Abstract
Signal residing on a graph topology, together with the graph, is named graph signal. For example, the social network generates graph signal with users’ attributes as signal and relationships between users as the underlying graph topology. Graph signal processing, as an emerging area, provides useful tools for graph signal. However, if the dimensionality of signal is high and the scale of graph is large, existing methods in graph signal processing may not work efficiently. In this research, I investigate new algorithms for high-dimensional graph signal processing on three problems: dimensionality reduction, graph learning and graph signal filtering. The proposed algorithms are applied to real graph signal data, including brain imaging data and point clouds data. The first part of this thesis discusses a dimensionality reduction algorithm for highdimensional graph signal with non-Gaussian noise. The proposed algorithm is robust to the noise in signal by making use of the graph as side information. Results show that the proposed algorithm outperforms other dimensionality reduction methods on real brain imaging data with complex noise. The second part of this thesis investigates a graph learning and low-rank components estimation algorithm for high-dimensional noisy graph signal. Based on graph smoothness assumption, the proposed algorithm estimates the graph and its low-rank components simultaneously from the signal with non-Gaussian noise. Experimental results indicate that the proposed algorithm outputs more reasonable brain connectivity from real brain imaging data compared with other graph learning methods and better low-rank components compared with other dimensionality reduction methods. The third part of this thesis proposes an efficient graph filter for large scale point clouds. By partitioning the large graph into small subgraphs, the proposed algorithm achieves similar filtering results with less time compared with the original whole graph-based convolution filter on real point clouds figure.