Abstract
Modern data analysis tasks typically involve large sets of structured data (graph signals) that reside on real-world networks such as social networks, biological gene networks, and transportation networks. Graphs are appealing mathematical tools that are used in various fields to represent, process, visualize, and analyze structured data effectively. In many practical cases, data samples are unorganized, and the underlying graph topology (representing the relationship between data entities) might (often) not be explicitly available. It is thus desirable to identify or learn a graph topology from data. Typically, graph learning is an ill-posed problem since multiple solutions may exist associating a graph with the data. To address the key challenge of defining a criterion to relate data to the sought graph topology, this thesis goes beyond the well-studied signal smoothness or graph sparsity priors to consider the exploitation of graph topological properties to improve graph learning performance. Several types of graph topological patterns are studied, and we show how constraints can be imposed directly on the learned graphs so as to promote certain topology properties that are desirable for some applications. First, inspired by a specific application domain (e.g., community detection), we develop a novel method that can learn a graph with overlapping community structure. Our method encompasses and leverages the community structure information, along with attributes such as sparsity and signal smoothness to capture the intrinsic relationships between data entities, such that the estimated graph can optimally fit the data. Besides, the learned graph can be used to further improve the detection of network communities. As a result, our algorithm can not only learn the graph with a good data fit, but also detect the underlying network communities with a high quality. Next, we approach the problem of learning multiple graphs from heterogeneous graph signals. In such a problem, a widely employed assumption is that of a simple common component shared among multiple graphs. However, in practice, a more intricate topological pattern, comprising simultaneously of sparse, homogeneity and heterogeneity components, would exhibit in multiple graphs. To this end, we propose a general graph estimator based on a novel structured fusion regularization that enables us to jointly learn multiple graph Laplacian matrices with such complex topological properties, and enjoys both high computational efficiency and rigorous theoretical guarantee. Moreover, in the proposed regularization term, the topological pattern among graphs is characterized by a Gram matrix, which allows us to model various types of topological patterns in a flexible way. Third, we build on the setting in the second approach and study in general the problem of learning multiple graphs from unlabeled heterogeneous graph signals. Specifically, we develop a novel framework for learning mixtures of graphs from unlabeled heterogeneous graph signals, called GRACE. The GRACE unifies the graph signal clustering and multiple graph estimation into one optimization framework. Thoroughly statistical analysis shows that the graph topological properties can help improve the algorithm convergence rate and graph estimation accuracy. Theoretical guarantees and comprehensive experimental results for the proposed methods are presented, which shows that the exploitation of topological properties in graph learning makes it possible to capture complex and non-typical behavior of graph signals that cannot be explicitly handled just by observed data.