通过等距群嵌入在图上构造紧小波框架
Tight Wavelet Frames on Graphs via Isometric Group Embedding
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中文总结 AI 辅助
研究如何在图上构造紧小波框架，基于有限阿贝尔群的凯莱图等距嵌入，提出膨胀小波和谱带通小波两种构造方法，证明其紧框架恒等式等性质，给出多分辨率分解，展示全变换成本，还说明了信号扩展到宿主的规范方法及相关精度和能量集中情况。
中文摘要 AI 辅助
谱图小波将核应用于图拉普拉斯谱。在不规则图上，其分析函数继承非规范特征基，不构成紧框架，重建需求逆框架算子。本文基于有限阿贝尔群的凯莱图给出等距嵌入，在其上构造小波并限制到图上。提出两种构造，膨胀小波利用群自同构作为膨胀，仅存在于有复合循环因子的宿主上；谱带通小波在对偶频率幅度上使用归一化滤波器组，存在于所有宿主上，构成帕塞瓦尔（紧）框架，可精确重建图信号，具有平移协变性且在顶点和频率上联合定位。证明了紧框架恒等式和精确重建，给出多分辨率分解，展示了通过宿主快速傅里叶变换全变换成本为\(O(JN\log N)\)。对于适当嵌入，将信号扩展到宿主剩余部分的规范方法是离散调和扩展，它唯一地最小化宿主狄利克雷能量，并将零填充和对称扩展启发式方法作为其近似。在基准宿主上重建达到机器精度，带通原子\(89 - 99\%\)的能量集中在其中心的图距离为二的范围内。
英文摘要
Spectral graph wavelets apply a kernel to the graph Laplacian spectrum. On an irregular graph their analyzing functions inherit a non-canonical eigenbasis, they do not form a tight frame, and reconstruction requires inverting a frame operator. We take a different route, built on an exact substrate. 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 construct wavelets on the host and restrict them to the graph. Two constructions arise and we keep them separate. Dilation wavelets use a group automorphism as a dilation, reproducing the classical translate-dilate template but existing only on hosts with composite cyclic factors. Spectral band-pass wavelets use a normalized filter bank in the dual frequency magnitude; they exist on every host, form a Parseval (tight) frame, reconstruct any graph signal exactly via restriction, are translation-covariant, and localize jointly in vertex and frequency. We prove the tight-frame identity and exact reconstruction, give a multiresolution decomposition, and show the full transform costs O(JN log N) via the host fast Fourier transform. For a proper embedding we show the canonical way to complete a signal onto the host remainder is the discrete harmonic extension, which uniquely minimizes the host Dirichlet energy and places the zero-padding and symmetric-extension heuristics as approximations of it. On benchmark hosts reconstruction reaches machine precision and band-pass atoms concentrate 89-99 percent of their energy within graph-distance two of their center.
