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A Learned Generalized Geodesic Distance Function-Based Approach for Node Feature Augmentation on Graphs

Amitoz Azad, Yuan Fang

TL;DR

The paper addresses noisy real-world graphs by augmenting node features with learned generalized geodesic distances. It introduces LGGD, a two-stage approach that learns the parameters of a graph p-eikonal type distance via an ODE solver, allowing the distance features to incorporate both topology and node content. Key contributions include a hybrid learning framework for the potential ρ(x) and initial condition φ0(x), demonstrated improvements over non-learning baselines, and a dynamic label inclusion mechanism that updates features without retraining backbone GNNs. The findings show LGGD can compete with state-of-the-art augmentation methods and generalize across backbone architectures, offering practical benefits for fast adaptation to new labels and noisy graphs.

Abstract

Geodesic distances on manifolds have numerous applications in image processing, computer graphics and computer vision. In this work, we introduce an approach called `LGGD' (Learned Generalized Geodesic Distances). This method involves generating node features by learning a generalized geodesic distance function through a training pipeline that incorporates training data, graph topology and the node content features. The strength of this method lies in the proven robustness of the generalized geodesic distances to noise and outliers. Our contributions encompass improved performance in node classification tasks, competitive results with state-of-the-art methods on real-world graph datasets, the demonstration of the learnability of parameters within the generalized geodesic equation on graph, and dynamic inclusion of new labels.

A Learned Generalized Geodesic Distance Function-Based Approach for Node Feature Augmentation on Graphs

TL;DR

The paper addresses noisy real-world graphs by augmenting node features with learned generalized geodesic distances. It introduces LGGD, a two-stage approach that learns the parameters of a graph p-eikonal type distance via an ODE solver, allowing the distance features to incorporate both topology and node content. Key contributions include a hybrid learning framework for the potential ρ(x) and initial condition φ0(x), demonstrated improvements over non-learning baselines, and a dynamic label inclusion mechanism that updates features without retraining backbone GNNs. The findings show LGGD can compete with state-of-the-art augmentation methods and generalize across backbone architectures, offering practical benefits for fast adaptation to new labels and noisy graphs.

Abstract

Geodesic distances on manifolds have numerous applications in image processing, computer graphics and computer vision. In this work, we introduce an approach called `LGGD' (Learned Generalized Geodesic Distances). This method involves generating node features by learning a generalized geodesic distance function through a training pipeline that incorporates training data, graph topology and the node content features. The strength of this method lies in the proven robustness of the generalized geodesic distances to noise and outliers. Our contributions encompass improved performance in node classification tasks, competitive results with state-of-the-art methods on real-world graph datasets, the demonstration of the learnability of parameters within the generalized geodesic equation on graph, and dynamic inclusion of new labels.
Paper Structure (42 sections, 1 theorem, 19 equations, 4 figures, 1 table)

This paper contains 42 sections, 1 theorem, 19 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Mathematical Background
  4. Notation
  5. Gradient Operators
  6. Generalized Geodesic Distance Function
  7. Solving Eq. (\ref{['eq:time']}) with an ODE Solver
  8. Learning Eq. (\ref{['eq:time']}) with an ODE Solver
  9. Connection with Dijkstra
  10. Choosing Potential Function $\rho(x)$
  11. Proposed Approaches
  12. Generalized Geodesic Distances as Features
  13. Learned Generalized Geodesic Distances as Features
  14. Dynamic Inclusion of New Labels
  15. Experiments & Results
  16. ...and 27 more sections

Key Result

Proposition 1

For an unweighted graph with a constant potential function $\rho(x) = 1$, the Eq. (eq:p-eiko) with supremum norm (i.e.$p=\infty$) yields geodesic (shortest-path) distance function of Eq. (eq:dijk_0).

Figures (4)

  • Figure 1: The $n$ represents the number of random corrupted edges added to a given graph. The graph construction: 20,000 points (nodes) were randomly sampled from a unit ball in ${R}^2$. An $\epsilon$-neighborhood unweighted graph was constructed using these sampled points with $\epsilon=0.05$. All points within $\epsilon$ distance of the boundary of the unit ball are considered boundary nodes. Colors represent the distance from the boundary, with red indicating the boundary where the distance function is zero, and yellow indicating the maximum distance.
  • Figure 2: Generalized Geodesic Distances as Features
  • Figure 3: Learned Generalized Geodesic Distances as Features.
  • Figure 4: The top row shows the performance of LGGD (Learned Generalized Geodesic Distances) features across different datasets for various backbone models. The bottom row demonstrates the ability to incorporate new incoming labels without retraining the backbone model (see Sec. \ref{['sec:dynamic']}), as illustrated for three datasets. A green dot represents the results obtained after dynamically adding 10% new labels.

Theorems & Definitions (1)

  • Proposition 1