Structure in modern data and how to exploit it: some signal processing applications
Creators: name: Lodhi, Muhammad Asad
Publisher: name: No Publisher Supplied
Year: 2020

description: Modern applications in real-world scenarios generate data that are massive and often times highly structured. Exploiting this structure in an effective manner leads to improved performance, and reduced computational and memory complexities. Moreover, successful exploitation of this underlying structure also admits efficient data representation, superior inference capabilities, and scalable estimation with fewer samples. This dissertation investigates these advantages of structure exploitation in three applications: $(i)$ signal detection and classification under the union-of-subspaces model, $(ii)$ learning product graphs underlying smooth graph signals, and $(iii)$ distributed radar imaging under position errors and unsynchronized clocks. For detection under the union-of-subspaces model we derive the generalized likelihood ratio tests and bounds on the recovery performance under varying levels of knowledge about colored noise in the observations. We also make explicit the dependence of the performance metrics on the geometry of the subspaces comprising the union and of the colored noise. We validate the theoretical insights through numerical experiments on synthetic and real data.In regards to the product graph learning problem, we devise a method to learn structured graphs from data that are given in the form of product graphs. Product graphs arise naturally in many real-world datasets and provide an efficient and compact representation of large-scale graphs through several smaller factor graphs. We initially pose the graph learning problem as a linear program, which (on average) outperforms the state-of-the-art graph learning algorithms. Afterwards, we devise an alternating minimization-based algorithm aimed at learning various types of product graphs from data, and establish local convergence guarantees to the true solution. Finally the superior performance and reduced sample complexity of the proposed algorithm over existing methods are also validated through numerical simulations on synthetic and real datasets.Our final focus is on distributed radar imaging, which is essential for modern radar applications to enable high resolution imaging through a large synthetic aperture. This distributed setup suffers from two commons problems: $(i)$ access to imprecise antenna locations, and $(ii)$ clock mismatch between the distributed components, which adversely affects the final reconstruction of radar scene. We develop exact models to address both of these issues in the most general settings by modeling the errors as convolutions with 1-sparse spatial and temporal shifts. The radar scene reconstruction problems associated with the resulting forward models can then be expressed as nonconvex blind deconvolution problems, which can be solved through a block coordinate descent-based method. At each step of this method, each subproblem is convex and can be solved using accelerated proximal gradient methods like FISTA. Finally, we characterize the theoretical performance of the proposed method by deriving error bounds for the estimated unknowns, and through numerical simulations on synthetic data obtained under varying degrees of noise in the observations.