AI 中文总结
研究基于拉普拉斯或邻接移位特征向量的图信号处理的结构妥协问题，通过连通图到有限阿贝尔群凯莱图的等距嵌入，定义群嵌入图傅里叶变换，证明相关恒等式，比较两框架算子，提供精确规范结构。
AI 中文摘要
基于拉普拉斯或邻接移位特征向量的图信号处理存在三个结构妥协：特征基仅在简并特征空间内旋转时固定，移位不是等距变换，且不存在使滤波成为真正卷积的真正平移。本文开发了一种替代的调和分析方法，一次性消除了这三个问题。给定一个连通图到有限阿贝尔群的凯莱图的等距嵌入，在其上经典傅里叶分析可精确应用，我们从宿主特征定义群嵌入图傅里叶变换，将图信号提升到宿主并在那里进行处理。特征提供规范正交傅里叶基，群平移形成一族酉置换算子，滤波是真正的群卷积，卷积定理成立。我们证明了嵌入设置下的普兰切尔、卷积、平移协方差和采样恒等式，并比较了两个框架的移位和卷积算子。数值上，结构恒等式在机器精度下成立，在相同滤波协议下，一旦通过尊重平滑性的扩展填充宿主补集，群特征基与拉普拉斯特征基去噪效果相当。贡献在于精确、规范的结构，而非去噪优势。
英文摘要
Graph signal processing built on the eigenvectors of a Laplacian or adjacency shift inherits three structural compromises: the eigenbasis is fixed only up to rotation within degenerate eigenspaces, the shift is not an isometry, and there is no genuine translation under which filtering is a true convolution. We develop an alternative harmonic analysis that removes all three at once. Given an isometric embedding of a connected graph into a Cayley graph of a finite abelian group, a host on which classical Fourier analysis applies exactly, we define a group-embedding graph Fourier transform from the host characters, lift graph signals to the host, and process them there. The characters supply a canonical orthonormal Fourier basis; the group translations form a family of unitary permutation operators obeying an exact group law; and filtering is genuine group convolution, for which the convolution theorem holds as a theorem rather than a definition and which possesses an identity element. We prove the Plancherel, convolution, translation-covariance, and sampling identities in the embedded setting, and compare the shift and convolution operators of the two frameworks side by side. Numerically, the structural identities hold to machine precision; under a same-filter protocol the groupcharacter basis denoises equivalently to the Laplacian eigenbasis once the host complement is filled by a smoothness-respecting extension. The contribution is exact, canonical structure, not a denoising advantage.
Comments29 pages, 7 figures
