切比雪夫流形自适应
Chebyshev Manifold Adaptation
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中文总结 AI 辅助
研究提出ChebyMA参数高效自适应方法，通过切比雪夫多项式基多表面叠加采用权重矩阵。理论上证明其收敛性与表达优势，实验表明在CIFAR和自然语言处理数据集上，该方法比其他方法参数-精度表现更优，验证了理论基础。
中文摘要 AI 辅助
本文提出了一种名为ChebyMA（切比雪夫流形自适应）的新的参数高效自适应方法。ChebyMA通过在可学习坐标上评估的切比雪夫多项式基的多表面叠加来采用权重矩阵，并通过可训练系数矩阵进行组合，用高表达性的连续函数逼近取代标准线性投影。理论上，建立了逼近表达定理，从函数逼近理论角度证明单流形ChebyMA保证重构的弗罗贝尼乌斯范数误差收敛。此外，借鉴柯尔莫哥洛夫n宽度直觉，证明多流形叠加（S>1）在解耦高维复杂特征方面的表达优势。在计算机视觉CIFAR数据集（CIFAR-10、CIFAR-100）和自然语言处理（AG News、SST-2）数据集上的实验结果表明，与标准全参数微调、LoRA、TLoRA和StelLA相比，ChebyMA始终实现更好的参数-精度帕累托前沿。ChebyMA在测试数据集中显著优于其他测试方法，验证了其基于纯矢量化计算的通用性的坚实理论基础。
英文摘要
The paper presents a new parameter-efficient adaptation method called ChebyMA (Chebyshev Manifold Adaptation). ChebyMA adopts weight matrices through a multi-surface superposition of Chebyshev polynomial bases evaluated on learnable coordinates and combined via trainable coefficient matrices, replacing standard linear projections with highly expressive continuous function approximation. Theoretically, we establish an Approximation Expressivity Theorem, proving from the perspective of function approximation theory that single-manifold ChebyMA guarantees convergence in Frobenius norm error of reconstruction. Besides, drawing on Kolmogorov $n$-width intuition, we demonstrate the expressive advantages of multi-manifold superposition ($S > 1$) in decoupling high-dimensional complex features. Experimental results on Computer Vision CIFAR datasets(CIFAR-10, CIFAR-100)\cite{CIFAR} and Natural Language Processing (AG News, SST-2) datasets demonstrate that ChebyMA consistently achieves a superior parameter-accuracy Pareto front compared to standard full-parameter fine-tuning, LoRA\cite{LoRA}, TLoRA\cite{TLoRA}, and StelLA\cite{StelLA}. ChebyMA significantly outperforms other tested methods in tested datasets, validating its solid theoretical foundation for generality with purely vectorized computations.
