图卷积网络介绍及进展学习笔记

原版视频：https://www.bilibili.com/video/av39809391/

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Introduction

Definition

Methods

Applications

Problems and progress

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Introduction

• 数学上的卷积定义：

• CNN上的卷积：

  

  • CNN卷积的特点
    • Traslation invariance
    • Weight sharing
• 图所应用的场景（非欧几里得结构）

 欧几里得几何与非欧几里得几何

1、欧几里得几何，简称欧式几何。数学上，欧几里得几何是平面和三维空间中常见的几何，基于点线面假设。

2、非欧几里得几何是指不同于欧几里得几何学的几何体系，一般包含罗氏几何和黎曼几何

• Why not directly use a kernal like CNN?
  • Number of neighbors
  • Order of neighbors
• Question
  • How to define the convolution on non-Euclidean structures and extract features for machine leaning tasks?
  • Spectral vs. Spatial  谱还是空间

Definition

• graph
  • 
  • 代表顶点集
  • 代表边集
• adjacency 邻接矩阵
  • 
  • 代表i, j两点是否相连
• degree matrix 度矩阵
  • 
  • 
• Laplacian
  • 
  •     
  • is eigenvalues of L
  • is called the specturm of L
  • is an orthonormal basis of 

 

• 图卷积网络中的卷积由傅里叶变换转换而来
• Traditional Fourier transform:
  • 
  • 
  • 可看作空间中的基
• 转换过程
  • 
  • The graph Fourier transform of a signal  is defined as:
    • 
    • Its inverse is 
  • convolution:
    • 
  • convolution on graph:
    • 
      •  is element-wise Handamard product
      • 
    • A filter 
  • Then 

Methods

• Then focus on 
• 
• There are n parameters 
• computational complexity is      实际应用中，一个图网络的节点个数n通常是万级
• Question: fewer parameters, lower complexity

 

• 方法：多项式拟合
  • Chebyshev polynomial
    • 
    • 
    • An orthogonal basis for 
•     
• Then    
• complexity , K+1 parameters 

 

• 进一步的
  • Let K=1, it is then linear w.r.t. 
  • recover a rich class of convolutional filter functions by stacking multiple such layers
  • Approximate 
  • Let 
  • 
  • 
  • Renormaliztion: 

We can gereralize this definition to a signal  with C input channels and F filters or feature maps as follows:

Applications

• Graph Semi-supervised Classification
  • A few nodes are labeled while others are not. Assign labels to unlabeled nodes.
  • 
  • Evaluate the cross-entropy error over all labeled examples:
    • 
  • Train W using gradient desecnt
  • 
• Link Prediction
• Community detection
• Traffic Prediction
• Molecular properties

Problems and progress

• Memory requirement
  • 
• -> Stochastic Training
  • 
  • 思路：
    • 随机选取历史信息中的某些信息，选取的信息和历史信息一起生成下一层的信息
  • 

CONTACT INFORMATION

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