Tight and fast generalization error bound of graph embedding in metric space

• URL: http://arxiv.org/abs/2305.07971v1
• Date: Sat, 13 May 2023 17:29:18 GMT
• Title: Tight and fast generalization error bound of graph embedding in metric space
• Authors: Atsushi Suzuki, Atsushi Nitanda, Taiji Suzuki, Jing Wang, Feng Tian, and Kenji Yamanishi
• Abstract summary: We show that graph embedding in non-Euclidean metric space can outperform that in Euclidean space with much smaller training data than the existing bound has suggested. Our new upper bound is significantly tighter and faster than the existing one, which can be exponential to $R$ and $O(frac1S)$ at the fastest.
• Score: 54.279425319381374
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recent studies have experimentally shown that we can achieve in non-Euclidean metric space effective and efficient graph embedding, which aims to obtain the vertices' representations reflecting the graph's structure in the metric space. Specifically, graph embedding in hyperbolic space has experimentally succeeded in embedding graphs with hierarchical-tree structure, e.g., data in natural languages, social networks, and knowledge bases. However, recent theoretical analyses have shown a much higher upper bound on non-Euclidean graph embedding's generalization error than Euclidean one's, where a high generalization error indicates that the incompleteness and noise in the data can significantly damage learning performance. It implies that the existing bound cannot guarantee the success of graph embedding in non-Euclidean metric space in a practical training data size, which can prevent non-Euclidean graph embedding's application in real problems. This paper provides a novel upper bound of graph embedding's generalization error by evaluating the local Rademacher complexity of the model as a function set of the distances of representation couples. Our bound clarifies that the performance of graph embedding in non-Euclidean metric space, including hyperbolic space, is better than the existing upper bounds suggest. Specifically, our new upper bound is polynomial in the metric space's geometric radius $R$ and can be $O(\frac{1}{S})$ at the fastest, where $S$ is the training data size. Our bound is significantly tighter and faster than the existing one, which can be exponential to $R$ and $O(\frac{1}{\sqrt{S}})$ at the fastest. Specific calculations on example cases show that graph embedding in non-Euclidean metric space can outperform that in Euclidean space with much smaller training data than the existing bound has suggested.

Related papers

• Shedding Light on Problems with Hyperbolic Graph Learning [2.3743504594834635]
  Recent papers in the graph machine learning literature have introduced a number of approaches for hyperbolic representation learning. We take a careful look at the field of hyperbolic graph representation learning as it stands today. We find that a number of papers fail to diligently present baselines, make faulty modelling assumptions when constructing algorithms, and use misleading metrics to quantify geometry of graph datasets.
  arXiv  Detail & Related papers  (2024-11-11T03:12:41Z)
• Weighted Embeddings for Low-Dimensional Graph Representation [0.13499500088995461]
  We propose embedding a graph into a weighted space, which is closely related to hyperbolic geometry but mathematically simpler. We show that our weighted embeddings heavily outperform state-of-the-art Euclidean embeddings for heterogeneous graphs while using fewer dimensions.
  arXiv  Detail & Related papers  (2024-10-08T13:41:03Z)
• Normed Spaces for Graph Embedding [8.949780057954404]
  We show that normed spaces can embed finite metric spaces with surprisingly low theoretical bounds on distortion in low dimensions. Our work highlights the potential of normed spaces for geometric graph representation learning, raises new research questions, and offers a valuable tool for experimental mathematics in the field of finite metric space embeddings.
  arXiv  Detail & Related papers  (2023-12-03T20:21:08Z)
• Improving embedding of graphs with missing data by soft manifolds [51.425411400683565]
  The reliability of graph embeddings depends on how much the geometry of the continuous space matches the graph structure. We introduce a new class of manifold, named soft manifold, that can solve this situation. Using soft manifold for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets.
  arXiv  Detail & Related papers  (2023-11-29T12:48:33Z)
• 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)
• Geometry Contrastive Learning on Heterogeneous Graphs [50.58523799455101]
  This paper proposes a novel self-supervised learning method, termed as Geometry Contrastive Learning (GCL) GCL views a heterogeneous graph from Euclidean and hyperbolic perspective simultaneously, aiming to make a strong merger of the ability of modeling rich semantics and complex structures. Extensive experiments on four benchmarks data sets show that the proposed approach outperforms the strong baselines.
  arXiv  Detail & Related papers  (2022-06-25T03:54:53Z)
• Provably Accurate and Scalable Linear Classifiers in Hyperbolic Spaces [39.71927912296049]
  We propose a unified framework for learning scalable and simple hyperbolic linear classifiers. The gist of our approach is to focus on Poincar'e ball models and formulate the classification problems using tangent space formalisms. The excellent performance of the Poincar'e second-order and strategic perceptrons shows that the proposed framework can be extended to general machine learning problems in hyperbolic spaces.
  arXiv  Detail & Related papers  (2022-03-07T21:36:21Z)
• Geometric Graph Representation Learning via Maximizing Rate Reduction [73.6044873825311]
  Learning node representations benefits various downstream tasks in graph analysis such as community detection and node classification. We propose Geometric Graph Representation Learning (G2R) to learn node representations in an unsupervised manner. G2R maps nodes in distinct groups into different subspaces, while each subspace is compact and different subspaces are dispersed.
  arXiv  Detail & Related papers  (2022-02-13T07:46:24Z)
• ACE-HGNN: Adaptive Curvature Exploration Hyperbolic Graph Neural Network [72.16255675586089]
  We propose an Adaptive Curvature Exploration Hyperbolic Graph NeuralNetwork named ACE-HGNN to adaptively learn the optimal curvature according to the input graph and downstream tasks. Experiments on multiple real-world graph datasets demonstrate a significant and consistent performance improvement in model quality with competitive performance and good generalization ability.
  arXiv  Detail & Related papers  (2021-10-15T07:18:57Z)
• Highly Scalable and Provably Accurate Classification in Poincare Balls [40.82908295137667]
  We establish a unified framework for learning scalable and simple hyperbolic linear classifiers with provable performance guarantees. Our results include a new hyperbolic and second-order perceptron algorithm as well as an efficient and highly accurate convex optimization setup for hyperbolic support vector machine classifiers. Their performance accuracies on synthetic data sets comprising millions of points, as well as on complex real-world data sets such as single-cell RNA-seq expression measurements, CIFAR10, Fashion-MNIST and mini-ImageNet.
  arXiv  Detail & Related papers  (2021-09-08T16:59:39Z)

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.