Evaluating Gr-GCN++ for Node Classification Across Various Datasets: Results and Comparisons

Table of Links

Abstract and 1. Introduction

2. Preliminaries

3. Proposed Approach

   3.1 Notation

   3.2 Nueral Networks on SPD Manifolds

   3.3 MLR in Structure Spaces

   3.4 Neural Networks on Grassmann Manifolds

4. Experiments

5. Conclusion and References

A. Notations

B. MLR in Structure Spaces

C. Formulation of MLR from the Perspective of Distances to Hyperplanes

D. Human Action Recognition

E. Node Classification

F. Limitations of our work

G. Some Related Definitions

H. Computation of Canonical Representation

I. Proof of Proposition 3.2

J. Proof of Proposition 3.4

K. Proof of Proposition 3.5

L. Proof of Proposition 3.6

M. Proof of Proposition 3.11

N. Proof of Proposition 3.12

E NODE CLASSIFICATION

E.1 DATASETS

Airport (Chami et al., 2019) It is a flight network dataset from OpenFlights.org where nodes represent airports, edges represent the airline Routes, and node labels are the populations of the country where the airport belongs.

Pubmed (Namata et al., 2012b) It is a standard benchmark describing citation networks where nodes represent scientific papers in the area of medicine, edges are citations between them, and node labels are academic (sub)areas.

Cora (Sen et al., 2008) It is a citation network where nodes represent scientific papers in the area of machine learning, edges are citations between them, and node labels are academic (sub)areas.

The statistics of the three datasets are summarized in Tab. 11.

E.2 IMPLEMENTATION DETAILS

E.2.1 SETUP

E.2.2 GRASSMANN LOGARITHMIC MAP IN THE ONB PERSPECTIVE

The Grassmann logarithmic map in the ONB perspective is given (Edelman et al., 1998) by

E.2.3 GR-GCN++

E.2.4 GR-GCN-ONB

E.2.5 OPTIMIZATION

E.3 MORE EXPERIMENTAL RESULTS

E.3.1 ABLATION STUDY

Projector vs. ONB perspective More results of Gr-GCN++ and Gr-GCN-ONB are presented in Tabs. 12 and 13. As can be observed, Gr-GCN++ outperforms Gr-GCN-ONB in all cases. In particular, the former outperforms the latter by large margins on Airport and Cora datasets. Results show that while both the networks learn node embeddings on Grassmann manifolds, the choice of perspective for representing these embeddings and the associated parameters can have a significant impact on the network performance.

E.3.2 COMPARISON OF GR-GCN++ AGAINST STATE-OF-THE-ART METHODS

Tab. 14 shows results of Gr-GCN++ and some state-of-the-art methods on the three datasets. The hyperbolic networks outperform their SPD and Grassmann counterparts on Airport dataset with high hyperbolicity (Chami et al., 2019). This agrees with previous works (Chami et al., 2019; Zhang et al., 2022) that report good performances of hyperbolic embeddings on tree-like datasets. However, our network and its SPD counterpart SPD-GCN outperform their competitors on Pubmed and Cora datasets with low hyperbolicities. Compared to SPD-GCN, Gr-GCN++ always gives more consistent results.