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- Type of Document: Ph.D. Dissertation
- Language: Farsi
- Document No: 57926 (05)
- University: Sharif University of Technology
- Department: Electrical Engineering
- Advisor(s): Marvasti, Farokh; Amini, Arash
- Abstract:
- Graphical models are widely used in signal processing and machine learning, with Graph Signal Processing (GSP) becoming a trending research area. This branch of signal processing focuses on modeling and analyzing signals defined on networked structures. Graphs act as mathematical representations for data across various applications, from social networks to communication, sensor, and brain networks. Effective use of advanced GSP tools requires learning a suitable graph model for data representation. Undirected graphs typically depict similarity or mutual correlation between signal elements, while directed graphs reveal causal or dependency relationships, such as the temporal dependency of one component of a random process on its previous time components. Most existing methods for learning these structures rely on complete data statistics. However, many real-world applications involve incomplete data due to measurement limitations, leading to missing samples, outliers, or noisy observations. Furthermore, most existing approaches for learning graphs from data assume a Gaussian distribution, which limits their effectiveness for non-Gaussian, particularly heavy-tailed, distributions that are prone to outliers. Such distributions appear in a range of applications, including data analysis and processing in financial markets. This dissertation seeks to establish methods for learning graph structured models from incomplete data, focusing on robustness against noise and missing samples, as well as adaptability for heavy-tailed distributions. We first address the problem of learning static graph structures from incomplete data, specifically for time-varying or spatiotemporal graph signals. A model is introduced that, unlike traditional methods, incorporates both undirected and directed graphs to simultaneously capture spatial and temporal dependencies. Additionally, we propose a learning approach for this graph model that adapts to heavy-tailed distributions. Beyond signal processing, certain tasks in data mining and machine learning, such as data clustering, also benefit from graph-based representations. In this dissertation, we discuss graph representation-based approaches to data clustering and propose a method for clustering graph structures from incomplete data, which is robust against non-Gaussian data distributions. While the aforementioned graph models are effective, they assume static structures, whereas many real-world networks, such as social and financial networks, evolve over time. Discovering a graph model for data defined on such dynamic, time-varying networks is known as time-varying graph learning. Existing methods in this area lack robustness against noise, missing samples, outliers, and non-Gaussian distributions. This dissertation thus extends to learning time-varying graphs from incomplete data and proposes a method compatible with heavy-tailed distributions, such as those commonly encountered in financial markets. The proposed method is also capable of learning graphs with specific structures, making it applicable for data clustering tasks, which can subsequently provide valuable insights and aid in the design of investment strategies in financial markets
- Keywords:
- Graph Learning ; Graph Signal Processing ; Vector Autoregressive Model ; Time-Varying Graph ; Heavy-Tailed Distribution ; Graph-Based Clustering ; Gaussian Markov Random Field ; Incomplete Data
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