Learning with graph structured data
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Singh, Rahul
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Abstract
Graphs provide a natural way to represent information in structured form. A graph is a data structure describing a collection of entities, represented as nodes, and their pairwise relationships, represented as edges. When the entities in a graph are random variables, its gives rise to probabilistic graphical models (PGMs). PGMs provide a natural framework for the representation of complex systems and offer straightforward abstraction for the interactions within the systems. Reasoning with help of probabilistic graphical models allows us to answer inference queries with uncertainty following the framework of probability theory. General inference tasks can be to compute marginal probabilities, conditional probabilities of states of a system. Apart from the inference tasks in PGMs, another fundamental problem is learning the parameters of a candidate graphical model by extracting information from empirical observations.
Traditional methods in PGMs are concerned with the structured data generated with known individual’s association. When the structured data is generated by a large population of individuals with unknown individual’s association, it gives rise to collective graphical models (CGMs). Learning and inference from large population is a difficult task and the lack of individual measurement makes it even more challenging. We address the inference problems from aggregate data via its connections to multimarginal optimal transport theory and propose convergent algorithms for aggregate inference and learning. We further specialize our methods to simple yet popular hidden Markov models (HMMs) and Gaussian HMMs.
Another important problem with graph structured data is learning graph embeddings which has a vast variety of application domains including bioinformatics, social networks and recommendation systems. The standard deep learning techniques such as recurrent neural networks (RNNs) or convolutional neural networks (CNNs) cannot generalize to arbitrary graph structure. Recently, graph neural networks (GNNs) have been proposed to alleviate the limitations, however, in its current state it is far from being mature in both theory and applications. The existing GNNs methods cannot be directly applied to signed graphs (with positive as well as negative edges) due to computational irregularities. To this end, we propose spectral signed graph neural network designs for learning node embeddings for signed graphs. Furthermore, we introduce signed Magnetic Laplacian for spectral analysis of directed signed graphs and use it to propose new spectral GNN designs applicable to directed signed graphs.
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Date
2023-04-25
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Dissertation