Multi-View Signal Processing and Learning on Graphs

In modern information processing tasks we often need to deal with complex and “multi-view” data, that come from multiple sources of information with structures behind them. For example, observations about the same set of entities are often made from different angles, or, individual observations collected by different sensors are intrinsically related. This poses new challenges in developing novel signal processing and machine learning techniques to handle such data efficiently. In this thesis, we formulate the processing and analysis of multi-view data as signal processing and machine learning problems defined on graphs. Graphs are appealing mathematical tools for modeling pairwise relationships between entities in datasets. Moreover, they are flexible and adaptable to incorporate multiple sources of information with structures. For instance, multiple types of relationships between the same entities in a certain set can be modeled as a multi-layer graph, where the layers share the same set of vertices (entities) but have different edges (relationships) between them. Alternatively, information from one source can be modeled as signals defined on the vertex set of a graph that corresponds to another source. While the former setting is an extension of the classical graph-based learning in machine learning, the latter leads to the emerging research field of signal processing on graphs. In this thesis, we bridge the gap between the two fields by studying several problems related to the clustering, classification and representation of data of various forms associated with weighted and undirected graphs. First, we address the problem of analyzing multi-layer graphs and propose methods for clustering the vertices by efficiently combining the information provided by the multiple layers. To this end, we propose to combine the characteristics of individual graph layers using tools from subspace analysis on a Grassmann manifold. The resulting combination can be viewed as a low dimensional representation of the original data, which preserves the most important information from diverse relationships between entities. We use our algorithm in clustering methods and demonstrate that the proposed method is superior to baseline schemes and competitive to state-of-the-art techniques. Next, we approach the problem of combining different layers in a multi-layer graph from a different perspective. Specifically, we consider the eigenvectors of the graph Laplacian matrix of one layer as signals defined on the vertex set of another layer. We propose a novel method based on a graph regularization framework, which produces a set of “joint eigenvectors”, or a “joint spectrum”, shared by the two layers. We use this joint spectrum in clustering problems on multilayer graphs. Compared to our previous approach and most of the state-of-the-art techniques, a unique characteristic and potential advantage of this method is that, it allows us to combine individual layers based on their respective importance in a convincing way. Third, we build on the setting in the second approach and study in general the classification problem of signals on graphs. To this end, we adopt efficient signal representations defined in