论文解读《The Emerging Field of Signal Processing on Graphs》

感悟

  看完图卷积一代、二代,深感图卷积的强大,刚开始接触图卷积的时候完全不懂为什么要使用拉普拉斯矩阵( $L=D-W$),主要是其背后的物理意义。通过借鉴前辈们的论文、博客、评论逐渐对图卷积有了一定的了解,作为一个刚上研的博士生,深感得对图神经网络进行一个系统的学习。

  本篇论文得感谢论文 David I Shuman 作者及博主:纯牛奶爱酸牛奶 


Paper Information

  Authors:D. Shuman, S. Narang, P. Frossard, Antonio Ortega, P. Vandergheynst
  Sources:2012, IEEE Signal Processing Magazine
  Paper:Download 
  Code:Download
  2528 Citations, 75 References


Abstract

  The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs.

  图上信号处理的新兴领域,将代数和谱图理论的概念与计算谐波分析相结合,以处理图上的信号。

  本文将阐述该领域上的一些挑战,以及定义在  graph spectral domains  上的一些方法,这些方法和经典的 frequency domain  相似。本文同时还点明了在处理 graph signal 融合使用 irregular data of graph 的重要性(每每看论文都出现结合结构特性,但并没有有人能完全说清楚)。然后介绍一些 common operator ,比如说  filtering、translation、modulation、dilation、downsampling to the graph setting 。对已经提出的   localized, multiscale transforms  用于提取图数据的高维特征做了总结。


1 Introduction 

  • 图中每条边的权重,经常表示为两个顶点之间的相似度。对于节点之间的连接性以及权重大小,一般为数据本身就有的,如社交关系,另外一种为自己根据需求构建的,比如使用KNN 构建权重矩阵 $W$。这在本博客《谱聚类原理总结》已经做了详细介绍。
  • 这里相似性通常和距离成反比,采用聚类的思想:距离较远的两个点之间的边权重值较低,而距离较近的两个点之间的边权重值较高。
  • graph signal:The data on these graphs can be visualized as a finite collection of samples, with one sample at each vertex in the graph. Collectively, we refer to these samples as a graph signal.示例: Fig. 1.

    

1.1 The Main Challenges of Signal Processing on Graphs

  • 目前的研究现状:使用  wavelet, time-frequency, curvelet and other localized transforms 可以很好的将 Euclidean space 上的高维不同类数据区分开来,但对 non-Euclidean space 上的数据却无有效手段。
  • 传统的   signal processing techniques  忽略了 irregular data domain  。

   Challenges on Graph:

    • The unavoidable fact is that weighted graphs are irregular structures that lack a shift-invariant notion of translation.
      • 平移不变性(translation invariance):在欧式空间中,平移是一种几何变换,表示把一幅图像或空间中的每一个点在相同方向移动相同距离。用基础的分类结构比如 ResNet、Inception 给一只猫分类时,无论猫怎么扭曲、平移,最终识别出来的都是猫,输入怎么变形输出都不变这就是平移不变性,网络的层次越深这个特性会越明显。
      • 平移可变性(translation variance):针对目标检测的,比如一只猫从图片左侧移到了右侧,检测出的猫的坐标会发生变化就称为平移可变性。当卷积网络变深后最后一层卷积输出的  feature map  变小,物体在输入上的小偏移,经过 $N$ 层  pooling  后在最后的小 feature map 上会感知不到,这就是为什么  R-FCN  原文会说网络变深平移可变性变差。
      • CNN 无法处理 Non Euclidean Structure 的数据,通俗理解就是在拓扑图中每个顶点的相邻顶点数目都可能不同,那么当然无法用一个同样尺寸的卷积核来进行卷积运算。
    • Tramsform thr spatial domain data into spectral domain date.$F\left(\lambda_{i}\right)=\hat{f}\left(\lambda_{i}\right)=\sum \limits _{i=1}^{N} f(i) u(i)$
    • We need a method to generate a coarser version of the graph that somehow captures the structural properties embedded in the original graph.
    • We intuitively downsample a discrete-time signal by deleting every other data point,the process is as Fig.1.

  处理数据域的不规则性、图结构,在之前提到的应用中,可以表示很多顶点的特征。为了能很好地对数据的尺度进行缩放,对于图信号的处理技术应该使用局部操作,通过对每个顶点,计算顶点的邻居,或是和它很近的顶点的信息。

     The overarching challenges of processing signals on graphs:

    1. How to construct a weighted graph that captures the geometric structure of the underlying data domain;
    2. Incorporating the graph structure into localized transform methods;
    3. Leveraging the invaluable intuitions in Euclidean domains to deal with processing signals on graphs;
    4. Consider to improve the efficient of  localized transforms.

2 The graph spectral domain

2.1 Weighted Graphs and Graph Signals

  Defitions of graph:

    • Graph type:undirected, connected, weighted graph  $\mathcal{G}=\{\mathcal{V}, \mathcal{E}, \mathbf{W}\} $,
    • Vertice set:a finite set of vertices  $\mathcal{V}$  with  $|\mathcal{V}|=N$ ,
    • Edge set:a set of edges  $\mathcal{E} $,
    • Weighted adjacency matrix:Symbol as  $\mathbf{W}$ . If there is an edge  $e=(i, j)$  connecting vertices  $i$ and  $j$ , the entry  $W_{i, j}$  represents the weight of the edge; otherwise,  $W_{i, j}=0$ .
    • SSC(连通分量):If the graph  $\mathcal{G}$  is not connected and has  $M$  connected components $ (M>1)$ , we can separate signals on  $\mathcal{G}$  into  $M$  pieces corresponding to the  $M$  connected components, and independently process the separated signals on each of the subgraphs.

   According the need of application,a common way to construct a weight matrix is full connect which for construct a symmetric matrx.More detail you can refer to my blog 《谱聚类原理总结》.

    $W_{i, j}=\left\{\begin{array}{ll}\exp \left(-\frac{[\operatorname{dist}(i, j)]^{2}}{2 \theta^{2}}\right) & \text { if } \operatorname{dist}(i, j) \leq \kappa \\0 & \text { otherwise }\end{array}\right.$

  Where, $ dist(i, j)$ can be Euclidean distance between two vector $i$ and $j$. two leanable parmeter $\theta$ and $\kappa$.

  Another way two generate a weight matrix is KNN .

  $f: \mathcal{V} \rightarrow \mathbb{R}$ defined the value of verticles of the graph,the $i^{t h}$  component $f_i$ mean the $i^{t h}$ vertices value .

2.2  The Non-Normalized Graph Laplacian

  The non-normalized graph Laplacian, also called the combinatorial graph Laplacian, is defined as

    $L:=\mathbf{D}-\mathbf{W}$

  Where

    • $D$ is degree matrix which a diagonal matrix ;
    • $\mathbf{W}$ is weighted adjacency matrix .

   The graph Laplacian matrix is define as :

    $(L f)(i)=\sum \limits _{j \in \mathcal{N}_{i}} W_{i, j}[f(i)-f(j)]$

  If you want to know the background physical meaning of this Eq, you can refer to my another blog 《图神经网络基础二:谱图理论》.

  Where 

    • The neighborhood  $\mathcal{N}_{i}$  is the set of vertices connected to vertex  $i$  by an edge. More generally, we denote by  $\mathcal{N}(i, k)$  the set of vertices connected to vertex  $i$  by a path of  $k$  or fewer edges.(邻居定义为小于等于k-step可到达的点)

   Due to graph Laplacian matrix $L$ is a real symmetric matrix ,so it can do Laplace Spectral Decomposition as the following:

    $L \mu_{k}=\lambda_{k} \mu_{k}$

   Because $L$ is real symmetric matrix ,so it can be transformed as:

    $L=U \Lambda U^{-1}=U \Lambda U^{T}$

   Some note in here:

    • Orthonormal eigenvectors  $\left\{\mathbf{u}_{l}\right\}_{l=0,1, \ldots, N-1}$ ;
    • Non-negative eigenvalues $\left\{\lambda_{l}\right\}_{l=0,1, \ldots, N-1}$ ;
    • $\Lambda  $ is consist of the non-negative eigenvalues that are ordered as                                                                                                $0=\lambda_{0}<\lambda_{1} \leq \lambda_{2} \cdots \leq \lambda_{N-1}:=\lambda_{\max }$
    • We denote the entire spectrum by $\sigma(L):=\left\{\lambda_{0}, \lambda_{1}, \ldots, \lambda_{N-1}\right\}$ .

2.3  A Graph Fourier Transform and Notion of Frequency

   Classical fourier tramsform is :

    $\hat{f}(\xi):=\left\langle f, e^{2 \pi i \xi t}\right\rangle=\int_{\mathbb{R}} f(t) e^{-2 \pi i \xi t} d t$

  即:将函数  $f$  在特征函数(eigenfunctions)上表示出来,类似坐标点在平面上的形式。

  Eigenfunction with Laplace operator :

    $-\Delta\left(e^{2 \pi i \xi t}\right)=-\frac{\partial^{2}}{\partial t^{2}} e^{2 \pi i \xi t}=(2 \pi \xi)^{2} e^{2 \pi i \xi t}\quad \quad \quad \quad(2)$

   即:Eigenfunction 的负散度。参考《图神经网络基础二:谱图理论

  Analogously,we can define the Graph Fourier tramsform $\hat{f}$ of any function $\mathbf{f} \in \mathbb{R}^{N}$ on the vertices of the $G$  as the expansion of f in terms of the eigenvectors of the graph Laplacian:

    $\hat{f}\left(\lambda_{l}\right):=\left\langle\mathbf{f}, \mathbf{u}_{l}\right\rangle=\sum\limits _{i=1}^{N} f(i) u_{l}^{*}(i)\quad \quad \quad \quad(3)$

  The inverse of Graph Fourier tramsform is :

    $f(i)=\sum \limits _{l=0}^{N-1} \hat{f}\left(\lambda_{l}\right) u_{l}(i)\quad \quad \quad \quad(4)$

   More analysis about Classical Fourier tramsform:

    • The eigenvalues $\left\{(2 \pi \xi)^{2}\right\}_{\xi \in \mathbb{R}}$ in Eq.2 carry a specific notion of frequency  $\xi$.
    • If  $\xi$  close to zero (low frequencies), the associated complex exponential eigenfunctions are smooth

  In graph Laplacian, the eigenvalues and eigenvectors provide a similar notion of frequency. 

    • The Laplacian eigenvector  $\mathbf{u}_{0}$  associated with the eigenvalue $0$ is constant and equal to  $\frac{1}{\sqrt{N}}$  at each vertex. 特征值为 $0$ 时,对应的特征向量为归一化的单位向量,即每个分量为 $\frac{1}{\sqrt{N}}$ 。
    • 将 特征值 $\lambda_{l}$ 看作频率,$\lambda_{l}$  越小代表变化慢和周围节点相差不大。(其实就是平滑性度量)
    • The graph Laplacian eigenvectors associated with low frequencies  $\lambda_{l}$  vary slowly across the graph.即不同graph signal 的特征向量受特征值的影响。图中相似的节点,特征向量差别不大;不相似的节点,特征向量差别很大,而判断相似性是通过特征值来判断,特征值越小,节点和邻居节点的相似性越大。

   This is demonstrated in both Figure 2.

    

  分析

    • 对于第一个图,此时特征值 $\lambda_{0} $  为  $0$,特征向量为 $u_0$ ,可以看到每个 graph signal 都是 $\frac{1}{\sqrt{N}} $ (用高度衡量)。
    • 对于第二个图,此时特征向量为 $u_1$,可以看到每个 graph signal 区域附近 graph signal 的高度差不多。
    • 对于第三个图,此时特征向量为 $u_{50}$,可以看到每个 graph signal 区域附近 graph signal 的高度差很多。

    Figure 3, which shows the number of zero  crossings of each graph Laplacian eigenvector. The set of zero crossings of a signal f on a graph G is defined as:

    $\mathcal{Z}_{\mathcal{G}}(\mathbf{f}):=\{e=(i, j) \in \mathcal{E}: f(i) f(j)<0\}$

     

   It means that if $\lambda $ is more bigger ,the bigger the change.

2.4  Graph Signal Representations in Two Domains

   The graph Fourier transform (3) and its inverse (4) give us a way to equivalently represent a signal in two different domains: the vertex domain and the graph spectral domain.

  • A signal g in the vertex domain.
  • Define a signal $\hat{g}$ directly in the graph spectral domain.

   Figure 4 will show you the graph Fourier coefficients with different $\lambda_{l}$.

    

   Figure 4 告诉我们一个 graph signal 在不同的域上的等价表现形式。图4中展示的一个平缓信号的图傅里叶系数衰减的很快。这样的信号是可压缩的(compressible),因为可以通过调整一些图傅里叶系数来趋近他们。

2.5 Discrete Calculus and Signal Smoothness with Respect to the Intrinsic Structure of the Graph

  When we analyze signals, it is important to emphasize that properties such as smoothness are with respect to the intrinsic structure of the data domain.

  下面将介绍 smoothness function 。至于推导完全可以看《图神经网络基础二:谱图理论

  The edge derivative of a signal  $\mathbf{f}$  with respect to edge  $e=(i, j)$  at vertex  $i$  is defined as

    $\left.\frac{\partial \mathbf{f}}{\partial e}\right|_{i}:=\sqrt{W_{i, j}}[f(j)-f(i)]$

   The graph gradient of $f$ at vertex $i$ is the vector

    $\nabla_{i} \mathbf{f}:=\left[\left\{\left.\frac{\partial \mathbf{f}}{\partial e}\right|_{i}\right\}_{e \in \mathcal{E} \text { s.t. } e=(i, j) \text { for some }_{j \in \mathcal{V}}}\right]$

   Then the local variation at vertex $i$:

    $\begin{aligned}\left\|\nabla_{i} \mathbf{f}\right\|_{2} &:=\left[\sum \limits _{e \in \mathcal{E} \text { s.t. } e=(i, j) \text { for some } j \in \mathcal{V}}\left(\left.\frac{\partial \mathbf{f}}{\partial e}\right|_{i}\right)^{2}\right]^{\frac{1}{2}} \\&=\left[\sum \limits_{j \in \mathcal{N}_{i}} W_{i, j}[f(j)-f(i)]^{2}\right]^{\frac{1}{2}}\end{aligned}$

  This provides a measure of local smoothness of $f$ around vertex $i$,  as it is small when the function $f$ has similar values at $i$ and all neighboring vertices of $i$.

  For global smoothness of all nodes in the graph,we can define a discrete p-Dirichlet form of $f$ :

    $S_{p}(\mathbf{f}):=\frac{1}{p} \sum \limits _{i \in V}\left\|\nabla_{i} \mathbf{f}\right\|_{2}^{p}=\frac{1}{p} \sum\limits_{i \in V}\left[\sum_{j \in \mathcal{N}_{i}} W_{i, j}[f(j)-f(i)]^{2}\right]^{\frac{p}{2}}\quad \quad \quad \quad(5)$

  When  $p=1$, $S_{1}(\mathbf{f})$  is the total variation of the signal with respect to the graph. When  $p=2$ , we have

    $\begin{aligned}S_{2}(\mathbf{f}) &=\frac{1}{2} \sum\limits _{i \in V} \sum\limits_{j \in \mathcal{N}_{i}} W_{i, j}[f(j)-f(i)]^{2} \\&=\sum\limits_{(i, j) \in \mathcal{E}} W_{i, j}[f(j)-f(i)]^{2}\\&=\mathbf{f}^{\mathrm{T}} \text { Lf }\end{aligned}\quad \quad \quad \quad(6)$

 拉普拉斯算子

    $\begin{aligned}\Delta f(x) &=\frac{\partial^{2} f}{\partial x^{2}} \\&=f^{\prime \prime}(x) \\& \approx f^{\prime}(x)-f^{\prime}(x-1) \\& \approx[f(x+1)-f(x)]-[f(x)-f(x-1)] \\&=f(x+1)+f(x-1)-2 f(x)\end{aligned}$

图的拉普拉斯算子

    $\Delta f_{i}=\sum \limits _{j \in N_{i}}\left(f_{i}-f_{j}\right)$

  而如果边 $E_{i j}$ 具有权重 $W_{i j}$ 时,则有:

    $\Delta f_{i}=\sum\limits_{j \in N} W_{i j}\left(f_{i}-f_{j}\right)$

  对于任意向量  $f$,有:

    $\begin{aligned}f^{T} L f &=f^{T} D f-f^{T} W f \\&=\sum\limits_{i=1}^{N} d_{i} f_{i}^{2}-\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N} w_{i j} f_{i} f_{j} \\&=\frac{1}{2}\left(\sum\limits_{i=1}^{N} d_{i} f_{i}^{2}-2 \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N} w_{i j} f_{i} f_{j}+\sum\limits_{j=1}^{N} d_{j} f_{j}^{2}\right) \\&=\frac{1}{2}\left(\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N} w_{i j} f_{i}^{2}-2 \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N} w_{i j} f_{i} f_{j}+\sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N} w_{i j} f_{j}^{2}\right) \\&=\frac{1}{2} \sum\limits_{i=1}^{N} \sum\limits_{j=1}^{N} w_{i j}\left(f_{i}-f_{j}\right)^{2}\end{aligned}$

   $S_{2}(\mathbf{f})$  is known as the graph Laplacian quadratic form and the semi-norm  $\|\mathbf{f}\|_{\text {L }} $ (准范数)is defined as

    $\|\mathbf{f}\|_{L}:=\left\|L^{\frac{1}{2}} \mathbf{f}\right\|_{2}=\sqrt{\mathbf{f}^{\mathrm{T}} L \mathbf{f}}=\sqrt{S_{2}(\mathbf{f})}$

   $S_{2}(f)$ is small when the signal $f$ has similar values at neighboring vertices connected by an edge with a large weight.

   现在回到 graph Laplacian eigenvalues 和 eigenvectors :(Courant-Fischer Theorem)可以参考《极大极小定理

    $\lambda_{0}=\underset{\mathbf{f} \in \mathbb{R}^{N} \atop\|\mathbf{R}\|_{2}=1}{min} \;\left\{\mathbf{f}^{\mathrm{T}} L \mathbf{f}\right\}\quad \quad \quad \quad(7)$

  and 

    ${\large \lambda_{l}=\underset{\mathbf{f} \in \mathbb{R}^{N} \atop   \underset{\mathbf{f} \perp \operatorname{span}\left\{\mathbf{u}_{0}, \ldots, \mathbf{u}_{l-1}\right\}}{\|\mathbf{f}\|_{2}=1} }{min}    \left\{\mathbf{f}^{\mathrm{T}} L \mathbf{f}\right\}, l=1,2, \ldots, N-1}\quad \quad \quad \quad(8)$ 

  拉普拉斯矩阵可以定义图傅里叶变换(通过特征向量)以及平滑性的不同表示。并且图的连通性也编码进了拉普拉斯矩阵。 Example 1 展示了 smoothness 和一个图信号的谱内容是如何依赖于图的。

    

  Discussion: 上面的三个图,点同边不同,顶点域的图  the vertex domains 下面的三个图是对应的谱域 graph spectral domains 可以看出,smoothness and graph spectral content of the signa 取决于图的结构的,$G_1$ 的  signal  $f$  最平滑(smoothest),$G_3$  最不平滑。

2.6 Other Graph Matrices

Method1:

  A second popular option is to normalize each weight $W_{i, j}$ by a factor of $\frac{1}{\sqrt{d_{i} d_{j}}}$ .
  Doing so leads to the normalized graph Laplacian, which is defined as

     $\tilde{L}:=\mathbf{D}^{-\frac{1}{2}} \text { LD }^{-\frac{1}{2}}$

  equivalently

    $(\tilde{L}f)(i) = \frac{1}{\sqrt{d_i}} \sum \limits_{j \in \mathcal{N}_i} W_{i,j} \LARGE[\normalsize \frac{f(i)}{\sqrt{d_i}} - \frac{f(j)}{\sqrt{d_j}} \LARGE]$

   The eigenvalues  satisfy 

    $0 = \tilde{\lambda}_0 < \tilde{\lambda}_1 \leq ... \leq \tilde{\lambda}_{\text{max}} \leq 2$

   with $\tilde{\lambda}_{\text{max}} = 2$  if and only if $G$ is bipartite(二部图).

   The normalized and non-normalized graph Laplacians are Generalized graph Laplacians.

  A generalized graph Laplacian of a graph G is any symmetric matrix:

  • 如果这个矩阵中有边连接顶点 $i$ 和顶点 $j$ ,那么这个矩阵的  $entry([i,j])$  是负的,
  • 如果  $i \ne j$  并且  $i$  和顶点  $j$  不相连,那么为  $0$;
  • 如果  $i = j$  ,那么有可能是任何值。

Method2: random walk matrix

    $P:=D^{-1} W$

  $P_{ij}$  describes the probability of going from vertex $i$ to vertex $j$ in one step of a Markov random walk on the graph $G$.

  For connected, aperiodic graphs, each row of $P^{t}$ converges to the stationary distribution of the random walk as $t$ goes to infinity.

   Another type random walk matrix :asymmetric graph Laplacian

    $L_{a}:=\mathbf{I}_{N}-\mathbf{P}$

   Where 

    $\mathbf{I}_{N}$ is the $N \times N$  identity matrix.

  Note that  $L_{a}$  has the same set of eigenvalues as  $\tilde{L}$ , and if  $\tilde{\mathbf{u}}_{l}$  is an eigenvector of  $\tilde{L}$  associated with  $\tilde{\lambda}_{l}$ , then  $\mathbf{D}_{}^{-\frac{1}{2}} \tilde{\mathbf{u}}_{l}$  is an eigenvector of  $L_{a}$  associated with the eigenvalue  $\tilde{\lambda}_{l}$ .

   The normalized graph Laplacian has the nice properties that its spectrum is always contained in the interval [0, 2] and, for bipartite graphs.


3 Generalized Operators For Signals on Graphs

  In this section ,we will review different fundamental operations such as  filtering, translation, modulation, dilation, and downsampling to the graph setting.

3.1  Filtering

3.1.1 Frequency Filtering

  In classical signal processing,

    $\hat{f}_{\text {out }}(\xi)=\hat{f}_{\text {in }}(\xi) \hat{h}(\xi)$

  Where

    $\hat{h}(\cdot)$  is the transfer function of the filter.  

  This frequency filtering will combine the input signal as a linear combination of complex exponentials.其实就是将输入的  signal 使用基函数线性组合来表示。

  The inverse Fourier transform is as following:

    $f_{\text {out }}(t)={\mathcal{F}}^{-1}\left\{\hat{f}_{\text {in }}(\xi) \hat{h}(\xi)\right\}$

  equivalently

    $\begin{array}{l}f_{\text {out }}(t)&=\int_{\mathbb{R}} \hat{f}_{\text {in }}(\xi) \hat{h}(\xi) e^{2 \pi i \xi t} d \xi \quad  \quad\quad \quad\quad\quad\quad\quad(10)  \\&=\int_{\mathbb{R}} f_{i n}(\tau) h(t-\tau) d \tau=:\left(f_{i n} * h\right)(t)\;\;\quad\quad(11)  \end{array}$

Refer to my blog《图神经网络基础一:傅里叶级数与傅里叶变换

  $F(W)$ 是 $f(t) $ 的傅里叶变换

    $F(W)=\int_{-\infty}^{+\infty} f(t) e^{-i W t} \mathrm{~d} t$

  $f(t) $ 是傅里叶变换的逆变换。 

    $f(t)=\frac{1}{2 \pi} \int_{-\infty}^{+\infty} F(W) e^{i W t} \mathrm{~d} W$

     Frequency filtering or graph spectral filtering definition:

    $\hat{f}_{\text {out }}\left(\lambda_{l}\right)=\hat{f}_{\text {in }}\left(\lambda_{l}\right) \hat{h}\left(\lambda_{l}\right)\quad\quad\quad\quad(12)$

     Inverse graph Fourier transform

     $f_{\text {out }}(i)=\sum\limits _{l=0}^{N-1} \hat{f}_{i n}\left(\lambda_{l}\right) \hat{h}\left(\lambda_{l}\right) u_{l}(i)\quad\quad\quad\quad(13)$

Graph Laplacian

$ L=U\left[\begin{array}{cccc}\lambda_{0} & 0 & \cdots & 0 \\0 & \lambda_{1} & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & \lambda_{N-1}\end{array}\right] U^{-1}=U\left[\begin{array}{cccc}\lambda_{0} & 0 & \cdots & 0 \\0 & \lambda_{1} & \cdots & 0 \\\vdots & \vdots & \ddots & \vdots \\0 & 0 & \cdots & \lambda_{N-1}\end{array}\right] U^{T}$

   Graph Laplacian another expression:

    $\hat{h}(\mathbf{L}):=\mathbf{U}\left[\begin{array}{ccc}\hat{h}\left(\lambda_{0}\right) & & \mathbf{0} \\& \ddots & \\\mathbf{0} & & \hat{h}\left(\lambda_{N-1}\right)\end{array}\right] \mathbf{U}^{\mathrm{T}} .$

    $\mathbf{f}_{\text {out }}=\hat{h}(L) \mathbf{f}_{i n}$

  The basic graph spectral filtering in Eq.12 can be used to Gaussian smoothing, bilateral filtering, total variation filtering, anisotropic diffusion, and non-local means filtering.

  Example.2 中比较了 the classical Gaussian filter 与 graph spectral filtering。前者在平滑处理时,会将图片的边缘也 smooth ;而后者没有,这是因为图拉普拉斯矩阵包含了几何结构信息。

  Here,we talk about the common minimize mode (discrete regularization framework):

    $\min _{\mathbf{f}}\left\{\|\mathbf{f}-\mathbf{y}\|_{2}^{2}+\gamma S_{p}(\mathbf{f})\right\}$

  Where

    • $S_{p}(\mathbf{f})$   is the p-Dirichlet form of (5b). 
    • $f$ is the classfier 
    • $y$ is the ground-trueth 

3.1.2 Filtering in the Vertex Domain

   In the vertex domain,the output $f_{out}(i)$ at vertex is :

    $f_{\text {out }}(i)=b_{i, i} f_{\text {in }}(i)+\sum \limits _{j \in \mathcal{N}(i, K)} b_{i, j} f_{\text {in }}(j)\quad\quad\quad\quad(18)$

  It is a linear combination of the components of the input signal at vertices within a K-hop local neighborhood of vertex $i$:

  It also mean a localized linear transform.

  Now,we relate the spectral domain with vertex domain .When Eq.12  frequency filter is a order $K$ polynomial kernal,

    $\hat{h}\left(\lambda_{l}\right)=\sum\limits _{k=0}^{K} a_{k} \lambda_{l}^{k}$

  Where 

     Some constants  $\left\{a_{k}\right\}_{k=0,1, \ldots, K}$

  Attempt it to vertex domain in Eq.13:

    $\begin{aligned}f_{\text {out }}(i) &=\sum\limits_{l=0}^{N-1} \hat{f}_{\text {in }}\left(\lambda_{l}\right) \hat{h}\left(\lambda_{l}\right) u_{l}(i) \\&=\sum\limits_{j=1}^{N} f_{\text {in }}(j) \sum\limits_{k=0}^{K} a_{k} \sum\limits_{l=0}^{N-1} \lambda_{l}^{k} u_{l}^{*}(j) u_{l}(i) \\&=\sum\limits_{j=1}^{N} f_{\text {in }}(j) \sum\limits_{k=0}^{K} a_{k}\left(L^{k}\right)_{i, j}\end{aligned}$

   当 vertex $i$ 和 vertex $j$ 的距离大于 $K$时 ,有

    $\left(\mathrm{L}^{k}\right)_{i, j}=0$

  此时将 Eq18 中的系数写成

    $b_{i, j}:=\sum \limits _{k=d_{\mathcal{G}}(i, j)}^{K} a_{k}\left(L^{k}\right)_{i, j}$

3.2 Convolution

  We can not dirrectly define a convolution product into graph setting($h(t-\tau)$),so we try to replace the  complex exponentials with graph Laplacian eigenvectors.

    $(f * h)(i):=\sum\limits _{l=0}^{N-1} \hat{f}\left(\lambda_{l}\right) \hat{h}\left(\lambda_{l}\right) u_{l}(i)\quad\quad\quad\quad(20)$

3.3 Translation

  在图域中没法直接定义“平移”概念,因此仍需要通过谱域定义平移。
  时域平移可以视为信号与在延时  $t$  上的脉冲  $δ$的卷积结果,因此顶点域的平移  $n$  可以视为信号与在顶点  $n$  上的脉冲  $δ$  的卷积,而  $δ$  的图傅里叶变换即为顶点  $n$  上特征向量之和。

    $\left(T_{n} g\right)(i):=\sqrt{N}\left(g * \delta_{n}\right)(i) \stackrel{20}{=} \sqrt{N} \sum \limits _{l=0}^{N-1} \hat{g}\left(\lambda_{l}\right) u_{l}^{*}(n) u_{l}(i)\quad\quad\quad(21)$

    where

    $\delta_{n}(i)=\left\{\begin{array}{ll}1 & \text { if } i=n \\0 & \text { otherwise }\end{array}\right.\quad\quad\quad(22)$

  但是,我们一般不认为这是“图上的平移”,而是认为这是图谱域上的核(核即前文提到的信号 kernel )操作,要将核  $g(·)$ 平移到顶点 $n$ 上,则需要在核 $g$ 的每一项上乘上对应的特征向量 $u_l(n)$,再反变换回顶点域。其次,系数  $\sqrt{N}$  是为了保证平移算子保持了原信号的均值不变。再次,信号 $g(·)$ 的平滑程度控制了平移后信号在顶点n附近的局部性(localization),即随着顶点 $i$ 与 $n$ 距离增大(最短路径跳数),其幅值下降的程度。最后,广义平移算子并非等距映射,due to the possible localization of the graph Laplacian eigenvectors(谁能解释一下)。

   看不下去了...........心情好了接着看

 

posted @ 2022-01-20 10:26  Learner-  阅读(260)  评论(0编辑  收藏  举报