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 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.

New Fractional Ambiguity Function Integrated with CNN-Based Machine Learning for Signal Classification

A new fractional ambiguity function (NFrAF) derived from the fractional Fourier transform is introduced as a generalization of the classical ambiguity function. The fundamental analytical properties of the NFrAF, including symmetry, marginality, and Moyal type identities, are rigorously established. After verifying its ability to detect and localize monocomponent and multicomponent linear frequency modulated (LFM) signals, the NFrAF is integrated into a convolutional neural network based machine learning framework for signal classification. Owing to its superior time frequency resolution and localization, the NFrAF provides a more informative input representation than conventional methods such as the spectrogram and classical ambiguity function. Experimental results on simulated datasets demonstrate consistent improvements in classification accuracy, highlighting the effectiveness of the proposed representation for data driven signal analysis.