稀疏图的谱推断方法：理论与应用
Spectral Inference Methods on Sparse Graphs: Theory and Applications
摘要：
在一个前所未有的数据泛滥的时代（大部分是非结构化的），图形被证明越来越有用，在整个科学领域，作为一种灵活的抽象来捕捉复杂对象之间的复杂关系。在这种网络的研究中出现的主要挑战之一是推断宏观，大规模的属性影响了大量的对象，仅基于其基本成分之间的微观相互作用。统计物理学的创立正是为了从相互作用粒子的理想化模型中恢复宏观热力学热力学定律，它为解决这种复杂网络提供了重要的见解。在这篇论文中，我们使用来自无序系统的统计物理学的方法来设计和研究新的图上的推理算法。我们的重点是谱方法，精心选择的矩阵的某些特征向量的基础上，稀疏图，只包含少量的信息。我们开发了一个原始的理论，光谱推断的基础上放松各种平均场自由能优化。因此，我们的方法是完全概率的，并与更传统的动机的基础上优化的成本函数。我们说明了我们的方法在各种问题上的效率，包括社区检测，随机相似性为基础的聚类，矩阵完成。
英文摘要：
In an era of unprecedented deluge of (mostly unstructured) data, graphs are proving more and more useful, across the sciences, as a flexible abstraction to capture complex relationships between complex objects. One of the main challenges arising in the study of such networks is the inference of macroscopic, large-scale properties affecting a large number of objects, based solely on the microscopic interactions between their elementary constituents. Statistical physics, precisely created to recover the macroscopic laws of thermodynamics from an idealized model of interacting particles, provides significant insight to tackle such complex networks. In this dissertation, we use methods derived from the statistical physics of disordered systems to design and study new algorithms for inference on graphs. Our focus is on spectral methods, based on certain eigenvectors of carefully chosen matrices, and sparse graphs, containing only a small amount of information. We develop an original theory of spectral inference based on a relaxation of various mean-field free energy optimizations. Our approach is therefore fully probabilistic, and contrasts with more traditional motivations based on the optimization of a cost function. We illustrate the efficiency of our approach on various problems, including community detection, randomized similarity-based clustering, and matrix completion.
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