Graph filters have been used to implement distributedly specific linear transformations such as fast consensus or projections onto the low-rank space of graph-bandlimited signals. This is particularly useful in network setting because graph filters can be implemented locally.
Graph signal processing how to#
We investigate how to design graph filters, which are a generalization of classical time-invariant systems, to implement (or at least approximate) a pre-specified linear transformation. ii) Graph Filter Design and Identification More specifically, we study the problem of inducing a known graph signal using as input a graph signal that is nonzero only for a small subset of seeding nodes, which is then percolated (interpolated) across the graph using a graph filter.
Hence, we also study the problem of reconstructing bandlimited graph signals via low-pass filters. Interestingly, and unlike the case for time-signals, the ideal interpolators for graph signals are not low-pass graph filters, generating a decoupling between the processes of sampling and low-pass interpolation for general graphs. Moreover, we also show that the proposed method shares similarities with the classical sampling and interpolation of time-varying signals. We present a new sampling method that accounts for the graph structure, can be run at a single node and only requires access to information of neighboring nodes. Most of the current efforts have been focused on using the value of the signal observed at a subset of nodes to recover the signal in the entire graph. The underlying assumption is that such signals are bandlimited, i.e., they admit a sparse representation in a (frequency) domain which is related to the structure of the graph where they reside. The goal of this project is to investigate the sampling and posterior recovery of signals that are defined in the nodes of a graph. Sampling is a cornerstone problem in classical signal processing. In the past two years, I worked on multiple GSP projects which, for ease of understanding, I further divide into the following four subcategories. Transversal to the particular application, the general goal of GSP is to contribute to the advancement of the understanding of network data by redesigning traditional tools originally conceived to study signals defined on regular domains (such as time-varying signals) and extend them to analyze signals on the more complex graph domain. A plethora of graph-supported signals exist in different engineering and science fields, with examples ranging from gene expression patterns defined on top of gene networks to the spread of epidemics over a social network. This is the matter addressed in the field of graph signal processing, where the notions of, e.g., frequency and linear filtering are extended to signals supported on graphs. In other occasions, the network defines an underlying notion of proximity, but the object of interest is a signal defined on top of the graph, i.e., data associated with the nodes of the network. Sometimes networks have intrinsic value and are themselves the object of study. Graph signal processing (GSP) is an exciting emerging field in which I have been working since my fourth year at UPenn, where I get to combine everything I learned about networks with my knowledge of classical signal processing. I can categorize the work I have done into the following three classes, with networks being the common denominator. Indeed, my whole work has been built around networks and this allowed me to collaborate with people from varying backgrounds from mathematicians and computer scientists to medical doctors and literature professors. My way of achieving this in today’s highly specialized research environment is by studying networks, since these are ubiquitous data structures transversal to multiple fields and areas of expertise. Ideally, I would like to contribute to the understanding of multiple fields of knowledge, including engineering, economics, sociology, and medicine.
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