On the Completeness of Invariant Geometric Deep Learning Models

Invariant models, one important class of geometric deep learning models, are capable of generating meaningful geometric representations by leveraging informative geometric features. These models are characterized by their simplicity, good experimental results and computational efficiency. However, their theoretical expressive power still remains unclear, restricting a deeper understanding of the potential of such models. In this work, we concentrate on characterizing the theoretical expressiveness of invariant models. We first rigorously bound the expressiveness of the most classical invariant model, Vanilla DisGNN (message passing neural networks incorporating distance), restricting its unidentifiable cases to be only those highly symmetric geometric graphs. To break these corner cases' symmetry, we introduce a simple yet E(3)-complete invariant design by nesting Vanilla DisGNN, named GeoNGNN. Leveraging GeoNGNN as a theoretical tool, we for the first time prove the E(3)-completeness of three well-established geometric models: DimeNet, GemNet and SphereNet. Our results fill the gap in the theoretical power of invariant models, contributing to a rigorous and comprehensive understanding of their capabilities. Experimentally, GeoNGNN exhibits good inductive bias in capturing local environments, and achieves competitive results w.r.t. complicated models relying on high-order invariant/equivariant representations while exhibiting significantly faster computational speed.

Latent Graph Diffusion: A Unified Framework for Generation and Prediction on Graphs

In this paper, we propose the first framework that enables solving graph learning tasks of all levels (node, edge and graph) and all types (generation, regression and classification) with one model. We first propose Latent Graph Diffusion (LGD), a generative model that can generate node, edge, and graph-level features of all categories simultaneously. We achieve this goal by embedding the graph structures and features into a latent space leveraging a powerful encoder which can also be decoded, then training a diffusion model in the latent space. LGD is also capable of conditional generation through a specifically designed cross-attention mechanism. Then we formulate prediction tasks including regression and classification as (conditional) generation, which enables our LGD to solve tasks of all levels and all types with provable guarantees. We verify the effectiveness of our framework with extensive experiments, where our models achieve state-of-the-art or highly competitive results across generation and regression tasks.

PyTorch Geometric High Order: A Unified Library for High Order Graph Neural Network

We introduce PyTorch Geometric High Order (PyGHO), a library for High Order Graph Neural Networks (HOGNNs) that extends PyTorch Geometric (PyG). Unlike ordinary Message Passing Neural Networks (MPNNs) that exchange messages between nodes, HOGNNs, encompassing subgraph GNNs and k-WL GNNs, encode node tuples, a method previously lacking a standardized framework and often requiring complex coding. PyGHO's main objective is to provide an unified and user-friendly interface for various HOGNNs. It accomplishes this through streamlined data structures for node tuples, comprehensive data processing utilities, and a flexible suite of operators for high-order GNN methodologies. In this work, we present a detailed in-depth of PyGHO and compare HOGNNs implemented with PyGHO with their official implementation on real-world tasks. PyGHO achieves up to $50\%$ acceleration and reduces the code needed for implementation by an order of magnitude. Our library is available at \url{https://github.com/GraphPKU/PygHO}.

Facilitating Graph Neural Networks with Random Walk on Simplicial Complexes

Node-level random walk has been widely used to improve Graph Neural Networks. However, there is limited attention to random walk on edge and, more generally, on $k$-simplices. This paper systematically analyzes how random walk on different orders of simplicial complexes (SC) facilitates GNNs in their theoretical expressivity. First, on $0$-simplices or node level, we establish a connection between existing positional encoding (PE) and structure encoding (SE) methods through the bridge of random walk. Second, on $1$-simplices or edge level, we bridge edge-level random walk and Hodge $1$-Laplacians and design corresponding edge PE respectively. In the spatial domain, we directly make use of edge level random walk to construct EdgeRWSE. Based on the spectral analysis of Hodge $1$-Laplcians, we propose Hodge1Lap, a permutation equivariant and expressive edge-level positional encoding. Third, we generalize our theory to random walk on higher-order simplices and propose the general principle to design PE on simplices based on random walk and Hodge Laplacians. Inter-level random walk is also introduced to unify a wide range of simplicial networks. Extensive experiments verify the effectiveness of our random walk-based methods.

Distance-Restricted Folklore Weisfeiler-Leman GNNs with Provable Cycle Counting Power

The ability of graph neural networks (GNNs) to count certain graph substructures, especially cycles, is important for the success of GNNs on a wide range of tasks. It has been recently used as a popular metric for evaluating the expressive power of GNNs. Many of the proposed GNN models with provable cycle counting power are based on subgraph GNNs, i.e., extracting a bag of subgraphs from the input graph, generating representations for each subgraph, and using them to augment the representation of the input graph. However, those methods require heavy preprocessing, and suffer from high time and memory costs. In this paper, we overcome the aforementioned limitations of subgraph GNNs by proposing a novel class of GNNs -- $d$-Distance-Restricted FWL(2) GNNs, or $d$-DRFWL(2) GNNs. $d$-DRFWL(2) GNNs use node pairs whose mutual distances are at most $d$ as the units for message passing to balance the expressive power and complexity. By performing message passing among distance-restricted node pairs in the original graph, $d$-DRFWL(2) GNNs avoid the expensive subgraph extraction operations in subgraph GNNs, making both the time and space complexity lower. We theoretically show that the discriminative power of $d$-DRFWL(2) GNNs strictly increases as $d$ increases. More importantly, $d$-DRFWL(2) GNNs have provably strong cycle counting power even with $d=2$: they can count all 3, 4, 5, 6-cycles. Since 6-cycles (e.g., benzene rings) are ubiquitous in organic molecules, being able to detect and count them is crucial for achieving robust and generalizable performance on molecular tasks. Experiments on both synthetic datasets and molecular datasets verify our theory. To the best of our knowledge, our model is the most efficient GNN model to date (both theoretically and empirically) that can count up to 6-cycles.

P-vectors: A Parallel-Coupled TDNN/Transformer Network for Speaker Verification

Typically, the Time-Delay Neural Network (TDNN) and Transformer can serve as a backbone for Speaker Verification (SV). Both of them have advantages and disadvantages from the perspective of global and local feature modeling. How to effectively integrate these two style features is still an open issue. In this paper, we explore a Parallel-coupled TDNN/Transformer Network (p-vectors) to replace the serial hybrid networks. The p-vectors allows TDNN and Transformer to learn the complementary information from each other through Soft Feature Alignment Interaction (SFAI) under the premise of preserving local and global features. Also, p-vectors uses the Spatial Frequency-channel Attention (SFA) to enhance the spatial interdependence modeling for input features. Finally, the outputs of dual branches of p-vectors are combined by Embedding Aggregation Layer (EAL). Experiments show that p-vectors outperforms MACCIF-TDNN and MFA-Conformer with relative improvements of 11.5% and 13.9% in EER on VoxCeleb1-O.

From Relational Pooling to Subgraph GNNs: A Universal Framework for More Expressive Graph Neural Networks

Relational pooling is a framework for building more expressive and permutation-invariant graph neural networks. However, there is limited understanding of the exact enhancement in the expressivity of RP and its connection with the Weisfeiler Lehman hierarchy. Starting from RP, we propose to explicitly assign labels to nodes as additional features to improve expressive power of message passing neural networks. The method is then extended to higher dimensional WL, leading to a novel $k,l$-WL algorithm, a more general framework than $k$-WL. Theoretically, we analyze the expressivity of $k,l$-WL with respect to $k$ and $l$ and unifies it with a great number of subgraph GNNs. Complexity reduction methods are also systematically discussed to build powerful and practical $k,l$-GNN instances. We theoretically and experimentally prove that our method is universally compatible and capable of improving the expressivity of any base GNN model. Our $k,l$-GNNs achieve superior performance on many synthetic and real-world datasets, which verifies the effectiveness of our framework.

Improving Graph Neural Networks on Multi-node Tasks with Labeling Tricks

In this paper, we provide a theory of using graph neural networks (GNNs) for \textit{multi-node representation learning}, where we are interested in learning a representation for a set of more than one node such as a link. Existing GNNs are mainly designed to learn single-node representations. When we want to learn a node-set representation involving multiple nodes, a common practice in previous works is to directly aggregate the single-node representations obtained by a GNN. In this paper, we show a fundamental limitation of such an approach, namely the inability to capture the dependence among multiple nodes in a node set, and argue that directly aggregating individual node representations fails to produce an effective joint representation for multiple nodes. A straightforward solution is to distinguish target nodes from others. Formalizing this idea, we propose \text{labeling trick}, which first labels nodes in the graph according to their relationships with the target node set before applying a GNN and then aggregates node representations obtained in the labeled graph for multi-node representations. The labeling trick also unifies a few previous successful works for multi-node representation learning, including SEAL, Distance Encoding, ID-GNN, and NBFNet. Besides node sets in graphs, we also extend labeling tricks to posets, subsets and hypergraphs. Experiments verify that the labeling trick technique can boost GNNs on various tasks, including undirected link prediction, directed link prediction, hyperedge prediction, and subgraph prediction. Our work explains the superior performance of previous node-labeling-based methods and establishes a theoretical foundation for using GNNs for multi-node representation learning.

Is Distance Matrix Enough for Geometric Deep Learning?

Graph Neural Networks (GNNs) are often used for tasks involving the geometry of a given graph, such as molecular dynamics simulation. While the distance matrix of a graph contains the complete geometric structure information, whether GNNs can learn this geometry solely from the distance matrix has yet to be studied. In this work, we first demonstrate that Message Passing Neural Networks (MPNNs) are insufficient for learning the geometry of a graph from its distance matrix by constructing families of geometric graphs which cannot be distinguished by MPNNs. We then propose $k$-DisGNNs, which can effectively exploit the rich geometry contained in the distance matrix. We demonstrate the high expressive power of our models and prove that some existing well-designed geometric models can be unified by $k$-DisGNNs as special cases. Most importantly, we establish a connection between geometric deep learning and traditional graph representation learning, showing that those highly expressive GNN models originally designed for graph structure learning can also be applied to geometric deep learning problems with impressive performance, and that existing complex, equivariant models are not the only solution. Experimental results verify our theory.

Neural Common Neighbor with Completion for Link Prediction

Despite its outstanding performance in various graph tasks, vanilla Message Passing Neural Network (MPNN) usually fails in link prediction tasks, as it only uses representations of two individual target nodes and ignores the pairwise relation between them. To capture the pairwise relations, some models add manual features to the input graph and use the output of MPNN to produce pairwise representations. In contrast, others directly use manual features as pairwise representations. Though this simplification avoids applying a GNN to each link individually and thus improves scalability, these models still have much room for performance improvement due to the hand-crafted and unlearnable pairwise features. To upgrade performance while maintaining scalability, we propose Neural Common Neighbor (NCN), which uses learnable pairwise representations. To further boost NCN, we study the unobserved link problem. The incompleteness of the graph is ubiquitous and leads to distribution shifts between the training and test set, loss of common neighbor information, and performance degradation of models. Therefore, we propose two intervention methods: common neighbor completion and target link removal. Combining the two methods with NCN, we propose Neural Common Neighbor with Completion (NCNC). NCN and NCNC outperform recent strong baselines by large margins. NCNC achieves state-of-the-art performance in link prediction tasks.