SRBGCN: Tangent space-Free Lorentz Transformations for Graph Feature Learning
Authors: Abdelrahman Mostafa
Wei Peng
Guoying Zhao
Venue: Submitted to ICLR 2023 (2023)
Keywords: fully hyperbolic network
Lorentz transformations
boost and rotation
graph convolutional networks
hyperbolic rotations

Abstract: Hyperbolic graph convolutional networks have been successfully applied to represent complex graph data structures. However, optimization on Riemannian manifolds is nontrivial thus most of the existing hyperbolic networks build the network operations on the tangent space of the manifold, which is a Euclidean local approximation. This distorts the learnt features, limits the representation capacity of the network and makes it hard to optimize the network. In this work, we introduce a fully hyperbolic graph convolutional network (GCN), referred to as SRBGCN, which performs neural computations such as feature transformation and aggregation directly on the manifold, using manifold-preserving Lorentz transformations that include spatial rotation (SR) and boost (B) operations. Experiments conducted on static graph datasets for node classification and link prediction tasks validate the performance of the proposed method.

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SRBGCN: TANGENT SPACE-FREE LORENTZ TRANS- FORMATIONS FOR GRAPH FEATURE LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Hyperbolic graph convolutional networks have been successfully applied to repre- sent complex graph data structures. However, optimization Riemannian mani- folds is nontrivial thus most of the existing hyperbolic networks build the network operations the tangent space of the manifold, which is Euclidean local ap- proximation. This distorts the learnt features, limits the representation capacity of the network and makes it hard optimize the network. In this work, we introduce a fully hyperbolic graph convolutional network (GCN), referred to as SRBGCN, which performs neural computations such as feature transformation and aggrega- tion directly on the manifold, using manifold manifold-preserving Lorentz transformations that include spatial rotation (SR) and boost (B) operations. Experiments con- ducted on static graph datasets for node classification and link prediction tasks validate the performance of the proposed method.
INTRODUCTION
Graph convolutional networks (GCNs) were proposed to make use of the graph topology and model the spatial relationship between graph nodes, hence generalizing the convolution operation to graph & Welling (2017); Defferrard et al. (2016). Initially, the proposed models were built in the Euclidean space Hamilton et al. (2017): Zhang et al. (2018); Velickovic et al. (2019) which not the natural space for embedding graph data and produces distorted feature representations Nickel & Kiela (2188); Chami et al. 2019). Hyperbolic spaces are more suitable for representing graph data as the space volume is increasing exponentially which is perfect for embedding tree-like data struc- tures that also grow exponentially with the depth of the tree whereas the space grows polynomially for the Euclidean space. Motivated by this, recent works built GCNs in the hyperbolic space to take advantage of the hyperbolic geometry properties Chami et al. (2019) Liu et al. (2019). The hyper- bolic graph convolutional networks (HGCNs) achieved better performance than the corresponding Euclidean ones which shows the effectiveness of using the hyperbolic space to model hicrarchical data structures and graph data.
However, these works performed the network operations in the tangent space of the manifold which Euclidean local approximation the manifold point. The Euclidean network operations such as feature transformation and feature aggregation are not manifold- soaffolddpreeeviing and can not be directly applied on the manifold, that why these methods resort to the tangent space. However, using tangent space may limit the representation capabilities of the hyperbolic networks which caused by distortion specially as most of these works used the tangent space at the origin. In this work, we propose full prrefelii-p Lorentz feature transformations using both boost and spatial rotation operations to build SRBGCN fully in the hyperbolic space without resorting to the tangent space. Experiments conducted node classification and link prediction tasks static graph datasets show the effectiveness of our proposed method. SRBGCN has a good physical interpretation and can be used build deep networks with more representation capacity and less distorted features.
2 RELATED WORK
Chami et al. (2019) proposed HGCNs where networks operations are performed the tangent space of the manifold. They were able achieve better performance than the Fuclidean analogs node

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classification and link prediction tasks. Concurrently to this work, Liu al. (011 proposed the Hyperbolic Graph Neural Networks (HGNNs) which performed well on graph classification tasks. Several models have been proposed using different hyperbolic models especially the Lorentz and the
Poincaré models for different other tasks such as image segmentation Gulcehre et al. (2018), word
embeddings Tifrea al.) (2018), human action recognition Peng al. (2020) text classffication Zhu et al. (2020), machine translation Shimizu et al. (2020): Gulcehre et al. (2018). knowledge graph embeddings Chami et al. (2020) and so on. Gu al. (018)) built a two-stream network for Euclidean and hyperbolic features (operations on tangent space) and used an interaction module to
enhance the learnt feature representations in the two geometries. Peng et al. (202T) presented comprehensive survey for hyperbolic networks.
Zhang et al. (2021b) rebuilt the network operations in HGCNs to guarantee that the learnt features follow the hyperbolic geometry and used the Lorentz centroid Ratcliffe et al. (1994); Law et al. (2019) for aggregating the features. Zhang et al. (2021a) used attention modules the hyperbolic space to build hyperbolic networks. Dai et al. (021)) built a hyperbolic network by imposing the orthogonal constraint on a sub-matrix of the translormation matrix (subspace transformation), They used the same number of learnable parameters for the feature transformation step the networks built on the tangent space, however the orthogonal constraint ensured that the transformation is and they did not need learn the parameters the tangent space. They used the Finstein midpoint method defined in the Klein model [Ugarr (2005) for the feature aggregation step. Chen et al. (2022) used normalization procedure to keep the points on the manifold. They also use normalization for the feature aggregation step. The idea is similar to learning a general transformation matrix performing aggregation for spherical embeddings and then normalizing the resulting features to have the norm of the sphere radius. In this work, we introduce full space manifold-
manifold-preserving preserving transformation matrix in SRBGCN without the need for normalization keep the points on the manifold.
3 BACKGROUND
3.1 GRAPHI CONVOLUTIONAL NETWORKS
... Un is the of graph nodes with & represent- ing the set of graph edges. The edge set & can be encoded in an adjacency matrix A E R=*a where Aij [0, 1] if there is link between e; and tj otherwise, A,j 0. Each node v; has feature vector in the Euclidean space na Rdd of dimension and X is the of features for all the nodes in the graph.
A static graph G {V.E) where V fui, .......
The feature transformation step in GCNs can be formulated as:
Y X' BB'
where W is the weight matrix corresponding to the input X' at layer and B is the bias translation matrix. The weight matrix acts as a linear transformer whereas the optional bias matrix makes the transformation affine.
Then the feature aggregation from neighboring nodes step with nonlinear activation applied can be formulated as:
x =DDD -IDD-/DD/A
where a is an activation function. D D- + +IDD-'/a is the normalized adjacency matrix to nor- malize nodes weights in the neighboring set. D is diagonal matrix where D 1 + Σ, AM and
I is the identity matrix to keep identity features. X++1 represents the output of layer / which can be the input to the next layer 1 + 1. A GCN is built by stacking number of those layers.
Clearly, the linear transformation matrix W can not be used in hyperbolic networks as this uncon- strained transformation matrix will not keep points the manifold i.c. not manifold preserving transformations. The same applies for the aggregation step as the Euclidean mean operation is not manifold-preserving.
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a) Circular rotation
b) Hyperbolic rotation
Figure 1:
Regular/ circular rotation vs hyperbolic rotation/ squeeze mapping. points are color coded for illustration purposes.
3.2 HYPERBOLIC ROTATIONS/SQUTEEZE MAPPING
regular rotation is a linear map that preserves the Euclidean inner product (-.-) IRd x Rdd
RR where v)e = >i= Fill whereas a hyperbolic rotation or a squeeze mapping is a linear map that preserves (- Regular rotations can be realized by trigonometric functions whereas hyperbolic rotations can be realized by hyperbolic functions which are related to their trigonometric counterparts through complex angles. Intuitively, regular rotations can be thought of regular rotation to the axes whereas hyperbolic rotations are rotations in the hyperbolic sense and can be thought of as squeezing the axes (see Figure [] for visualization), hence the name. The following matrix is squeeze mapping that keeps the points rotating hyperbola (HH).)):
coosh w sinh w
sinh w cosh
L(w) =
(3)
A hyperbolic geometry review is provided in the appendix to make the paper self-contained.
4 SPATIAL ROTATION BooST GRAPH CONVOLUTIONAL NETWORK(SRBGCN)
In this part, we show how to build SRBGCN fully in the hyperbolic space. A manifold-presenvve Lorentz transformation is used for feature transformation that includes the boost and spatial rotation operations.
4.1 LORENTZ MRATIOOORMATION
The linear transformation matrix used Equation 1 for Euclidean features can not used directly for hyperbolic features this linear transformation unconstrained matrix not manifold-preservinn for the hyperbolic space. Instead, Lorentz transformation matrix A should satisfy the following constraint:
AT gc = 9c
where A E (dd+1)x(d+1) and T represents the transpose operation of the matrix. A Lorentz transfor- mation matrix is a matrix that is orthogonal with respect to the Minkowski metric ge and belongs to the Lorentz group. When Ağ is positive (the first element in the transformation matrix), the mapping remains the upper half of the hyperboloid. Taking the determinant of this equation, obtain (det A)2 = i.c. (det A) = +1. The set of matrices with (det A) = 1 and Ag > are referred to the proper Lorentz group 10t
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4.2 BOOST AND SPATIAL ROTATION
The Lorentz transformation can be decomposed into boost and spatial rotation operations Durney Moretti (2002) by polar decomposition. The boost matrix symmetric semi-posieee comi-positive and the spatial rotation matrix is unitary. The spatial rotation operation rotates the spatial coordinates and the boost operation moves point along the time coordinate without rotating the spatial coordinates.
Intuitively, the subspace manifold formed by the spatial axes is a (d 1)-dimensional sphere for d-dimensional hyperboloid at given level zu zo K since Σ =; ziu" K is sphere for any value d9 E 20 K. Hence, regular rotation transformation matrix represents the spatial rotation operation in this subspace manifold. The spatial rotation matrix is given by:
1 0
0
P
(S)
Q (d+1)x(d+1)
where Q belongs to the special orthogonal group SO(d) i.e. QTQ I. It can easily verified that P is Lorentz transformation matrix which satisfies Equation
Proof: From Equation
4
have:
1
-1
0
1
- 0 = 9c
pd PTgcP
Pc
Q
QTIQ
The rotation matrix Q in Equation 5 can be realized using different representations such as the basic rotations using Trigonometric functions (d degrees of freedom in this case), axis and angle (d + degrees of freedom) using the Gram-Schmine orthonormalization process. We enforce the orthogonalization constraint on the spatial rotation feature transformation matrix as the angles in the other two methods form discontinuous search space Zhou al. (2019) and also there singularities in this search space (gimbal lock problem).
The boost operation be realized using hyperbolic rotation squeeze mapping. Since the squeeze mapping matrix in Equation 3 satisfies the constraint 4 we can use such transformation matrix in the hyperbolic feature transformation step. d basic hyperbolic rotations in Equation for each spatial axis with the time axis can be used to realize the boost operation. hyperbolic rotation representation using a hyperbolic rotation axis ni and hyperbolic rotation parameter w (regular circular rotation can be realized also using an axis and an angle for rotation) in a d-dimensional hyperboloid can be represented by the transformation matrix:
more compact
cosh ws (sinhw)ny
(sinh yn cosh wyms & na) (a+ (d+1)x(d+1)
L
I I-
where @ represents the outer product operation. The hyperbolic rotation plane is the plane parallel the plane spanned by the normal vector na (a normalized linear combination of the spatial axes) and the axis zo (referred to the time axis in special relativity). Note that when is canonical basis vector, the resulting matrix will similar to one in Equation 3 after padding with zeros for other canonical spatial basis vectors.
Proof: Let L
a
b
where C = C as L is a symmetric matrix.
ba
Caxd
From Equation
have:
L gcL
a
0
a
b
d
ba
I
ba C
ba
ba
-a² + ba ba
-bba + Ch.
-1 0
abba + Cba
ba 8 by + C
= 96
0
I

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Tangent space methods
Normalization method
Our method
ZAA
lnd by
T
Figure 2: Left: the transformation step in the methods that rely on the tangent space of the manifold.
Middle: the normalization method used for feature transformation Chen et al. (2022). Right: transformation method using circular regular rotation for the spatial rotation operation and hyperbolic rotation for the boost operation.
So, we get -a² + bd bu
and using the hyperbolic identity: cosh w + sinh? w -1, we have = cosh w and by = (sinh w)na where na is a unit vector. To solve for C, we have ba @ ba C² I. So, we get:
C² = I (sinh² w)md @ I+ (-1 + cosh w)nd @
=I+ (1 cosh w + 2 cosh 2 cosh w w d nd
II - co na @ na 2 cosh w ma @ nd (I (1 - coshw)m @ na)²
and using - -oba + Cba = O, we omit the negative solution and accept the positive one. So, we have C= (1 cosh w m @ n, the solution.
Since and P represent the boost and spatial rotation operations, respectively. We use the following Lorentz transformation matrix the feature transformation matrix for the hyperbolic space:
M = PL
To show that M is Lorentz transformation matrix, we have (PL)"gg(PP)) (P gcP) L' ggL 92 since both P and L are Lorentz transformation matrices as shown before.
Figure 2 shows visualization comparison between different methods. SRBGCN is fully hyperbolic as the boost and spatial rotation operations are intrinsic manifold prreefrvi-p transformation.
4.3 FEATURE TRANSFORMATION AND AGGREGATION IN SRBGCN
The feature transformation step in SRBGCN in contrast to the Euclidean one in Equation | can be formulated as:
Y X X
(8)
where the subscript represents features the hyperboloid. To get the initial features representa- tion on the hyperboloid, the exponential map (Equation 12) at the origin is used to map the features from the tangent space at the origin to the hyperboloid.
For the feature aggregation step, we use the Lorentz centroid Ratcliffe et al. (1994) Law et al. (0199
which minimizes the squared Lorentzian distance. It can be computed as:
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Tangent space methods
Lorentz centroid
TIPR
Figure 3: Left: the aggregation step the tangent space methods. Right: the Lorentz centroid.
A1+1 VK EjeNsty wasy A. |1| EjENs()) Wis 1llllel
(9)
where -1/K is the constant negative curvature (K > 0) .NS(i) is the neighbor set for node i which includes the node itself and Wiy the weight between nodes and j in the normalized adjacency matrix. Figure 3 shows visualization of different aggregation techniques used in hyperbolic net- works.
The margin ranking loss is used the loss function defined as:
Loss mar ( d + m,0)
(10)
where m is the non-negative margin hyperparameter. For node classification tasks where the goal to predict the label of given node in a graph, d is the distance between node and its correct class whereas d is distance between the node and wrong class. We calculate the hyperbolic distances between the node and a set of trainable centroids softmax classifier for node classification. For link prediction tasks where the goal is to predict the existence of links between graph nodes, d is the distance between connected nodes where links exist and à is the distance for negative samples. We the Fermi-Dirac decoder Tifrea et al. (0119) to calculate the probability for link existence between nodes.
the manifold and feed the distances vector to
5 EXPERIMENTS
Experiments are conducted on four publicly available datasets: Disease, Airport, PubMed and Cora. TTe datasets description, evaluation metrics and the evaluation results with comparison other methods are presented in the following subsections.
5.1 DATASETS DESCRIPTION
The Disease dataset is constructed by using the SIR disease spreading model Anderson & May (1992) and the label indicates whether the node is infected not. In the Airport dataset, the nodes represent airports with the population of the city as the label and the edges represents the flight routes between different cities. PubMed and Cora are citation network datasets where the nodes representing the scientific papers with the academic area as the label and the edges representing the existence of citations between papers. Table [ shows the statistics for all datasets. The 5- hyperbolicity represents the Gromovs hyperbolicity and reported on these datasets by (Caami et al. 2019). The lower the 5-hyperholicity value, the closer the graph to tree i.e. more hyperbolic

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Table 1: Datasets statistics
Dataset
nodes
node features
nodes classes
edges
8-hyperbolicity
Disease
1044
1000
2
1043
0
Airport 3188
18631
PubMed 19717
500
88651
3.5
Cora
2708
1433
5429
11
Table Evaluation results and comparison with other methods. ROC AUC results are reported for Link Prediction (LP) tasks and F1 scores reported for Node Classification (NC) tasks. The latent feature representation dimension set to 16 for fair comparison.
Dataset
Disease
Airport
PubMed
Coru
Method
LP
NC
NC
LP
NC
NC
Euclidean
GCN
64.7.20.5
69.7=10.4
89.3:10.4
81.410.6
91.1+00.5
78.110.2
90.41:.2
81.340.3
GAT
69.81003
70.41004
90.5.10.3
81.5
91.2.10.1
79.0100.3
93.7.0..1
83.01...7
SAGE
65.910..
69.1100.6
90.4.10.5
82.110.5
86.2.1100
77.412.2
85.5100.
77.912.4
SGC
65.1+0.2
69.5++0.2
89.8+0.3
80.6+0.1
94.1+-0.0
78.9+0.0
91.5+0.1
81.0+0.1
Hyberpolic
HGCN
91.2:10.6
82.8:0.88
96.4110.1
90.6:10.2
96.1
78.4+0.44
93.140.4
81.3:0.66
HAT
91.8=0.55
83.6=20.9
96.0++0.3
78.6+0.5
93.0+0.3
83.1+0.6
LGCN
96.6110.6
84.410.8
96.0.10.6
90.911.7
96.810.11
78.610.77
93.610..3
83,310,7
HYPONET 96.8+-0.4
96.0+1.0
97.3++0.3
90.9++1.4
95.8+0.2
78.0+1.0
93.6+0.3
80.2+
SRBGCN
97.3=002
93.0=20.4
97.3+0.00
91.6++0.9
97.240.0
79.1+0.33
95.2+10.0
82,940.2
where a tree structure has a 5-hyperbolicity value of zero as the case for the Disease dataset. The higher the d-hyperbolicity value, the closer the graph to complete graph.
5.2 EVALUATION METRICS
F1 score is used the evaluation metric for node classification tasks and the Area Under Curve (AUC) the evaluation metric for link prediction tasks. 10 independent runs are performed and the mean and the standard deviation for each experiment are reported with the same data splits as in Errica et al. (2020) that were used in previous works. The code will publicly available be used by other researchers in the community to encourage further developments in this field. The code was developed using PyTorch Paszke et al. (2019) and experiments were run on Tesla P100 and Tesla T4 GPUs.
5.3 EVALUATION RESULTS AND COMPARISONS WITH OTHER GRAPH METHODS
2 shows the performance of different methods including Euclidean and hyperbolic ones. The Eu- clidean methods include GCN Kipf & Welling (017). GAT Velickovic et al. (211)), SAGE Hamil- ton al. (2017) and SGC Wu et al. (2019). HGCN et al. (2019), HAT Zhang al. (2021a). LGCN Zhang et al. 2022 1b) and HYPONET Chen et al. (2022) are the hyperbolic methods used in the comparison. The hyperbolic methods outperform the Euclidean ones specially for the tree-like Disease dataset which has a zero 5-hyperbolicity. This proves the effectiveness of using the hyper- bolic space to model graph data specially graphs with small Gromovs hyperbolicity. As shown in the table, SRBGCN outperforms all other methods for most of the benchmarks. Even for the other benchmarks that SRBGCN do not get the best performance using a latent representation of 16, our method can still get comparable results that are much better than other methods in an efficient way. This is a clear evidence of the advantage of learning the graph features directly in the hyperbolic space. It is worth mentioning that HAT and LGCN methods use techniques such as attention mod- ules to improve the performance. Our method simple yet very effective. For the disease dataset that has tree structure with depth of 4, our method achieved very good performance using latent feature representation dimension or 8 compared to other methods which shows the effectiveness of using the intrinsic transformations specially tree-like datasets. This in turn help in build-

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Table 3: Comparison between different methods using different dimensions (dim) the Disease and Cora datasets for the node classification task.
Dataset
dim
GAT
HGCN
HAT
LGCN
HYPONET
SRBGCN
49.411..3
73.2.16.5
87.413.1
91.043.8
93.1
Disease
8
76.7+0.77
81.5+1.3
82.3+1.2
82.9#1.2
92.9#1.0
93.3=0.4
Cora
64
83.1
82.1
83.1
83.5+0.5
81.5.10.9
83.
Table 4: Ablation study on different datasets to show effect of using only the spatial rotation (SR) transformation operation and both the spatial rotation and the boost (SR and B) operations. ROC AUC results are reported for Link Prediction (LP) tasks and scores are reported for Node Classification (NC) tasks.
Dataset
Disease
Airport
PubMed
Cora
Transformation
LP
NC
LP
NC
LP
NC
LP
NC
SR only
Y)_- X pi
97.2+0.3
92.1+000
97.3-0. 0.0
89.7+
97,2+0.0
78.8+0.4
95.2++0.0
81.4+0.4
SR and
Y = P'P'
97.3+0.2
93.3+0.4
96.8 +0.0
91.6+0.9
97,2+0.0
79.1
94.3++.00
83.8+0.3
ing more compact models for such datasets. For larger and more complicated datasets with higher d-hyperbolicity such as Cora dataset, our method achieved better performance than other methods using higher dimensional latent space to embed the features for such more complicated datasets. Table shows this comparison between the different methods.
5.4 ABLATION STUDY
We present an ablation study to show the effectiveness of using both the boost operation and the spatial rotation operation which are the decomposition of the Lorentz transformation matrix. Table 4 shows the performance comparison between using the transformation Y = X p' and using the full transformation Y X ''L'. Using both the boost and the spatial rotation operations i.e. the full Lorentz transformation usually gives better performance and increases the model expressiveness of the hyperbolic network specially for the node classification tasks with deeper networks.
5.5 DISTORTION
The average distortion can be computed as: 1 Σi, (med)22)2111 where nedk,j the normalized "ndd..
embedding distance between nodes and j and ngdi,j is the normalized graph distance between them. The distortion reflects how close the structure of the graph is preserved in the graph embed- dings and the closer the value to zero, the less-distorted the features. Table 5 shows the distortion using GCN, HGCN and SRBGCN methods on the Disease and Airport datasets. The Euclidean GCN method introduces a lot of distortion compared to the hyperbolic ones. At the same time, our method has less distortion than the HGCN method which resorts the tangent space perform net- work operations. This shows the effectiveness of using the intrinsic Lorentz transformations to build
Table 5: Distortion values for Disease and Airport datasets.
Dataset
GCN
HGCN
SRBGCN
Disease
67.924:544.1 175.024216.90
1.05
0.35-1.0.03
Airport
1.394-0.64
0.27++0.00
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Initial embeddings
(b)
(c) HGCN
(d)
Figure 4: The initial embeddings and the learnt embeddings using different methods on the whole Disease dataset for the link prediction (top row) and the node classification task (bottom row).
fully hyperbolic networks. Figure 4 shows the initial embeddings and the learnt embeddings in the last layer using these methods on the whole Disease dataset. For the link prediction task (top row), is visually clear that the hyperbolic methods preserve the structure of the tree dataset better than the Euclidean method. Moreover, the hierarchies learnt from our method is much clearer than the one learnt by tangent-space methods. The visualization and distortion value show that our method generates higher quality features with less distortion. The ability to build better features with less distortion leads to higher performance which shows the superiority of our method. Similarly, for the node classification task (bottom row), our method separates the 2 classes of the Disease dataset more efficiently.
6 CONCLUSION
In this work, we presented tangent-free full Lorentz transformation layer using the polar decompo- sition of the Lorentz transformation into both the boost operation the spatial rotation operation. Our results show the effectiveness of using hyperbolic space model graph data and to learn less distorted useful features which can be used to build more expressive and compact models for graph learning tasks. We hope this work can be extended cover and build unified framework that include other geometries such as Euclidean and Spherical spaces order to increase the flexibility of the model to embed more complicated structures.
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John G Ratcliffe, S Axler, and KA Ribet. Foundations of hyperbolic manifolds, volume 149. Springer, 1994.
Ryohei Shimizu, Yusuke Mukuta, and Tatsuya Harada. Hyperbolic neural networks++. arXiv preprint arXiv:2006.088 00, 2020.
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Alexandru Tifrea, Gary Bécigneul, and Octavian- Eugen Ganea. Poincar 'e glove: embeddings. arXiv preprint arXiv: 2018.
Hyperbolic word
Alexandru Tifrea, Gary Bécigneul, and Octavian-Eugen Ganea. Poincare glove: Hyperbolic word embeddings. In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA. May 6-9, 2019. OpenReview.net, 2019. htth::t:: http:::://opeereviiee
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Petar Velickovic, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Liò, and Yoshua Bengio. Graph attention networks. 6th International Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 May 3, 2018, Conference Truck Proceedings.
OpenReview.net, 2018. URL BtLhttps:// wwwp///eeeeeeeeeeeeeoeeeeeooeeem / net
Petar Velickovic, William Fedus, William L. Hamilton, Pietro Liö, Yoshua Bengio, and R Devon Hjelm. Deep graph infomax. ICLR (Poster), 2(3):4, 2019.
Felix Wu, Amauri H. Souza Jr., Tianyi Zhang, Christopher Fifty. Tao Yu, and Kilian Q. Weinberger. Simplifying graph convolutional networks. In Kamalika Chaudhuri and Ruslan Salakhutdinov (eds.), Proceedings of the 36th International Conference on Machine Learning, ICMI, 2019, 9- June 2019, Long Beach, California, USA, volume 97 of Proceedings of Machine Learning Research, pp. 6861-6871. PMLR, 2019. URL http://pooceednngg.........e.. wul 9e. html
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A APPENDIX
A.I
HYPERBOLIC GEOMETRY REVIEW
We give a quick review about hyperbolic geometry to make the paper self-contained.
A.1.1 TOPOLOGICAL SPACE AND TOPOLOGICAL HOMEOMORPHISM
A topological space a geometrical space which has the notion of closeness. The closeness can, but not necessarily, be measured by the notion of distance determine if points close to each other. A homeomorphism is a continuous one-to-one mapping function or bicontinuous function between topological spaces that has continuous inverse function.
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Hyperboloid
Hyperroole
K.kin
ecard
P-line
Figure 5: Projection of hyperbolic geodesic from H onto the Klein disk and the Poincaré disk.
A.1.2 MANIFOLD AND TANGENT SPACE
A d-dimensional Manifold M (which can be embedded in Rd+1) is topological space which can be locally approximated by d-dimensional Euclidean space RR For any point x .Md, there is homeomorphism between the neighbourhood of and the Euclidean space Rd. Lines and circles are examples of one-dimensional manifolds. Planes and spheres are examples two-dimensional manifolds which are called surfaces. The notion of manifold is generalization of surfaces in any dimension d. The tangent space TM at point x MM is a d-dimensional hyperplane which is embedded in Id++1 that locally approximates the manifold M around the point
A.1.3 RIEMANNIAN METRIC AND RIEMANNIAN MANTFOLD
A Riemannian metric is used to define geometric notions on the manifold such as distances, angles, areas or volumes. It a collection smoothly varying inner products on tangent spaces, Gr T..Md x TM R. A Riemannian manifold can then be defined as (Md.))
A.1.4 CURVATURE AND GEODESICS
A curvature measures how much curve deviates from being a straight line. Fuclidean spaces have zero curvature whereas non-Fuclidean spaces have non-zero curvature For example, spheres have constant positive curvatures whereas hyperbolic spaces have constant negative curvatures. Geodesics are the generalizations shortest paths in graphs or lines in Euclidean geometry to non-Euclidean geometry. These are the curves that give the shortest paths between pairs of points.
A.1.5 HYPERBOLIC SPACE
A hyperbolic space is Riemannian manifold with constant negative curvature. Many models have been proposed to model a hyperbolic space such as the Lorentz model (also called the hyperboloid model), the Poincaré model and the Klein model. The Lorentz model is the upper sheet of two- sheeted hyperboloid. The Poincaré model and the Klein model are the projections of the Lorentz model onto the hyperplanes 2g 0 and To 1, respectively. There bijection functions to map between different hyperbolic models as they are all isomorphic. F shows the three different models which model the hyperbolic space.
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Table 6: Hyperparameters used for network training.
Dataset
Disease
Airport
PubMed
Cora
Parameter
LP
NC
LP
NC
LP
NC
LP
NC
Learning rate
0.001
0.005
0.5
0.2
0.05
0.04
0.001
0.001
Number of layers
6
1
2
5
3
Weight decay
0.0
0.0
lc-05
0.0
0.0
0.01
lc-04
0.01
Dropout
0.0
0.0
0.0
0.6
0.5
0.8
0.7
0.9
Margin
2
0.1
2
0.1
1
0.1
2
Normalize features
1
1
0
1
1
1
Let (,,))c Rdd11 Rd+1 - represents the Lorentz-Minkowski inner product where (x, f) E TiYi - Iozl = x² Псd where dC - (éag(-1,1, 1..., 1) is a diagonal matrix that represents the Riemannian metric for the hyperbolic manifold. Let Hd.K be a dimensional hyperboloid model with constant negative curvature -1/K where K 0. Then we have:
m.K.K = fo g (1, - 0}
(11)
Note that vo > 0 to indicate the upper half of the hyperboloid manifold. In special relativity, is referred to as the time axis whereas the rest axes are called the spatial axes.
A.1.6 EXPONENTIAL AND LOGARITHMIC MAPS
The exponential and logarithmic maps are used map between the hyperbolic space and the tangent space and represent bijection between the tangent space at point and the hyperboloid. The exponential map maps a point e H.H..K where x l...K to the hyperboloid Hd.K such that # 0 and is defined as:
|olllc VK
) VK Ksnnh Mollc VK ||lllc
exp (v) cosh
(12)
where ||u|ic Vu, c is the norm of The logarithmic map maps point y E H...K to the tangent space T,, H.d.K centered at point E H...K such that x + y and is defined as:
y 1/K (x, cx
Log (y) di (x. y)
(13)
11 1/x (x. y) colle
where d/c (x.y) the Minkowskian distance between two points x and y Hd and is given by:
d (x. = / K arcosh((((n.)eek)
(14)
A.2 HYPERPARAMETERS DETAILS
The hyperparameters used for network training are shown in Table
13


Reviews:
[{"review_id": "o9XPEubpb0", "reviewer": "Reviewer_x9HY", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper argues that methods based on tangent space have many problems, and therefore, avoid adopting this method. This paper proposes an approach to manifold-preserving Lorentz transformation.", "summary_of_the_review": "This paper argues that methods based on tangent space have many problems, and therefore, avoid adopting this method. This paper proposes an approach to manifold-preserving Lorentz transformation. Although the paper is easy to follow, there is something that I am concerned about, including the claim, method, and experiments.", "strength_and_weaknesses": "## Weakness\n\n1. Unconvincing conclusions.\n\n(1) Indeed, $W\\otimes_H X$ cannot guarantee that the result still lives in the Lorentz model after transformation. However, the problem has been solved by LGCN; that is, we only need to transform the space-like dimension (with the origin as a reference point) or just use the rotation matrix. Any of these methods can map it back to the Lorentz model. Also, the first approach can contain the rotation and boost operations.\n\nTherefore, according to the transformation, the author cannot derive the conclusion—the method based on tangent space is not manifold-preserve. Similarly, we can utilize the same strategy to achieve the aggregation in the tangent space and successfully map it back to the Lorentz model.\n\n(2) The author claimed that \"This makes it hard to optimize deeper networks because of these back-and-forth mappings between the manifold and the tangent space,\" which could be incorrect since the deep network does not necessarily require frequent maps. For example, in image tasks, the network is deep [1][2]. Besides, it is not hard for us to optimize the HGNN. Please run the HGCN code directly using more than two layers.\n\n[1] Capturing implicit hierarchical structure in 3D biomedical images with self-supervised hyperbolic representations\n[2] Hyperbolic Image Segmentation\n\n2. Confusing experimental results.\n\n- Could the author explain why the performance of the disease and Pubmed reported is different from the original work HGCN? Why not take their original results but present the results from LGCN? \n- The usage of dimension in Table 3 seems to cherry-pick. Why is the disease compared in 8 and 4 dimensions while the Cora is compared in 64?\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "1: The main claims of the paper are incorrect or not at all supported by theory or empirical results.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "Not applicable"}, "extra_text": {}}, {"review_id": "fB99lZshEx", "reviewer": "Reviewer_Ht7h", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "Similar to refs [A,B], the paper proposes a new version of Hyperbolic Graph Convolutional Network (HGCN). The main difference with refs [A,B] is that, instead of learning a linear map in the tangent space of the manifold, the paper proposes to learn linear operators directly in the extrinsic geometry of the Lorentzian ambient space of the hyperboloid. \nThe learned linear operators use some boost and spatial rotation operations and directly map to the hyperboloid. \n\n[A] Ines Chami, Zhitao Ying, Christopher Ré, and Jure Leskovec. Hyperbolic graph convolutional neural\nnetworks. NeurIPS 2019.\n\n[B] Qi Liu, Maximilian Nickel, and Douwe Kiela. Hyperbolic graph neural networks, NeurIPS 2019.", "summary_of_the_review": "The proposed method is interesting and could be useful to the community. What is missing is a clear discussion about the differences with refs [A,B]. What is possible in refs [A,B] that is not possible in the proposed approach and vice versa? What are the reasons of the improved performance of the method? Either a theoretical or empirical justification would be useful to readers.", "strength_and_weaknesses": "- The main strength of the paper is the simplicity of the approach which is well-explained. I believe the proposed approach could be a useful \"trick\" to know in the community that works on hyperbolic representations.  \n\n- The weaknesses of the paper are missing theoretical comparisons with refs [A,B]. In particular:\n\n1) Can both the boost and spatial rotation operations also be expressed by existing approaches (when working in the tangent space)? If the answer is yes, does the proposed approach act as a regularizer by limiting the nature of possible operations?\n\n2) What are the operations that the proposed approach can or cannot perform?\n\n3) Related to the above questions, in the introduction, it is mentioned: \"However, these works [A,B] performed the network operations in the tangent space of the manifold which is a Euclidean local approximation to the manifold at a point since the network operations such\nas feature transformation and feature aggregation are not manifold-preserving. This makes it hard\nto optimize deeper networks because of these back and forth mappings between the manifold and the tangent space, and limits the representation capabilities of the hyperbolic networks caused by distortion specially most of these works used the tangent space at the origin.\" The tangent space is supposed to preserve the metric of the manifold due to the Levi-Civita connection of the manifold. I then do not think that the issue is due to performing operations in the tangent space. What are the reasons of the worse performance of the baselines? Is it because of the approximations of the mappings from the manifold to the tangent space? Or because of operations that are performed or not performed in the learned linear mappings?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "4: All of the claims and statements are well-supported and correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "YMkvSOm2YY", "reviewer": "Reviewer_ehJR", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "This paper proposes a version of hyperbolic GCN based Lorentz model without resorting to the tangent space. Some of the layer operations exist in prior work. Experiments are conducted on standard small-scale citation networks and tree datasets. ", "summary_of_the_review": "It's a straightforward usage of the fully hyperbolic NNs paper in GCN applications. Many pieces in the paper are missing and lack of enough novelty, see above.", "strength_and_weaknesses": "There are recently many efforts on removing the tangent space of hyperbolic neural networks. This paper used the Lorentz boost and rotation to operate directly on the Lorentz manifold. \n\nStrength. The presentation and logic of the paper are easy to follow. \nThe ablation study is interesting and shows the learnt structure information from the model. \n\nWeakness.\n1. Lack of novelty and contribution. The Lorentz boost and rotation is already well-characterized and studied in [1], particularly the proposed fully hyperbolic linear layer in [1] contains all Lorentz rotation and boost matrices. In fact, the graph convolution operation is essentially a hyperbolic linear layer. One can just take the linear layer from [1] and make a similar tangent space free Lorentz based hyperbolic GCN, which is pretty much done in the paper. \n\n2. Limited experiments and lack of reproducibility. Experiments are performed on 4 standard small scale datasets, which are 4 highly overfitted datasets. For example, the disease and airport, they are so small datasets. From my experience, I can always get a very high accuracy, but with a large variance during multiple runs. No code or supplementary is provided for the model. Plus I can get higher accuracies with some baselines, for example, using HGCN on pubmed node classification, $80.2\\pm 0.3$ with 10 runs. It's really hard to derive meaningful conclusion from the experiments. \n\n------ 2.1 where is the training hyper-parameter? how many layers of the model, any regularization, hidden dimension? How many epochs? I am aware that many hyperbolic NNs report results of running thousand epochs with early stopping, while standard Euclidean GCNs, SGC report accuracy of 100 or 200 epochs. Is the comparison fair? \n\n3. Some claims are vague or inaccurate. For example, \n\n\"most of the existing hyperbolic networks build the network operations on the tangent space of the manifold, which is a Euclidean local approximation. This distorts the learnt features, limits the representation capacity of the network\"\n\nI don't think this holds true, generally the exp/log map is local approximation of the manifold. However, in the hyperbolic space, they are bijection between the tangent space and the hyperbolic space. Usage of exp map at origin is not a problem with respect to representation capacity, as one can use parallel transport to move it along the manifold, which is already introduced and used in many prior works. \n\n------ 3.1 It claimed that \"the full Lorentz transformation ... increase the model expressiveness ... with deeper networks.\" I failed to find evidence in the paper supporting this claim, in particular, personally I haven't seen any successful deep hyperbolic GCN style structure to work well even in existing/prior work. \n\n4. The word and text needs more polish. For example, \n\n\"This makes it hard to optimize deeper networks because of these back and forth mappings between the manifold and the tangent space, and limits the representation capabilities of the hyperbolic networks caused by distortion specially most of these works used the tangent space at the origin. \"\n\n[1] Chen et al. 2022, Fully hyperbolic neural networks. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "1: The contributions are neither significant nor novel.", "empirical_novelty_and_significance": "1: The contributions are neither significant nor novel."}, "extra_text": {}}, {"review_id": "H5oGms0y4d", "reviewer": "Reviewer_8Du2", "summary": "", "questions": "", "limitations": "", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "The paper looks at graph hyperbolic neural networks to learn maps from $\\mathbb{L}^n \\to \\mathbb{L}^d$ (where $\\mathbb{L}^d$ is $d-1$ dimensional Lorentz manifold. In particular, given a graph $G$ we learn features for each node using such a map and then use this for node classification and link prediction.\n\nThe novelty in the paper is in the parameterization of the maps from $\\mathbb{L}^n \\to \\mathbb{L}^d$ (where $\\mathbb{L}^d$. In particular, they use the parameterization given by Moretti 2002. They then show that such networks outperform other networks on a variety of standard datasets. \n\nThey also do an ablation study and show that their method learns near isometries. ", "summary_of_the_review": "In summary, I think the idea of doing hyperbolic graph neural networks without using the tangent space is important since it increases the flexibility, and stability of the method as well as reduces computational costs. \n\nHowever, it is not clear to be (due to the presentation) the novelty of the paper, the correctness of some of the claims, and the reproducibility of the results. I think these issues are fixable, but very much do need to be fixed. ", "strength_and_weaknesses": "**Strengths**\n---\n\n1) The method captures all Lorentz transformation. This is in contrast to Chen et al 2022 for which they show their parameterization captures all rotations and all boosts but not necessarily both together in one layer. However, this distinction may not be too important. \n\n2) The method seems to outperform other methods. But the important are small and may vanish with changing hyperparameters. \n\n**Weaknesses**\n---\n\n1) It is not clear to me how novel the method is. The decomposition into boosts and rotations is known (Moretti 2002). Further for the proofs in the texts, there are no statements, but statements being proved should be known (Moretii 2002, Chen et al 2022). \n\n2) There quite a few missing details including training details, statements of theorems among other things. ", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "2: Several of the paper’s claims are incorrect or not well-supported.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject