图形傅里叶变换：一种稳定的近似
Graph Fourier Transform: A Stable Approximation
摘要：
在图信号处理（GSP）中，数据依赖性由图表示，该图的节点标记数据并且边捕获节点之间的依赖性。该图由加权邻接矩阵$A$表示，该矩阵在GSP中推广了离散信号处理（DSP）移位运算符$z^{-1}$。移位$A$的（右）本征向量（图形频谱分量）对角化$A$并导致提供图形信号的图形频谱表示的图形傅立叶基$F$。图形傅里叶基$F$（的矩阵）的逆是图形傅里叶变换（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。本文通过求解一个非凸优化问题，给出了一种计算$F$和$F^{-1}$的精确逼近，同时保证了它们的数值稳定性的方法。为了解决非凸性，我们提出了一种算法，稳定的图形傅立叶基算法（SGFA），我们证明了近似$F$每次迭代的精度呈指数级增加。同样，我们可以将SGFA应用于$A^H$，因此，近似图移位$A$的稳定左特征向量，并直接计算GFT。我们评估经验的质量SGFA通过将其应用到图的变化$A$从两个现实世界的问题，2004年美国政治博客图和曼哈顿路线图，进行了全面的研究，不同的SGFA参数之间的权衡。我们还证实了我们的结论，通过应用SGFA非常稀疏和非常密集的有向Erd \H os-Rényi图。
英文摘要：
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 we prove to exponentially increase 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 Erd\H os-Rényi graphs.
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