixture of Experts improves over strong MoE-LoRA and spectral baselines -- including FlyLoRA, FourierMoE, and HMoRA -- while keeping the active-parameter budget small, and analysis shows that the learned orders specialize by task and layer in interpretable ways.
Subjects:
Machine Learning (cs.LG)
Cite as: arXiv:2607.00162 [cs.LG]
(or arXiv:2607.00162v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2607.00162
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Tom Saliencro [view email] [v1] Tue, 30 Jun 2026 20:39:55 UTC (494 KB)
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