基于信息熵的图信号多尺度节点频率表征及应用研究

Big data generated from networks generally has an underlying stable structure, which is related to the complexity and diversity of the data. Graph signals is a powerful tool for depicting these data. The existing multiscale representation methods for graph signals have disadvantages in information unbalance and low adaptivity. To overcome these problems, this project proposes to investigate the entropies and vertex-frequency properties of graph signals, and then establish a framework of entropy based adaptive multiscale representation method for graph signals. The research topics of the project include: 1) For the information unbalance, we plan to utilize the heavy light decomposition of trees, and construct the unbalance reduction algorithm for weighted graphs. Under the consistent frequency assumption, we devote to construct frequency balanced representation of graph signals. 2) Based on topic 1), we then investigate the properties of graph signal entropies, which are then utilized to construct an adaptive multiscale decomposition method for signals on weighted graphs. 3) We propose to apply the results and methods on network data processing, including graph signal compression, de-noising, and community detection. In conclusion, the project will develop a theory of graph signal entropy, construct an efficient graph signal representation framework, and provide methodologies for the analysis and processing of signals with complex structures.

网络大数据在表现出复杂性和多样性的同时也具有潜在的稳定结构，图信号是描述该结构关系的有力工具。目前图信号多尺度表示方法存在信息失衡与自适应性等问题，课题研究图信号信息熵及其节点频率性质，提出基于信息熵的图信号自适应多尺度表示方法。研究内容包括：1）对于采样失衡和频率失衡等信息失衡问题，首先以最大生成树的重轻分解为突破口，构造加权图上的采样降失衡方法；其次基于频率一致性假设，实现有效的频率均衡，揭示图信号信息失衡的本质；2）在研究图信号的节点频率性质的基础上给出图信号的多尺度节点频率表征，揭示图信号信息熵的本质，进而依据尺度间信息熵的变化，构造图信号的多尺度自适应分解；3）以智能网络大数据应用为背景，将所提出的理论与方法应用于解决网络数据的压缩、去噪以及网络社区发现等问题。课题研究的意义在于拓展图信号信息熵理论，建立高效的图信号表示方法，为解决网络数据分析与处理中的实际问题提供新的工具。

图信号是描述复杂数据及其结构关系的有力工具。课题深入研究图信号的节点频率性质，结合规则域信号表示方法及图卷积网络模型，提出图信号表示新理论与其应用方法。课题针对图信号的节点频率表示方法，提出了多窗图傅里叶变换及多窗图傅里叶框架构造方法，给出基于环分解的图信号滤波器设计方法，提高图信号表征的自适应性；课题同时研究基于框架理论的信号表示方法，构建基于半内积框架的信号表示理论，给出一般信号在框架扰动、信息丢失情况下的重构方法；在此理论基础上，课题围绕图卷积网络、图滤波器、图加权的等方法，研究基于子空间方法及图加权技术的人脸识别模型，设计基于图卷积网络及图滤波器的3D点云分类与生成算法，并开展基于图学习方法的交通流与疫情数据分析。项目研究成果为图信号的表示方法提供理论参考；同时为相关学科的应用研究提供理论支撑与方法借鉴。