Umbilical hypersurfaces of Minkowski spaces

In this paper, by the Gauss equation of the induced Chern connection for Finsler submanifolds, we prove that if M is an umbilical hypersurface of a Minkowski space $(V^{n+1}, overline{F})$, then either M is a Riemannian space form or a locally Minkowski space

The special hypersurfaces of Minkowski space

Let x:(Mn,F) (Vn+1, F̄) be a simply connected hypersurface in a Minkowski space (Vn+1, F̄). In this paper, using the Gauss formula of Chern connection on Finsler submanifolds, we shall prove that if x(p) is normal to Tp(M) (∀ p M), then M with the induced metric is isometric to the standard Euclidean sphere

The Devil is in the Data: Learning Fair Graph Neural Networks via Partial Knowledge Distillation

Graph neural networks (GNNs) are being increasingly used in many high-stakes tasks, and as a result, there is growing attention on their fairness recently. GNNs have been shown to be unfair as they tend to make discriminatory decisions toward certain demographic groups, divided by sensitive attributes such as gender and race. While recent works have been devoted to improving their fairness performance, they often require accessible demographic information. This greatly limits their applicability in real-world scenarios due to legal restrictions. To address this problem, we present a demographic-agnostic method to learn fair GNNs via knowledge distillation, namely FairGKD. Our work is motivated by the empirical observation that training GNNs on partial data (i.e., only node attributes or topology data) can improve their fairness, albeit at the cost of utility. To make a balanced trade-off between fairness and utility performance, we employ a set of fairness experts (i.e., GNNs trained on different partial data) to construct the synthetic teacher, which distills fairer and informative knowledge to guide the learning of the GNN student. Experiments on several benchmark datasets demonstrate that FairGKD, which does not require access to demographic information, significantly improves the fairness of GNNs by a large margin while maintaining their utility.Comment: Accepted by WSDM 202

Scaling Up, Scaling Deep: Blockwise Graph Contrastive Learning

Oversmoothing is a common phenomenon in graph neural networks (GNNs), in which an increase in the network depth leads to a deterioration in their performance. Graph contrastive learning (GCL) is emerging as a promising way of leveraging vast unlabeled graph data. As a marriage between GNNs and contrastive learning, it remains unclear whether GCL inherits the same oversmoothing defect from GNNs. This work undertakes a fundamental analysis of GCL from the perspective of oversmoothing on the first hand. We demonstrate empirically that increasing network depth in GCL also leads to oversmoothing in their deep representations, and surprisingly, the shallow ones. We refer to this phenomenon in GCL as long-range starvation', wherein lower layers in deep networks suffer from degradation due to the lack of sufficient guidance from supervision (e.g., loss computing). Based on our findings, we present BlockGCL, a remarkably simple yet effective blockwise training framework to prevent GCL from notorious oversmoothing. Without bells and whistles, BlockGCL consistently improves robustness and stability for well-established GCL methods with increasing numbers of layers on real-world graph benchmarks. We believe our work will provide insights for future improvements of scalable and deep GCL frameworks.Comment: Preprint; Code is available at https://github.com/EdisonLeeeee/BlockGC

SAILOR: Structural Augmentation Based Tail Node Representation Learning

Graph Neural Networks (GNNs) have achieved state-of-the-art performance in representation learning for graphs recently. However, the effectiveness of GNNs, which capitalize on the key operation of message propagation, highly depends on the quality of the topology structure. Most of the graphs in real-world scenarios follow a long-tailed distribution on their node degrees, that is, a vast majority of the nodes in the graph are tail nodes with only a few connected edges. GNNs produce inferior node representations for tail nodes since they lack structural information. In the pursuit of promoting the expressiveness of GNNs for tail nodes, we explore how the deficiency of structural information deteriorates the performance of tail nodes and propose a general Structural Augmentation based taIL nOde Representation learning framework, dubbed as SAILOR, which can jointly learn to augment the graph structure and extract more informative representations for tail nodes. Extensive experiments on public benchmark datasets demonstrate that SAILOR can significantly improve the tail node representations and outperform the state-of-the-art baselines.Comment: Accepted by CIKM 2023; Code is available at https://github.com/Jie-Re/SAILO

Spectral Adversarial Training for Robust Graph Neural Network

Recent studies demonstrate that Graph Neural Networks (GNNs) are vulnerable to slight but adversarially designed perturbations, known as adversarial examples. To address this issue, robust training methods against adversarial examples have received considerable attention in the literature. \emph{Adversarial Training (AT)} is a successful approach to learning a robust model using adversarially perturbed training samples. Existing AT methods on GNNs typically construct adversarial perturbations in terms of graph structures or node features. However, they are less effective and fraught with challenges on graph data due to the discreteness of graph structure and the relationships between connected examples. In this work, we seek to address these challenges and propose Spectral Adversarial Training (SAT), a simple yet effective adversarial training approach for GNNs. SAT first adopts a low-rank approximation of the graph structure based on spectral decomposition, and then constructs adversarial perturbations in the spectral domain rather than directly manipulating the original graph structure. To investigate its effectiveness, we employ SAT on three widely used GNNs. Experimental results on four public graph datasets demonstrate that SAT significantly improves the robustness of GNNs against adversarial attacks without sacrificing classification accuracy and training efficiency.Comment: Accepted by TKDE. Code availiable at https://github.com/EdisonLeeeee/SA