The definition of graph Fourier transform is a fundamental issue in graph signal processing. Conventional graph Fourier transform is defined through the eigenvectors of the graph Laplacian matrix, which minimize the ${\ell }_2$ norm signal variation. In this paper, we propose a generalized definition of graph Fourier transform based on the ${\ell }_1$ norm variation minimization. We obtain a necessary condition satisfied by the ${\ell }_1$ Fourier basis, and provide a fast greedy algorithm to approximately construct the ${\ell }_1$ Fourier basis. Numerical experiments are also provided to compare the differences between the Laplacian basis, ${\ell }_1$ basis and greedy basis.

中文翻译：

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图傅立叶变换基于ℓ _1个范变差最小化

图傅立叶变换的定义是图信号处理中的一个基本问题。传统的图傅立叶变换是通过图拉普拉斯矩阵的特征向量定义的，该特征向量使${\ell }_2$规范信号变化。在本文中，我们提出了基于图的傅立叶变换的广义定义${\ell }_{\mathrm{1个}}$规范变化最小化。我们获得了满足以下条件的必要条件：${\ell }_{\mathrm{1个}}$ 傅立叶基础，并提供一种快速贪婪算法来近似构造 ${\ell }_{\mathrm{1个}}$傅立叶基础。还提供了数值实验来比较拉普拉斯算子之间的差异，${\ell }_{\mathrm{1个}}$ 基础和贪婪的基础。