[论文解读] Characterization and inference of weighted graph topologies from observations of diffused signals.

Bastien Pasdeloup, Vincent Gripon+3 more|arXiv (Cornell University)|May 9, 2016

Complex Network Analysis Techniques参考文献 46被引用 6

一句话总结

本文提出了一种方法，通过将扩散信号建模为独立初始源并随时间演化，从观测到的扩散信号中推断加权图的拓扑结构。研究表明，可行图空间构成一个凸多面体，并利用线性规划选择稀疏或简单的图，在理想情况下可实现精确恢复，在观测有限时也能保持高精度，该方法在合成数据和音频数据上均得到验证。

ABSTRACT

In the field of signal processing on graphs, a key tool to process signals is the graph Fourier transform. Transforming signals into the spectral domain is accomplished using the basis of eigenvectors of the graph Laplacian matrix. Such a matrix is dependent on the topology of the graph on which the signals are defined. Therefore it is of paramount importance to have a graph available. We consider the problem of inferring the graph topology from observations of signals. In this article we model the observations as being measured after a few steps of diffusing signals that are initially mutually independent and have independent entries. We propose a way to characterize and recover the possible graphs used for this diffusion. We show that the space of feasible matrices on which mutually independent signals can have evolved, based on the observations, is a convex polytope, and we propose two strategies to choose a point in this space as the topology estimate. These strategies enforce the selection of graphs that are, respectively, simple or sparse. Both methods can be formulated as linear programs and hence solved efficiently. Experiments suggest that in the ideal case when we are able to retrieve the exact covariance matrix of the signal, we are able to recover the graph on which the signals were diffused, with no error. We also apply our method to a synthetic case in which a limited number of signals are obtained by diffusion on a known graph, and show that we are able to recover a graph that is very close to the ground truth one, with an error decreasing as the number of observations increases. Finally, we illustrate our method on audio signals, and show that we are able to recover a graph that is globally one-dimensional.

研究动机与目标

• 解决从经历扩散过程的观测信号中重建底层图拓扑的挑战。
• 将信号观测建模为来自初始独立源的若干步扩散结果。
• 表征与观测信号协方差一致的可行图拉普拉斯矩阵集合。
• 开发高效的推理策略，从可行集中选择拓扑估计，优先考虑稀疏性或简洁性。
• 在合成数据和真实音频信号上验证该方法，展示随着观测数量增加，恢复精度显著提升。

提出的方法

• 将观测信号建模为来自具有独立同分布条目的相互独立初始信号的若干步扩散结果。
• 将问题表述为识别图拉普拉斯矩阵，使得观测到的信号协方差与扩散过程相匹配。
• 证明与观测协方差一致的可行拉普拉斯矩阵集合构成一个凸多面体。
• 提出两种线性规划公式：一种用于选择最稀疏的可行图，另一种用于选择最简洁的图（例如，边权最小化）。
• 利用图拉普拉斯矩阵的特征分解定义图傅里叶变换，并将其与扩散信号的谱域表示关联。
• 通过利用可行解的多面体结构，高效地使用凸优化求解推理问题。

实验结果

研究问题

• RQ1 在扩散过程中，哪些图拓扑可能生成观测到的信号协方差？
• RQ2 我们能否高效地从可行集中识别出一个反映稀疏性或简洁性等结构先验的单一图拓扑？
• RQ3 当信号协方差已知精确时，我们能多准确地恢复真实图？
• RQ4 在有限样本设置下，随着观测到的扩散信号数量增加，恢复精度如何提升？
• RQ5 该方法能否从真实世界信号（如音频）中恢复出有意义且可解释的图结构？

主要发现

• 在已知信号协方差矩阵精确值的理想情况下，该方法可零误差地恢复真实底层图拓扑。
• 在有限数量的观测扩散信号下，随着观测数量增加，恢复误差减小，表现出一致性。
• 将该方法应用于音频信号时，成功恢复出全局一维的图结构，与音频的时间特性相符。
• 所提出的用于稀疏与简洁图选择的线性规划公式计算高效，且能生成高质量的拓扑估计。
• 可行图集合被证明为一个凸多面体，从而支持稳健且结构化的推理。
• 在合成数据上的实证结果表明，估计的图非常接近真实图，且在样本量更高时性能进一步提升。