Advances in graph signal processing: Graph filtering and network identification

To the surprise of most of us, complexity in nature spawns from simplicity. No matter how simple a basic unit is, when many of them work together, the interactions among these units lead to complexity. This complexity is present in the spreading of diseases, where slightly different policies, or conditions,might lead to very different results; or in biological systems where the interactions between elements maintain the delicate balance that keep life running. Fortunately, despite their complexity, current advances in technology have allowed us to have more than just a sneak-peak at these systems. With new views on how to observe such systems and gather data, we aimto understand the complexity within.
One of these new views comes from the field of graph signal processing which provides models and tools to understand and process data coming from such complex systems. With a principled view, coming from its signal processing background, graph signal processing establishes the basis for addressing problems involving data defined over interconnected systems by combining knowledge from graph and network theory with signal processing tools. In this thesis, our goal is to advance the current state-of-the-art by studying the processing of network data using graph filters, the workhorse of graph signal processing, and by proposing methods for identifying the topology (interactions) of a network from network measurements.
To extend the capabilities of current graph filters, the network-domain counterparts of time-domain filters, we introduce a generalization of graph filters. This new family of filters does not only provide more flexibility in terms of processing networked data distributively but also reduces the communications in typical network applications, such as distributed consensus or beamforming. Furthermore, we theoretically characterize these generalized graph filters and also propose a practical and numerically-amenable cascaded implementation.
As allmethods in graph signal processingmake use of the structure of the network, we require to know the topology. Therefore, identifying the network interconnections from networked data is much needed for appropriately processing this data. In this thesis, we pose the network topology identification problem through the lens of system identification and study the effect of collecting information only from part of the elements of the network. We show that by using the state-space formalism, algebraic methods can be applied to the network identification problem successfully. Further, we demonstrate that for the partially-observable case, although ambiguities arise, we can still retrieve a coherent network topology leveraging state-of-the-art optimization techniques.