Gaussian processes on graphs

In the ever-growing field of machine learning research, the use of graphs has recently gathered significant interest for modelling on data with relational structures. Graphs and network-based data now exist ubiquitously in the real world, with examples including social networks, transportation, financial exchanges, and brain networks. Therefore, developing models on graphs is essential to allow users to understand and predict the complex nature observed in everyday phenomena. Currently, there is an abundance of literature on graph neural networks, but limited options are available that are probabilistic and Bayesian. In addressing this issue, we develop a series of Gaussian processes (GPs) for graph data in this thesis. Building GPs on graphs is now more feasible thanks to the emergence of graph signal processing, providing us with the tools to handle graph-structured information and smoothness modelling. The first problem we tackle is predicting the evolution of signals with a multi-output Gaussian process. We use kernels defined from the graph Laplacian with learnable spectral filters to predict with the smoothness level that matches the data. We then turn our focus to semi-supervised classification, designing three models for this task each emphasizing on a particular approach: multi-scale modelling, transductive learning, and sheaf modelling. The first approach provides a novel utilization of wavelets on graphs to fully exploit their ability to capture multi-scale properties in the data. Next, we present a unified definition of kernels on graphs with transductive properties, aiming to utilize the distribution of the full dataset to better inform the prediction. This naturally suits semi-supervised problems on graphs where training and testing nodes are generally connected and available at the same time. Finally, we introduce sheaves as a higher order representation of graphs, to design GPs with stronger separation power by learning additional topological structures. Collectively, this thesis represents not only a valuable contribution to the study of GPs for discrete and non-Euclidean data, but also useful alternatives to the more broadly used graph neural networks

Combining Subgoal Graphs with Reinforcement Learning to Build a Rational Pathfinder

In this paper, we present a hierarchical path planning framework called SG-RL (subgoal graphs-reinforcement learning), to plan rational paths for agents maneuvering in continuous and uncertain environments. By "rational", we mean (1) efficient path planning to eliminate first-move lags; (2) collision-free and smooth for agents with kinematic constraints satisfied. SG-RL works in a two-level manner. At the first level, SG-RL uses a geometric path-planning method, i.e., Simple Subgoal Graphs (SSG), to efficiently find optimal abstract paths, also called subgoal sequences. At the second level, SG-RL uses an RL method, i.e., Least-Squares Policy Iteration (LSPI), to learn near-optimal motion-planning policies which can generate kinematically feasible and collision-free trajectories between adjacent subgoals. The first advantage of the proposed method is that SSG can solve the limitations of sparse reward and local minima trap for RL agents; thus, LSPI can be used to generate paths in complex environments. The second advantage is that, when the environment changes slightly (i.e., unexpected obstacles appearing), SG-RL does not need to reconstruct subgoal graphs and replan subgoal sequences using SSG, since LSPI can deal with uncertainties by exploiting its generalization ability to handle changes in environments. Simulation experiments in representative scenarios demonstrate that, compared with existing methods, SG-RL can work well on large-scale maps with relatively low action-switching frequencies and shorter path lengths, and SG-RL can deal with small changes in environments. We further demonstrate that the design of reward functions and the types of training environments are important factors for learning feasible policies.Comment: 20 page

Gaussian processes on graphs via spectral kernel learning

We propose a graph spectrum-based Gaussian process for prediction of signals defined on nodes of the graph. The model is designed to capture various graph signal structures through a highly adaptive kernel that incorporates a flexible polynomial function in the graph spectral domain. Unlike most existing approaches, we propose to learn such a spectral kernel defined on a discrete space. In addition, this kernel has the interpretability of graph filtering achieved by a bespoke maximum likelihood learning algorithm that enforces the positivity of the spectrum. We demonstrate the interpretability of the model through synthetic experiments from which we show various ground truth spectral filters can be accurately recovered, and the adaptability translates to improved predictive performances compared to the baselines on real-world graph data of various characteristics

Factors Influencing Purchase Intention on Mobile Shopping Web Site in China and South Korea: An Empirical Study

The research objective of this study is to analyze the factors that influence consumers\u27 perceptions of using mobile commerce services for online shopping in China and South Korea using ordered logistic regression analysis. We constructed the research model using the three dimensions of individual characteristics, shopping motivations and the characteristics of mobile shopping. We discovered that shopping frequency, utilitarianism, instant connectivity, and personalized information push positively impact the customers’ intention to use mobile phones in China. The results of the marginal effects indicated that the behavioral intentions of Chinese consumers increased when shopping frequency and instant connectivity increased. In addition, when utilitarianism and the personalized information push reach certain values, the shopping intention of online customers in China will decrease. Likewise, shopping frequency, hedonism, utilitarianism, instant connectivity, and SNS (Social Networking Services) accessibility positively affect the intention to use the Internet for m-shopping of South Korean consumers. In addition, the results regarding the marginal effects suggested that the intention to use m-shopping services on m-shopping web site of South Korean consumers increased as shopping frequency, hedonism, and instant connectivity increased. However, South Korean consumers\u27 adoption intention will decrease when utilitarianism and SNS accessibility reach certain values. These results provide important implications for mobile commerce literature and practice

Graph classification Gaussian processes via spectral features

Graph classification aims to categorise graphs based on their structure and node attributes. In this work, we propose to tackle this task using tools from graph signal processing by deriving spectral features, which we then use to design two variants of Gaussian process models for graph classification. The first variant uses spectral features based on the distribution of energy of a node feature signal over the spectrum of the graph. We show that even such a simple approach, having no learned parameters, can yield competitive performance compared to strong neural network and graph kernel baselines. A second, more sophisticated variant is designed to capture multi-scale and localised patterns in the graph by learning spectral graph wavelet filters, obtaining improved performance on synthetic and real-world data sets. Finally, we show that both models produce well calibrated uncertainty estimates, enabling reliable decision making based on the model predictions