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Structural and Positional Encodings can significantly improve the performance of Graph Neural Networks in downstream tasks. Recent literature has begun to systematically investigate differences in the structural properties that these approaches encode, as well as performance trade-offs between them. However, the question of which structural properties yield the most effective encoding remains open. In this paper, we investigate this question from a geometric perspective. We propose a novel structural encoding based on discrete Ricci curvature (Local Curvature Profiles, short LCP) and show that it significantly outperforms existing encoding approaches. We further show that combining local structural encodings, such as LCP, with global positional encodings improves downstream performance, suggesting that they capture complementary geometric information. Finally, we compare different encoding types with (curvature-based) rewiring techniques. Rewiring has recently received a surge of interest due to its ability to improve the performance of Graph Neural Networks by mitigating over-smoothing and over-squashing effects. Our results suggest that utilizing curvature information for structural encodings delivers significantly larger performance increases than rewiring.

Prompting methods play a crucial role in enhancing the capabilities of pre-trained large language models (LLMs). We explore how contrastive prompting (CP) significantly improves the ability of large language models to perform complex reasoning. We demonstrate that LLMs are decent contrastive reasoners by simply adding "Let's give a correct and a wrong answer." before LLMs provide answers. Experiments on two large language models show that zero-shot contrastive prompting improves performance on a range of arithmetic, commonsense, and symbolic reasoning tasks without any hand-crafted few-shot examples, such as increasing the accuracy on GSM8K from 35.9% to 88.8% and AQUA-RAT from 41.3% to 62.2% with the state-of-the-art GPT-4 model. Our method not only surpasses zero-shot CoT and few-shot CoT in most arithmetic and commonsense reasoning tasks but also can seamlessly integrate with existing prompting methods, resulting in improved or comparable results when compared to state-of-the-art methods. Our code is available at //github.com/yao8839836/cp

We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

This work presents a unified approach for collision avoidance using Collision-Cone Control Barrier Functions (CBFs) in both ground (UGV) and aerial (UAV) unmanned vehicles. We propose a novel CBF formulation inspired by collision cones, to ensure safety by constraining the relative velocity between the vehicle and the obstacle to always point away from each other. The efficacy of this approach is demonstrated through simulations and hardware implementations on the TurtleBot, Stoch-Jeep, and Crazyflie 2.1 quadrotor robot, showcasing its effectiveness in avoiding collisions with dynamic obstacles in both ground and aerial settings. The real-time controller is developed using CBF Quadratic Programs (CBF-QPs). Comparative analysis with the state-of-the-art CBFs highlights the less conservative nature of the proposed approach. Overall, this research contributes to a novel control formation that can give a guarantee for collision avoidance in unmanned vehicles by modifying the control inputs from existing path-planning controllers.

2D-based Industrial Anomaly Detection has been widely discussed, however, multimodal industrial anomaly detection based on 3D point clouds and RGB images still has many untouched fields. Existing multimodal industrial anomaly detection methods directly concatenate the multimodal features, which leads to a strong disturbance between features and harms the detection performance. In this paper, we propose Multi-3D-Memory (M3DM), a novel multimodal anomaly detection method with hybrid fusion scheme: firstly, we design an unsupervised feature fusion with patch-wise contrastive learning to encourage the interaction of different modal features; secondly, we use a decision layer fusion with multiple memory banks to avoid loss of information and additional novelty classifiers to make the final decision. We further propose a point feature alignment operation to better align the point cloud and RGB features. Extensive experiments show that our multimodal industrial anomaly detection model outperforms the state-of-the-art (SOTA) methods on both detection and segmentation precision on MVTec-3D AD dataset. Code is available at //github.com/nomewang/M3DM.

Recently, a considerable literature has grown up around the theme of Graph Convolutional Network (GCN). How to effectively leverage the rich structural information in complex graphs, such as knowledge graphs with heterogeneous types of entities and relations, is a primary open challenge in the field. Most GCN methods are either restricted to graphs with a homogeneous type of edges (e.g., citation links only), or focusing on representation learning for nodes only instead of jointly propagating and updating the embeddings of both nodes and edges for target-driven objectives. This paper addresses these limitations by proposing a novel framework, namely the Knowledge Embedding based Graph Convolutional Network (KE-GCN), which combines the power of GCNs in graph-based belief propagation and the strengths of advanced knowledge embedding (a.k.a. knowledge graph embedding) methods, and goes beyond. Our theoretical analysis shows that KE-GCN offers an elegant unification of several well-known GCN methods as specific cases, with a new perspective of graph convolution. Experimental results on benchmark datasets show the advantageous performance of KE-GCN over strong baseline methods in the tasks of knowledge graph alignment and entity classification.

This paper presents a new approach for assembling graph neural networks based on framelet transforms. The latter provides a multi-scale representation for graph-structured data. With the framelet system, we can decompose the graph feature into low-pass and high-pass frequencies as extracted features for network training, which then defines a framelet-based graph convolution. The framelet decomposition naturally induces a graph pooling strategy by aggregating the graph feature into low-pass and high-pass spectra, which considers both the feature values and geometry of the graph data and conserves the total information. The graph neural networks with the proposed framelet convolution and pooling achieve state-of-the-art performance in many types of node and graph prediction tasks. Moreover, we propose shrinkage as a new activation for the framelet convolution, which thresholds the high-frequency information at different scales. Compared to ReLU, shrinkage in framelet convolution improves the graph neural network model in terms of denoising and signal compression: noises in both node and structure can be significantly reduced by accurately cutting off the high-pass coefficients from framelet decomposition, and the signal can be compressed to less than half its original size with the prediction performance well preserved.

Graph Neural Networks (GNNs) draw their strength from explicitly modeling the topological information of structured data. However, existing GNNs suffer from limited capability in capturing the hierarchical graph representation which plays an important role in graph classification. In this paper, we innovatively propose hierarchical graph capsule network (HGCN) that can jointly learn node embeddings and extract graph hierarchies. Specifically, disentangled graph capsules are established by identifying heterogeneous factors underlying each node, such that their instantiation parameters represent different properties of the same entity. To learn the hierarchical representation, HGCN characterizes the part-whole relationship between lower-level capsules (part) and higher-level capsules (whole) by explicitly considering the structure information among the parts. Experimental studies demonstrate the effectiveness of HGCN and the contribution of each component.

Label Propagation (LPA) and Graph Convolutional Neural Networks (GCN) are both message passing algorithms on graphs. Both solve the task of node classification but LPA propagates node label information across the edges of the graph, while GCN propagates and transforms node feature information. However, while conceptually similar, theoretical relation between LPA and GCN has not yet been investigated. Here we study the relationship between LPA and GCN in terms of two aspects: (1) feature/label smoothing where we analyze how the feature/label of one node is spread over its neighbors; And, (2) feature/label influence of how much the initial feature/label of one node influences the final feature/label of another node. Based on our theoretical analysis, we propose an end-to-end model that unifies GCN and LPA for node classification. In our unified model, edge weights are learnable, and the LPA serves as regularization to assist the GCN in learning proper edge weights that lead to improved classification performance. Our model can also be seen as learning attention weights based on node labels, which is more task-oriented than existing feature-based attention models. In a number of experiments on real-world graphs, our model shows superiority over state-of-the-art GCN-based methods in terms of node classification accuracy.

We introduce an effective model to overcome the problem of mode collapse when training Generative Adversarial Networks (GAN). Firstly, we propose a new generator objective that finds it better to tackle mode collapse. And, we apply an independent Autoencoders (AE) to constrain the generator and consider its reconstructed samples as "real" samples to slow down the convergence of discriminator that enables to reduce the gradient vanishing problem and stabilize the model. Secondly, from mappings between latent and data spaces provided by AE, we further regularize AE by the relative distance between the latent and data samples to explicitly prevent the generator falling into mode collapse setting. This idea comes when we find a new way to visualize the mode collapse on MNIST dataset. To the best of our knowledge, our method is the first to propose and apply successfully the relative distance of latent and data samples for stabilizing GAN. Thirdly, our proposed model, namely Generative Adversarial Autoencoder Networks (GAAN), is stable and has suffered from neither gradient vanishing nor mode collapse issues, as empirically demonstrated on synthetic, MNIST, MNIST-1K, CelebA and CIFAR-10 datasets. Experimental results show that our method can approximate well multi-modal distribution and achieve better results than state-of-the-art methods on these benchmark datasets. Our model implementation is published here: //github.com/tntrung/gaan