In the following sections, we provide additional details with respect to various elements of the paper which could not be fully expanded upon in the main paper. This begins with an in depth explanation of the proposed dataset, including its exact contents, and the manner in which they were produced. Finally, a comprehensive examination of the prediction of vision charts is provided, again with detailed explanationsofarchitectures,experimentalprocedures,hyper-parameters,andadditionalresults. This includes both the methods by which each component of the dataset was produced,anditsexactcontents. These are CAD objects andsopossess geometry andtexture information.

An individual vertex is fully identified through its feature, which works as the vertex attribute. 's neighborhood, including both the internal connectivity information and contained vertex features Here we show more details about the graph datasets used in our experiments of both graph classification and vertex classification. Note that, three social network datasets, IMDB-B, IMDB-M and COLLAB do not provide specific vertex features, where the vertex dimension is denoted as 1 and the maximum vertex degrees are shown in addition. The detailed information of graph datasets used in the experiments of vertex classificationDataset Cora Citeseer Pubmed # V ertices 2708 3327 19717 # Edges 5429 4732 44338 # Classes 7 6 3 V ertex Dimension 1433 3703 500 # Train V ertices (full-sup.) The classification results are illustrated in Figure 1.

Point signatures based on the Laplacian operators on graphs, point clouds, and manifolds have become popular tools in machine learning for graphs, clustering, and shape analysis. In this work, we propose a novel point signature, the power spectrum signature, a measure on $\mathbb{R}$ defined as the squared graph Fourier transform of a graph signal. Unlike eigenvectors of the Laplacian from which it is derived, the power spectrum signature is invariant under graph automorphisms. We show that the power spectrum signature is stable under perturbations of the input graph with respect to the Wasserstein metric. We focus on the signature applied to classes of indicator functions, and its applications to generating descriptive features for vertices of graphs. To demonstrate the practical value of our signature, we showcase several applications in characterizing geometry and symmetries in point cloud data, and graph regression problems.

Hypergraph neural networks (HGNNs) effectively model complex high-order relationships in domains like protein interactions and social networks by connecting multiple vertices through hyperedges, enhancing modeling capabilities, and reducing information loss. Developing foundation models for hypergraphs is challenging due to their distinct data, which includes both vertex features and intricate structural information. We present Hyper-FM, a Hypergraph Foundation Model for multi-domain knowledge extraction, featuring Hierarchical High-Order Neighbor Guided Vertex Knowledge Embedding for vertex feature representation and Hierarchical Multi-Hypergraph Guided Structural Knowledge Extraction for structural information. Additionally, we curate 10 text-attributed hypergraph datasets to advance research between HGNNs and LLMs. Experiments on these datasets show that Hyper-FM outperforms baseline methods by approximately 13.3\%, validating our approach. Furthermore, we propose the first scaling law for hypergraph foundation models, demonstrating that increasing domain diversity significantly enhances performance, unlike merely augmenting vertex and hyperedge counts. This underscores the critical role of domain diversity in scaling hypergraph models.

SLAM is a foundational technique with broad applications in robotics and AR/VR. SLAM simulations evaluate new concepts, but testing on resource-constrained devices, such as VR HMDs, faces challenges: high computational cost and restricted sensor data access. This work proposes a sparse framework using mesh geometry projections as features, which improves efficiency and circumvents direct sensor data access, advancing SLAM research as we demonstrate in VR and through numerical evaluation.

Spatio-Temporal graph convolutional networks were originally introduced with CNNs as temporal blocks for feature extraction. Since then LSTM temporal blocks have been proposed and shown to have promising results. We propose a novel architecture combining both CNN and LSTM temporal blocks and then provide an empirical comparison between our new and the pre-existing models. We provide theoretical arguments for the different temporal blocks and use a multitude of tests across different datasets to assess our hypotheses.

Influence maximization is key topic in data mining, with broad applications in social network analysis and viral marketing. In recent years, researchers have increasingly turned to machine learning techniques to address this problem. They have developed methods to learn the underlying diffusion processes in a data-driven manner, which enhances the generalizability of the solution, and have designed optimization objectives to identify the optimal seed set. Nonetheless, two fundamental gaps remain unsolved: (1) Graph Neural Networks (GNNs) are increasingly used to learn diffusion models, but in their traditional form, they often fail to capture the complex dynamics of influence diffusion, (2) Designing optimization objectives is challenging due to combinatorial explosion when solving this problem. To address these challenges, we propose a novel framework, DeepSN. Our framework employs sheaf neural diffusion to learn diverse influence patterns in a data-driven, end-to-end manner, providing enhanced separability in capturing diffusion characteristics. We also propose an optimization technique that accounts for overlapping influence between vertices, which helps to reduce the search space and identify the optimal seed set effectively and efficiently. Finally, we conduct extensive experiments on both synthetic and real-world datasets to demonstrate the effectiveness of our framework.

The paper proposes to learn graph representations from visual data via graph convolutional unit (GCU). It transforms a 2D feature maps extracted from a neural network to a sample-dependent graph, where pixels with similar features form a vertex and edges measure affinity of vertices in a feature space. Then graph convolutions are applied to pass information along the edges of the graph and update the vertex features. Finally, the updated vertex features are projected back to 2D grids based on the pixel-to-vertex assignment. GCU can be integrated into existing networks allowing end-to-end training and capturing long-range dependencies among regions (vertices).

The rapid advancement of medical technology has led to an exponential increase in multi-modal medical data, including imaging, genomics, and electronic health records (EHRs). Graph neural networks (GNNs) have been widely used to represent this data due to their prominent performance in capturing pairwise relationships. However, the heterogeneity and complexity of multi-modal medical data still pose significant challenges for standard GNNs, which struggle with learning higher-order, non-pairwise relationships. This paper proposes GAMMA-PD (Graph-based Analysis of Multi-modal Motor Impairment Assessments in Parkinson's Disease), a novel heterogeneous hypergraph fusion framework for multi-modal clinical data analysis. GAMMA-PD integrates imaging and non-imaging data into a "hypernetwork" (patient population graph) by preserving higher-order information and similarity between patient profiles and symptom subtypes. We also design a feature-based attention-weighted mechanism to interpret feature-level contributions towards downstream decision tasks. We evaluate our approach with clinical data from the Parkinson's Progression Markers Initiative (PPMI) and a private dataset. We demonstrate gains in predicting motor impairment symptoms in Parkinson's disease. Our end-to-end framework also learns associations between subsets of patient characteristics to generate clinically relevant explanations for disease and symptom profiles. The source code is available at https://github.com/favour-nerrise/GAMMA-PD.