On the Analysis of Non-euclidean data: Sparsification, Classification and Generation

In light of the groundbreaking achievements of convolutional neural networks (CNNs) in 2D image processing, there has been a pronounced interest in adapting these methods to non-Euclidean data, such as graphs and 3D geometric data. Point clouds, in particular, present unique challenges as they are sparse, unordered, and locality-sensitive, making the adaptation of CNNs to point cloud processing a non-trivial task. Similar challenges are encountered in the context of graph data. Consequently, the exploration of extending successful neural processing paradigms from 2D images to these non-Euclidean domains has emerged as a vibrant and dynamic research area. This thesis focuses on advancing graph neural networks (GNNs) and analyzing 3D point clouds, emphasizing sparsification, classification and generation. For graph neural networks, a significant contribution is the introduction of Sparse Graph Attention Networks (SGAT), integrating a sparse attention mechanism into graph attention networks (GATs) through $L_0$-norm regularization. SGAT excels in edge removal (50\%-80\% on large graphs), enhancing interpretability without compromising performance on assortative graphs and improving it on disassortative graphs. In 3D point cloud analysis, an autoregressive approach, APSNet, formulates task-oriented point cloud sampling as a sequential generation process, and develops an attention-based point cloud sampling network that optimally samples 8 points out of 1024, tailoring the process for tasks like 3D point cloud classification, reconstruction, and registration. Extending into a non-autoregressive method, PTSNet, a point transformer, utilizes a transformer-based dynamic query generator. This innovation enables PTSNet to capture long-range correlations, mitigating issues like gradient vanishing and reducing duplicate samples compared to LSTM-based methods. Lastly, the thesis proposes GDPNet, first hybrid Generative and Discriminative PointNet, extending the Joint Energy-based Model (JEM) for point cloud generation and classification. GDPNet retains strong discriminative power of modern PointNet classifiers, while generating point cloud samples rivaling state-of-the-art generative approaches