Table of Links Abstract and 1. Introduction Preliminaries


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


Experiments


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. Authors:
(1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr);
(2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr);
(3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. Table of Links Abstract and 1. Introduction Abstract and 1. Introduction Preliminaries 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 Experiments Conclusion and References Preliminaries Preliminaries Preliminaries 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 Proposed Approach 3.1 Notation 3.1 Notation 3.2 Nueral Networks on SPD Manifolds 3.2 Nueral Networks on SPD Manifolds 3.3 MLR in Structure Spaces 3.3 MLR in Structure Spaces 3.4 Neural Networks on Grassmann Manifolds 3.4 Neural Networks on Grassmann Manifolds Experiments Experiments Experiments Conclusion and References Conclusion and References Conclusion and References A. Notations A. Notations B. MLR in Structure Spaces B. MLR in Structure Spaces C. Formulation of MLR from the Perspective of Distances to Hyperplanes C. Formulation of MLR from the Perspective of Distances to Hyperplanes D. Human Action Recognition D. Human Action Recognition E. Node Classification E. Node Classification F. Limitations of our work F. Limitations of our work G. Some Related Definitions G. Some Related Definitions H. Computation of Canonical Representation H. Computation of Canonical Representation I. Proof of Proposition 3.2 I. Proof of Proposition 3.2 J. Proof of Proposition 3.4 J. Proof of Proposition 3.4 K. Proof of Proposition 3.5 K. Proof of Proposition 3.5 L. Proof of Proposition 3.6 L. Proof of Proposition 3.6 M. Proof of Proposition 3.11 M. Proof of Proposition 3.11 N. Proof of Proposition 3.12 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. Airport (Chami et al., 2019) 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. Pubmed (Namata et al., 2012b) 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. Cora (Sen et al., 2008) The statistics of the three datasets are summarized in Tab. 11. E.2 IMPLEMENTATION DETAILS E.2.1 SETUP E.2.1 SETUP E.2.2 GRASSMANN LOGARITHMIC MAP IN THE ONB PERSPECTIVE 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.2.3 GR-GCN++ E.2.4 GR-GCN-ONB E.2.4 GR-GCN-ONB E.2.5 OPTIMIZATION E.2.5 OPTIMIZATION E.3 MORE EXPERIMENTAL RESULTS E.3.1 ABLATION STUDY 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. Projector vs. ONB perspective E.3.2 COMPARISON OF GR-GCN++ AGAINST STATE-OF-THE-ART METHODS 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. Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). Authors: Authors: (1) Xuan Son Nguyen, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (xuan-son.nguyen@ensea.fr); (2) Shuo Yang, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (son.nguyen@ensea.fr); (3) Aymeric Histace, ETIS, UMR 8051, CY Cergy Paris University, ENSEA, CNRS, France (aymeric.histace@ensea.fr). This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. This paper is available on arxiv under CC by 4.0 Deed (Attribution 4.0 International) license. available on arxiv available on arxiv

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

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