Convolutional layers in graph neural networks are a fundamental type of layer which output a representation or embedding of each graph vertex. The representation typically encodes information about the vertex in question and its neighbourhood. If one wishes to perform a graph centric task, such as graph classification, this set of vertex representations must be integrated or pooled to form a graph representation.
In this article we propose a novel pooling method which maps a set of vertex representations to a function space representation. This method is distinct from existing pooling methods which perform a mapping to either a vector or sequence space. Experimental graph classification results demonstrate that the proposed method generally outperforms most baseline pooling methods and in some cases achieves best performance.
Michigan State University（密歇根州立大学）；Pennsylvania State University（宾夕法尼亚州立大学）；IBM T. J. Watson Research Center（IBM T·J·沃森研究中心）
Graph neural networks, which generalize deep neural network models to graph structured data, have attracted increasing attention in recent years. They usually learn node representations by transforming, propagating and aggregating node features and have been proven to improve the performance of many graph related tasks such as node classification and link prediction. To apply graph neural networks for the graph classification task, approaches to generate the \textit{graph representation} from node representations are demanded. A common way is to globally combine the node representations. Howev…
Convolutional graph networks are used in particle physics for effective event reconstructions and classifications. However, their performances can be limited by the considerable amount of sensors used in modern particle detectors if applied to sensor-level data. We present a pooling scheme that uses partitioning to create pooling kernels on graphs, similar to pooling on images. Partition pooling can be used to adopt successful image recognition architectures for graph neural network applications in particle physics. The reduced computational resources allow for deeper networks and more extensi…
fortiss GmbH（fortiss有限公司）；An-Institut Technische Universität München（慕尼黑工业大学附属研究所）
Graph Neural Network (GNN) research has concentrated on improving convolutional layers, with little attention paid to developing graph pooling layers. Yet pooling layers can enable GNNs to reason over abstracted groups of nodes instead of single nodes. To close this gap, we propose a graph pooling layer relying on the notion of edge contraction: EdgePool learns a localized and sparse hard pooling transform. We show that EdgePool outperforms alternative pooling methods, can be easily integrated into most GNN models, and improves performance on both node and graph classification.
Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph n…
NORCE, the Norwegian Research Centre（挪威研究中心NORCE）；Università della Svizzera italiana（意大利语区大学（瑞士））；Politecnico di Milano（米兰理工大学）
Spectral clustering (SC) is a popular clustering technique to find strongly connected communities on a graph. SC can be used in Graph Neural Networks (GNNs) to implement pooling operations that aggregate nodes belonging to the same cluster. However, the eigendecomposition of the Laplacian is expensive and, since clustering results are graph-specific, pooling methods based on SC must perform a new optimization for each new sample. In this paper, we propose a graph clustering approach that addresses these limitations of SC. We formulate a continuous relaxation of the normalized minCUT problem an…
Graph neural networks (GNNs) have been used effectively in different applications involving the processing of signals on irregular structures modeled by graphs. Relying on the use of shift-invariant graph filters, GNNs extend the operation of convolution to graphs. However, the operations of pooling and sampling are still not clearly defined and the approaches proposed in the literature either modify the graph structure in a way that does not preserve its spectral properties, or require defining a policy for selecting which nodes to keep. In this work, we propose a new strategy for pooling and…
University of Technology Sydney（悉尼科技大学）；Monash University（莫纳什大学）
Graph convolutional networks (GCNs) have recently become one of the most powerful tools for graph analytics tasks in numerous applications, ranging from social networks and natural language processing to bioinformatics and chemoinformatics, thanks to their ability to capture the complex relationships between concepts. At present, the vast majority of GCNs use a neighborhood aggregation framework to learn a continuous and compact vector, then performing a pooling operation to generalize graph embedding for the classification task. These approaches have two disadvantages in the graph classificat…
Indian Institute of Technology, Guwahati（印度理工学院古瓦哈提分校）；Indian Institute of Science, Bangalore（印度科学学院班加罗尔分校）
Graph Neural Networks (GNN) have been shown to work effectively for modeling graph structured data to solve tasks such as node classification, link prediction and graph classification. There has been some recent progress in defining the notion of pooling in graphs whereby the model tries to generate a graph level representation by downsampling and summarizing the information present in the nodes. Existing pooling methods either fail to effectively capture the graph substructure or do not easily scale to large graphs. In this work, we propose ASAP (Adaptive Structure Aware Pooling), a sparse an…
Deep learning methods for graphs have seen rapid progress in recent years with much focus awarded to generalising Convolutional Neural Networks (CNN) to graph data. CNNs are typically realised by alternating convolutional and pooling layers where the pooling layers subsample the grid and exchange spatial or temporal resolution for increased feature dimensionality. Whereas the generalised convolution operator for graphs has been studied extensively and proven useful, hierarchical coarsening of graphs is still challenging since nodes in graphs have no spatial locality and no natural order. This …
University of New South Wales（新南威尔士大学）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数学科学研究所）；Zhejiang Normal University（浙江师范大学）；Princeton University（普林斯顿大学）；University of California, Los Angeles（加州大学洛杉矶分校）；City University of Hong Kong（香港城市大学）
Deep Graph Neural Networks (GNNs) are useful models for graph classification and graph-based regression tasks. In these tasks, graph pooling is a critical ingredient by which GNNs adapt to input graphs of varying size and structure. We propose a new graph pooling operation based on compressive Haar transforms -- HaarPooling. HaarPooling implements a cascade of pooling operations; it is computed by following a sequence of clusterings of the input graph. A HaarPooling layer transforms a given input graph to an output graph with a smaller node number and the same feature dimension; the compressiv…
This work generalizes graph neural networks (GNNs) beyond those based on the Weisfeiler-Lehman (WL) algorithm, graph Laplacians, and diffusions. Our approach, denoted Relational Pooling (RP), draws from the theory of finite partial exchangeability to provide a framework with maximal representation power for graphs. RP can work with existing graph representation models and, somewhat counterintuitively, can make them even more powerful than the original WL isomorphism test. Additionally, RP allows architectures like Recurrent Neural Networks and Convolutional Neural Networks to be used in a theo…
Institute of Exact and Applied Sciences (EA7484), University of New Caledonia（精确与应用科学研究所（EA7484），新喀里多尼亚大学）；Big Data Institute, College of Computer Science and Software Engineering（大数据研究所，计算机科学与软件工程学院）；Shenzhen University, China（深圳大学，中国）
We propose PiNet, a generalised differentiable attention-based pooling mechanism for utilising graph convolution operations for graph level classification. We demonstrate high sample efficiency and superior performance over other graph neural networks in distinguishing isomorphic graph classes, as well as competitive results with state of the art methods on standard chemo-informatics datasets.
Graph pooling is an essential ingredient of Graph Neural Networks (GNNs) in graph classification and regression tasks. For these tasks, different pooling strategies have been proposed to generate a graph-level representation by downsampling and summarizing nodes' features in a graph. However, most existing pooling methods are unable to capture distinguishable structural information effectively. Besides, they are prone to adversarial attacks. In this work, we propose a novel pooling method named as {HIBPool} where we leverage the Information Bottleneck (IB) principle that optimally balances the…
Graph neural networks have emerged as a powerful model for graph representation learning to undertake graph-level prediction tasks. Various graph pooling methods have been developed to coarsen an input graph into a succinct graph-level representation through aggregating node embeddings obtained via graph convolution. However, most graph pooling methods are heavily node-centric and are unable to fully leverage the crucial information contained in global graph structure. This paper presents a cross-view graph pooling (Co-Pooling) method to better exploit crucial graph structure information. The …
Graph neural networks have achieved great success in learning node representations for graph tasks such as node classification and link prediction. Graph representation learning requires graph pooling to obtain graph representations from node representations. It is challenging to develop graph pooling methods due to the variable sizes and isomorphic structures of graphs. In this work, we propose to use second-order pooling as graph pooling, which naturally solves the above challenges. In addition, compared to existing graph pooling methods, second-order pooling is able to use information from …
Graph neural networks have been widely used on modeling graph data, achieving impressive results on node classification and link prediction tasks. Yet, obtaining an accurate representation for a graph further requires a pooling function that maps a set of node representations into a compact form. A simple sum or average over all node representations considers all node features equally without consideration of their task relevance, and any structural dependencies among them. Recently proposed hierarchical graph pooling methods, on the other hand, may yield the same representation for two differ…
Recent work in graph neural networks (GNNs) has led to improvements in molecular activity and property prediction tasks. Unfortunately, GNNs often fail to capture the relative importance of interactions between molecular substructures, in part due to the absence of efficient intermediate pooling steps. To address these issues, we propose LaPool (Laplacian Pooling), a novel, data-driven, and interpretable hierarchical graph pooling method that takes into account both node features and graph structure to improve molecular representation. We benchmark LaPool on molecular graph prediction and unde…
Geometry-Aware Edge Pooling for Graph Neural Networks
Helmholtz Munich（海德堡-穆恩大学）；Technical University of Munich（慕尼黑技术大学）；Université de Montréal（蒙特利尔大学）；Mila - Quebec AI Institute（魁北克人工智能研究所）；Université de Fribourg（弗里堡大学）
Graph pooling has been increasingly considered for graph neural networks (GNNs) to facilitate hierarchical graph representation learning. Existing graph pooling methods commonly consist of two stages, i.e., selecting the top-ranked nodes and removing the rest nodes to construct a coarsened graph representation. However, local structural information of the removed nodes would be inevitably dropped in these methods, due to the inherent coupling of nodes (location) and their features (signals). In this paper, we propose an enhanced three-stage method via lifting, named LiftPool, to improve hierar…
Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）；University of Chinese Academy of Sciences（中国科学院大学）；Paradigm. Inc.（第四范式）；Department of Electronic Engineering, Tsinghua University（清华大学电子工程系）；Lenovo（联想）
Graph classification is an important problem with applications across many domains, like chemistry and bioinformatics, for which graph neural networks (GNNs) have been state-of-the-art (SOTA) methods. GNNs are designed to learn node-level representation based on neighborhood aggregation schemes, and to obtain graph-level representation, pooling methods are applied after the aggregation operation in existing GNN models to generate coarse-grained graphs. However,due to highly diverse applications of graph classification, and the performance of existing pooling methods vary on different graphs. I…
Shanghai Jiao Tong University（上海交通大学）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院（EPFL））
With the advent of data science, the analysis of network or graph data has become a very timely research problem. A variety of recent works have been proposed to generalize neural networks to graphs, either from a spectral graph theory or a spatial perspective. The majority of these works however focus on adapting the convolution operator to graph representation. At the same time, the pooling operator also plays an important role in distilling multiscale and hierarchical representations but it has been mostly overlooked so far. In this paper, we propose a parameter-free pooling operator, calle…
Physical Pooling Functions in Graph Neural Networks for Molecular Property Prediction
RWTH Aachen University（亚琛工业大学）；Delft University of Technology（代尔夫特理工大学）；Forschungszentrum Jülich GmbH（于利希研究中心）；JARA Center for Simulation and Data Science (CSD)（JARA模拟与数据科学中心）
Graph neural networks (GNN) extends deep learning to graph-structure dataset. Similar to Convolutional Neural Networks (CNN) using on image prediction, convolutional and pooling layers are the foundation to success for GNN on graph prediction tasks. In the initial PAN paper, it uses a path integral based graph neural networks for graph prediction. Specifically, it uses a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It further generalizes such conv…
In several machine learning tasks for graph structured data, the graphs under consideration may be composed of a varying number of nodes. Therefore, it is necessary to design pooling methods that aggregate the graph representations of varying size to representations of fixed size which can be used in downstream tasks, such as graph classification. Existing graph pooling methods offer no guarantee with regards to the similarity of a graph representation and its pooled version. In this work, we address this limitation by proposing FlowPool, a pooling method that optimally preserves the statistic…
Graph Neural Networks achieve state-of-the-art performance on a plethora of graph classification tasks, especially due to pooling operators, which aggregate learned node embeddings hierarchically into a final graph representation. However, they are not only questioned by recent work showing on par performance with random pooling, but also ignore completely higher-order connectivity patterns. To tackle this issue, we propose HoscPool, a clustering-based graph pooling operator that captures higher-order information hierarchically, leading to richer graph representations. In fact, we learn a prob…
Temple University（天普大学）；University of Texas at Dallas（德克萨斯大学达拉斯分校）
Graph neural networks (GNNs) have demonstrated a significant success in various graph learning tasks, from graph classification to anomaly detection. There recently has emerged a number of approaches adopting a graph pooling operation within GNNs, with a goal to preserve graph attributive and structural features during the graph representation learning. However, most existing graph pooling operations suffer from the limitations of relying on node-wise neighbor weighting and embedding, which leads to insufficient encoding of rich topological structures and node attributes exhibited by real-worl…
Georgia State University（佐治亚州立大学）；University of Wisconsin–Madison（威斯康星大学麦迪逊分校）
Graph Neural networks (GNNs) have recently become a powerful technique for many graph-related tasks including graph classification. Current GNN models apply different graph pooling methods that reduce the number of nodes and edges to learn the higher-order structure of the graph in a hierarchical way. All these methods primarily rely on the one-hop neighborhood. However, they do not consider the higher- order structure of the graph. In this work, we propose a multi-channel Motif-based Graph Pooling method named (MPool) captures the higher-order graph structure with motif and local and global g…
Seoul National University（首尔国立大学）；AIGENDRUG Co. Ltd.（AIGENDRUG有限公司）
While graph neural networks (GNNs) have been successful for node classification tasks and link prediction tasks in graph, learning graph-level representations still remains a challenge. For the graph-level representation, it is important to learn both representation of neighboring nodes, i.e., aggregation, and graph structural information. A number of graph pooling methods have been developed for this goal. However, most of the existing pooling methods utilize k-hop neighborhood without considering explicit structural information in a graph. In this paper, we propose Structure Prototype Guided…
University of Pittsburgh（匹兹堡大学）；Intel Labs（英特尔实验室）；Lehigh University（里海大学）
Recent years have witnessed the emergence and flourishing of hierarchical graph pooling neural networks (HGPNNs) which are effective graph representation learning approaches for graph level tasks such as graph classification. However, current HGPNNs do not take full advantage of the graph's intrinsic structures (e.g., community structure). Moreover, the pooling operations in existing HGPNNs are difficult to be interpreted. In this paper, we propose a new interpretable graph pooling framework - CommPOOL, that can capture and preserve the hierarchical community structure of graphs in the graph r…
MaxCutPool: differentiable feature-aware Maxcut for pooling in graph neural networks
Alma Mater Studiorum - University of Bologna（博洛尼亚大学）；UiT the Arctic University of Norway（挪威北极大学）；Fondazione Istituto Italiano di Tecnologia（意大利技术研究院）；NORCE Norwegian Research Centre AS（NORCE挪威研究中心）
Accurate traffic conditions prediction provides a solid foundation for vehicle-environment coordination and traffic control tasks. Because of the complexity of road network data in spatial distribution and the diversity of deep learning methods, it becomes challenging to effectively define traffic data and adequately capture the complex spatial nonlinear features in the data. This paper applies two hierarchical graph pooling approaches to the traffic prediction task to reduce graph information redundancy. First, this paper verifies the effectiveness of hierarchical graph pooling methods in tra…
Università della Svizzera italiana（瑞士意大利语大学）；UiT the Arctic University of Norway（挪威北极大学）；NORCE, The Norwegian Research Centre（挪威NORCE研究中心）；Politecnico di Milano（米兰理工大学）
Information Sciences Institute, University of Southern California（南加州大学信息科学研究所）；Michigan State University（密歇根州立大学）；University of California, Los Angeles（加州大学洛杉矶分校）
The ability to capture complex linguistic structures and long-term dependencies among words in the passage is essential for discourse-level relation extraction (DRE) tasks. Graph neural networks (GNNs), one of the methods to encode dependency graphs, have been shown effective in prior works for DRE. However, relatively little attention has been paid to receptive fields of GNNs, which can be crucial for cases with extremely long text that requires discourse understanding. In this work, we leverage the idea of graph pooling and propose to use pooling-unpooling framework on DRE tasks. The pooling…
Graph-level representation learning is the pivotal step for downstream tasks that operate on the whole graph. The most common approach to this problem heretofore is graph pooling, where node features are typically averaged or summed to obtain the graph representations. However, pooling operations like averaging or summing inevitably cause massive information missing, which may severely downgrade the final performance. In this paper, we argue what is crucial to graph-level downstream tasks includes not only the topological structure but also the distribution from which nodes are sampled. Theref…
Aalto University（阿尔托大学）；Federal Institute of Ceará（塞阿拉联邦研究所）；University of Manchester（曼彻斯特大学）
Graph pooling is a central component of a myriad of graph neural network (GNN) architectures. As an inheritance from traditional CNNs, most approaches formulate graph pooling as a cluster assignment problem, extending the idea of local patches in regular grids to graphs. Despite the wide adherence to this design choice, no work has rigorously evaluated its influence on the success of GNNs. In this paper, we build upon representative GNNs and introduce variants that challenge the need for locality-preserving representations, either using randomization or clustering on the complement graph. Stri…
MDL-Pool: Adaptive Multilevel Graph Pooling Based on Minimum Description Length
Chair of Machine Learning for Complex Networks（复杂网络机器学习教授席位）；Center for Artificial Intelligence and Data Science (CAIDAS)（人工智能与数据科学中心）；Julius-Maximilians-Universität Würzburg（乌尔姆-马克西米利安大学）
McGill University（麦吉尔大学）；Université de Montréal（蒙特利尔大学）；HEC Montréal（蒙特利尔高等商学院）；Mila（米拉人工智能研究所）
Graph Neural Networks (GNNs) are attracting growing attention due to their effectiveness and flexibility in modeling a variety of graph-structured data. Exiting GNN architectures usually adopt simple pooling operations (eg. sum, average, max) when aggregating messages from a local neighborhood for updating node representation or pooling node representations from the entire graph to compute the graph representation. Though simple and effective, these linear operations do not model high-order non-linear interactions among nodes. We propose the Tensorized Graph Neural Network (tGNN), a highly exp…
Chongqing University（重庆大学）；Zhejiang University（浙江大学）；University of California, Los Angeles（加州大学洛杉矶分校）
With the advancement of sensing technology, multivariate time series classification (MTSC) has recently received considerable attention. Existing deep learning-based MTSC techniques, which mostly rely on convolutional or recurrent neural networks, are primarily concerned with the temporal dependency of single time series. As a result, they struggle to express pairwise dependencies among multivariate variables directly. Furthermore, current spatial-temporal modeling (e.g., graph classification) methodologies based on Graph Neural Networks (GNNs) are inherently flat and cannot aggregate hub data…
In this paper, we propose a novel pooling layer for graph neural networks based on maximizing the mutual information between the pooled graph and the input graph. Since the maximum mutual information is difficult to compute, we employ the Shannon capacity of a graph as an inductive bias to our pooling method. More precisely, we show that the input graph to the pooling layer can be viewed as a representation of a noisy communication channel. For such a channel, sending the symbols belonging to an independent set of the graph yields a reliable and error-free transmission of information. We show …
Graph neural networks (GNN) have been proven to be mature enough for handling graph-structured data on node-level graph representation learning tasks. However, the graph pooling technique for learning expressive graph-level representation is critical yet still challenging. Existing pooling methods either struggle to capture the local substructure or fail to effectively utilize high-order dependency, thus diminishing the expression capability. In this paper we propose HAP, a hierarchical graph-level representation learning framework, which is adaptively sensitive to graph structures, i.e., HAP …
Recently, there has been considerable research interest in graph clustering aimed at data partition using the graph information. However, one limitation of the most of graph-based methods is that they assume the graph structure to operate is fixed and reliable. And there are inevitably some edges in the graph that are not conducive to graph clustering, which we call spurious edges. This paper is the first attempt to employ graph pooling technique for node clustering and we propose a novel dual graph embedding network (DGEN), which is designed as a two-step graph encoder connected by a graph po…
Handong Global University（韩东国际大学）；Dept. of Information and Communication Engineering（信息与通信工程系）
Convolutional neural networks (CNNs) have achieved remarkable performance in many applications, especially in image recognition tasks. As a crucial component of CNNs, sub-sampling plays an important role for efficient training or invariance property, and max-pooling and arithmetic average-pooling are commonly used sub-sampling methods. In addition to the two pooling methods, however, there could be many other pooling types, such as geometric average, harmonic average, and so on. Since it is not easy for algorithms to find the best pooling method, usually the pooling types are assumed a priorit…
Image classification is considered, and a hierarchical max-pooling model with additional local pooling is introduced. Here the additional local pooling enables the hierachical model to combine parts of the image which have a variable relative distance towards each other. Various convolutional neural network image classifiers are introduced and compared in view of their rate of convergence. The finite sample size performance of the estimates is analyzed by applying them to simulated and real data.
We introduce a simple and effective method for regularizing large convolutional neural networks. We replace the conventional deterministic pooling operations with a stochastic procedure, randomly picking the activation within each pooling region according to a multinomial distribution, given by the activities within the pooling region. The approach is hyper-parameter free and can be combined with other regularization approaches, such as dropout and data augmentation. We achieve state-of-the-art performance on four image datasets, relative to other approaches that do not utilize data augmentati…
