Graph Neural Networks and 3-Dimensional Topology

• URL: http://arxiv.org/abs/2305.05966v2
• Date: Fri, 28 Jul 2023 16:33:57 GMT
• Title: Graph Neural Networks and 3-Dimensional Topology
• Authors: Pavel Putrov and Song Jin Ri
• Abstract summary: We consider the class of 3-manifolds described by plumbing graphs and use Graph Neural Networks (GNN) for the problem. We use supervised learning to train a GNN that provides the answer to such a question with high accuracy. We consider reinforcement learning by a GNN to find a sequence of Neumann moves that relates the pair of graphs if the answer is positive.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We test the efficiency of applying Geometric Deep Learning to the problems in low-dimensional topology in a certain simple setting. Specifically, we consider the class of 3-manifolds described by plumbing graphs and use Graph Neural Networks (GNN) for the problem of deciding whether a pair of graphs give homeomorphic 3-manifolds. We use supervised learning to train a GNN that provides the answer to such a question with high accuracy. Moreover, we consider reinforcement learning by a GNN to find a sequence of Neumann moves that relates the pair of graphs if the answer is positive. The setting can be understood as a toy model of the problem of deciding whether a pair of Kirby diagrams give diffeomorphic 3- or 4-manifolds.

Related papers

• Detecting Homeomorphic 3-manifolds via Graph Neural Networks [0.0]
  We study the homeomorphism problem for a class of graph-manifolds using Graph Neural Network techniques. We train and benchmark a variety of network architectures within a supervised learning setting by testing different combinations of two convolutional layers.
  arXiv  Detail & Related papers  (2024-09-01T12:58:09Z)
• How Graph Neural Networks Learn: Lessons from Training Dynamics [80.41778059014393]
  We study the training dynamics in function space of graph neural networks (GNNs) We find that the gradient descent optimization of GNNs implicitly leverages the graph structure to update the learned function. This finding offers new interpretable insights into when and why the learned GNN functions generalize.
  arXiv  Detail & Related papers  (2023-10-08T10:19:56Z)
• NodeFormer: A Scalable Graph Structure Learning Transformer for Node Classification [70.51126383984555]
  We introduce a novel all-pair message passing scheme for efficiently propagating node signals between arbitrary nodes. The efficient computation is enabled by a kernerlized Gumbel-Softmax operator. Experiments demonstrate the promising efficacy of the method in various tasks including node classification on graphs.
  arXiv  Detail & Related papers  (2023-06-14T09:21:15Z)
• Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs [122.06927400759021]
  We study the relationship between a graph neural network (GNN) and a manifold neural network (MNN) when the graph is constructed from a set of points sampled from the manifold. We prove non-asymptotic error bounds showing that convolutional filters and neural networks on these graphs converge to convolutional filters and neural networks on the continuous manifold.
  arXiv  Detail & Related papers  (2023-05-29T08:27:17Z)
• Node Embedding from Hamiltonian Information Propagation in Graph Neural Networks [30.42111062496152]
  We propose a novel graph information propagation strategy called Hamiltonian Dynamic GNN (HDG) HDG uses a Hamiltonian mechanics approach to learn node embeddings in a graph. We demonstrate the ability of HDG to automatically learn the underlying geometry of graph datasets.
  arXiv  Detail & Related papers  (2023-03-02T07:40:40Z)
• Training Graph Neural Networks on Growing Stochastic Graphs [114.75710379125412]
  Graph Neural Networks (GNNs) rely on graph convolutions to exploit meaningful patterns in networked data. We propose to learn GNNs on very large graphs by leveraging the limit object of a sequence of growing graphs, the graphon.
  arXiv  Detail & Related papers  (2022-10-27T16:00:45Z)
• The Exact Class of Graph Functions Generated by Graph Neural Networks [43.25172578943894]
  Graph Neural Network (GNN) whose output is identical to the graph function? In this paper, we fully answer this question and characterize the class of graph problems that can be represented by GNNs. We show that this condition can be efficiently verified by checking quadratically many constraints.
  arXiv  Detail & Related papers  (2022-02-17T18:54:27Z)
• Graph Neural Networks with Learnable Structural and Positional Representations [83.24058411666483]
  A major issue with arbitrary graphs is the absence of canonical positional information of nodes. We introduce Positional nodes (PE) of nodes, and inject it into the input layer, like in Transformers. We observe a performance increase for molecular datasets, from 2.87% up to 64.14% when considering learnable PE for both GNN classes.
  arXiv  Detail & Related papers  (2021-10-15T05:59:15Z)
• Implicit Graph Neural Networks [46.0589136729616]
  We propose a graph learning framework called Implicit Graph Neural Networks (IGNN) IGNNs consistently capture long-range dependencies and outperform state-of-the-art GNN models.
  arXiv  Detail & Related papers  (2020-09-14T06:04:55Z)
• XGNN: Towards Model-Level Explanations of Graph Neural Networks [113.51160387804484]
  Graphs neural networks (GNNs) learn node features by aggregating and combining neighbor information. GNNs are mostly treated as black-boxes and lack human intelligible explanations. We propose a novel approach, known as XGNN, to interpret GNNs at the model-level.
  arXiv  Detail & Related papers  (2020-06-03T23:52:43Z)
• Geom-GCN: Geometric Graph Convolutional Networks [15.783571061254847]
  We propose a novel geometric aggregation scheme for graph neural networks to overcome the two weaknesses. The proposed aggregation scheme is permutation-invariant and consists of three modules, node embedding, structural neighborhood, and bi-level aggregation. We also present an implementation of the scheme in graph convolutional networks, termed Geom-GCN, to perform transductive learning on graphs.
  arXiv  Detail & Related papers  (2020-02-13T00:03:09Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.