Analyzing the Effect of Embedding Norms and Singular Values to Oversmoothing in Graph Neural Networks

• URL: http://arxiv.org/abs/2510.06066v1
• Date: Tue, 07 Oct 2025 15:55:28 GMT
• Title: Analyzing the Effect of Embedding Norms and Singular Values to Oversmoothing in Graph Neural Networks
• Authors: Dimitrios Kelesis, Dimitris Fotakis, Georgios Paliouras,
• Abstract summary: We study the factors that contribute to the effect of oversmoothing in deep Graph Neural Networks (GNNs)<n>We derive layer-wise bounds on $MASED$, which aggregate to yield global upper and lower distance bounds.<n>We show that by reducing oversmoothing in deep networks, we can achieve better results in some tasks than using shallow ones.
• Score: 12.876488159688506
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we study the factors that contribute to the effect of oversmoothing in deep Graph Neural Networks (GNNs). Specifically, our analysis is based on a new metric (Mean Average Squared Distance - $MASED$) to quantify the extent of oversmoothing. We derive layer-wise bounds on $MASED$, which aggregate to yield global upper and lower distance bounds. Based on this quantification of oversmoothing, we further analyze the importance of two different properties of the model; namely the norms of the generated node embeddings, along with the largest and smallest singular values of the weight matrices. Building on the insights drawn from the theoretical analysis, we show that oversmoothing increases as the number of trainable weight matrices and the number of adjacency matrices increases. We also use the derived layer-wise bounds on $MASED$ to form a proposal for decoupling the number of hops (i.e., adjacency depth) from the number of weight matrices. In particular, we introduce G-Reg, a regularization scheme that increases the bounds, and demonstrate through extensive experiments that by doing so node classification accuracy increases, achieving robustness at large depths. We further show that by reducing oversmoothing in deep networks, we can achieve better results in some tasks than using shallow ones. Specifically, we experiment with a ``cold start" scenario, i.e., when there is no feature information for the unlabeled nodes. Finally, we show empirically the trade-off between receptive field size (i.e., number of weight matrices) and performance, using the $MASED$ bounds. This is achieved by distributing adjacency hops across a small number of trainable layers, avoiding the extremes of under- or over-parameterization of the GNN.

Related papers

• Rethinking Oversmoothing in Graph Neural Networks: A Rank-Based Perspective [5.482832675034467]
  We show that rank-based metrics consistently capture oversmoothing, whereas energy-based metrics often fail.<n> Notably, we reveal that a significant drop in the rank aligns closely with performance degradation, even in scenarios where energy metrics remain unchanged.
  arXiv  Detail & Related papers  (2025-02-07T00:55:05Z)
• Generalization of Geometric Graph Neural Networks with Lipschitz Loss Functions [84.01980526069075]
  We study the generalization capabilities of geometric graph neural networks (GNNs)<n>We prove a generalization gap between the optimal empirical risk and the optimal statistical risk of this GNN.<n>We verify this theoretical result with experiments on multiple real-world datasets.
  arXiv  Detail & Related papers  (2024-09-08T18:55:57Z)
• Asymptotics of Learning with Deep Structured (Random) Features [9.366617422860543]
  For a large class of feature maps we provide a tight characterisation of the test error associated with learning the readout layer. In some cases our results can capture feature maps learned by deep, finite-width neural networks trained under gradient descent.
  arXiv  Detail & Related papers  (2024-02-21T18:35:27Z)
• OrthoReg: Improving Graph-regularized MLPs via Orthogonality Regularization [66.30021126251725]
  Graph Neural Networks (GNNs) are currently dominating in modeling graphstructure data. Graph-regularized networks (GR-MLPs) implicitly inject the graph structure information into model weights, while their performance can hardly match that of GNNs in most tasks. We show that GR-MLPs suffer from dimensional collapse, a phenomenon in which the largest a few eigenvalues dominate the embedding space. We propose OrthoReg, a novel GR-MLP model to mitigate the dimensional collapse issue.
  arXiv  Detail & Related papers  (2023-01-31T21:20:48Z)
• ResNorm: Tackling Long-tailed Degree Distribution Issue in Graph Neural Networks via Normalization [80.90206641975375]
  This paper focuses on improving the performance of GNNs via normalization. By studying the long-tailed distribution of node degrees in the graph, we propose a novel normalization method for GNNs. The $scale$ operation of ResNorm reshapes the node-wise standard deviation (NStd) distribution so as to improve the accuracy of tail nodes.
  arXiv  Detail & Related papers  (2022-06-16T13:49:09Z)
• Mixed Graph Contrastive Network for Semi-Supervised Node Classification [63.924129159538076]
  We propose a novel graph contrastive learning method, termed Mixed Graph Contrastive Network (MGCN)<n>In our method, we improve the discriminative capability of the latent embeddings by an unperturbed augmentation strategy and a correlation reduction mechanism.<n>By combining the two settings, we extract rich supervision information from both the abundant nodes and the rare yet valuable labeled nodes for discriminative representation learning.
  arXiv  Detail & Related papers  (2022-06-06T14:26:34Z)
• Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility [18.531464406721412]
  This article studies the infinite-width limit of deep feedforward neural networks whose weights are dependent. Each hidden node of the network is assigned a nonnegative random variable that controls the variance of the outgoing weights of that node.
  arXiv  Detail & Related papers  (2022-05-17T09:14:32Z)
• Tight Bounds on the Smallest Eigenvalue of the Neural Tangent Kernel for Deep ReLU Networks [21.13299067136635]
  We provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU networks. In the finite-width setting, the network architectures we consider are quite general.
  arXiv  Detail & Related papers  (2020-12-21T19:32:17Z)
• Optimization and Generalization Analysis of Transduction through Gradient Boosting and Application to Multi-scale Graph Neural Networks [60.22494363676747]
  It is known that the current graph neural networks (GNNs) are difficult to make themselves deep due to the problem known as over-smoothing. Multi-scale GNNs are a promising approach for mitigating the over-smoothing problem. We derive the optimization and generalization guarantees of transductive learning algorithms that include multi-scale GNNs.
  arXiv  Detail & Related papers  (2020-06-15T17:06:17Z)
• Revealing the Structure of Deep Neural Networks via Convex Duality [70.15611146583068]
  We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of hidden layers. We show that a set of optimal hidden layer weights for a norm regularized training problem can be explicitly found as the extreme points of a convex set. We apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds.
  arXiv  Detail & Related papers  (2020-02-22T21:13:44Z)

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.