In graph signal processing (GSP), data dependencies are represented by a graph whose nodes label the data and the edges capture dependencies among nodes. The graph is represented by a weighted adjacency matrix $A$ that, in GSP, generalizes the Discrete Signal Processing (DSP) shift operator $z^{-1}$. The (right) eigenvectors of the shift $A$ (graph spectral components) diagonalize $A$ and lead to a graph Fourier basis $F$ that provides a graph spectral representation of the graph signal. The inverse of the (matrix of the) graph Fourier basis $F$ is the Graph Fourier transform (GFT), $F^{-1}$. Often, including in real world examples, this diagonalization is numerically unstable. This paper develops an approach to compute an accurate approximation to $F$ and $F^{-1}$, while insuring their numerical stability, by means of solving a non convex optimization problem. To address the non-convexity, we propose an algorithm, the stable graph Fourier basis algorithm (SGFA) that improves exponentially the accuracy of the approximating $F$ per iteration. Likewise, we can apply SGFA to $A^H$ and, hence, approximate the stable left eigenvectors for the graph shift $A$ and directly compute the GFT. We evaluate empirically the quality of SGFA by applying it to graph shifts $A$ drawn from two real world problems, the 2004 US political blogs graph and the Manhattan road map, carrying out a comprehensive study on tradeoffs between different SGFA parameters. We also confirm our conclusions by applying SGFA on very sparse and very dense directed Erdős-Rényi graphs.

中文翻译：

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图傅立叶变换：一个稳定的近似

在图信号处理 (GSP) 中，数据依赖关系由一个图表示，该图的节点标记数据，边捕获节点之间的依赖关系。图由加权邻接矩阵表示$A$ 在 GSP 中，概括了离散信号处理 (DSP) 移位运算符 $z^{-1}$. 移位的（右）特征向量$A$ （图谱分量）对角化 $A$ 并导致图形傅立叶基础 $F$它提供了图形信号的图形频谱表示。图傅立叶基的（矩阵的）逆$F$ 是图傅立叶变换（GFT）， $F^{-1}$. 通常，包括在现实世界的例子中，这种对角化在数值上是不稳定的。本文开发了一种计算精确近似值的方法$F$ 和 $F^{-1}$，同时通过解决非凸优化问题来确保其数值稳定性。为了解决非凸性问题，我们提出了一种算法，即稳定图傅立叶基算法 (SGFA)，该算法以指数方式提高逼近的准确性$F$每次迭代。同样，我们可以将 SGFA 应用于$A^H$ 因此，近似于图移位的稳定左特征向量 $A$并直接计算 GFT。我们通过将 SGFA 应用于图位移来凭经验评估 SGFA 的质量$A$从2004年美国政治博客图和曼哈顿路线图两个现实世界问题中提取，对不同SGFA参数之间的权衡进行了综合研究。我们还通过在非常稀疏和非常密集的有向 Erdős-Rényi 图上应用 SGFA 来证实我们的结论。