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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.
10.1109/ACCESS.2022.0092316

This work was supported in part by the Zhejiang Provincial Natural Science
Foundation of China under Grant LY21F020004, in part by the National Natural
Science Foundation of China Grant No. 61572243, in part by the Public Welfare
Technology Application Research Program Project of Lishui City under Grant
No.2022GYX12 and 2022GYX09, in part by the Intelligent Manufacturing
Technology Innovation Base Project of Lishui City under Grant HXZKA2022010, in
part by the Baishanzu National Park Scientific Research Program under Grant
No.2021KFLY07.

Corresponding author: Kenan Lou (nanekuol@126.com).

# Geometric Pooling: maintaining more useful information

HAO XU1,2  JIA LIU1,2  YANG SHEN1  KENAN LOU1  YANXIA BAO1  RUIHUA ZHANG1
SHUYUE ZHOU1,2  HONGSEN ZHAO1  ZHEN YE1  XINMIAO ZHU1,2  SHUAI WANG3  Lishui
University, Lishui 323000, China  Zhejiang Xiaomache Technology Co.,Ltd.,
Lishui 323000, China  CNRS, LPL, Aix Marseille University, Aix-en-Provence,
13100, France

###### Abstract

Graph Pooling technology plays an important role in graph node classification
tasks. Sorting pooling technologies maintain large-value units for pooling
graphs of varying sizes. However, by analyzing the statistical characteristic
of activated units after pooling, we found that a large number of units
dropped by sorting pooling are negative-value units that contain useful
information and can contribute considerably to the final decision. To maintain
more useful information, we proposed a novel pooling technology, called
Geometric Pooling (GP), containing the unique node features with negative
values by measuring the similarity of all node features. We reveal the
effectiveness of GP from the entropy reduction view. The experiments were
conducted on TUdatasets to show the effectiveness of GP. The results showed
that the proposed GP outperforms the SOTA graph pooling technologies by $1\%$
$\sim$ $5\%$ with fewer parameters.

###### Index Terms:

Graph Neural Networks, Pooling, Similarity.

=-21pt

##  I Introduction

The graph neural networks (GNNs) [3, 19] have been applied and developed in
many domains, such as Biomedicine [6, 9], Semantic Segmentation [14], Pose
Estimation [29, 26, 27, 28, 25], and Recommender System [30]. GNNs are able to
extract good representations from graph-structured data in non-Euclidean
space. Mimicking Convolutional Neural Networks (CNNs), researchers designed
GNNs sequentially containing convolutional filters, activation functions, and
pooling operators to extract features with non-linearity and invariance.
Nevertheless, the typical pooling technology in CNNs, such as max-pooling and
average-pooling, requires that the processed data are structured (i.e., the
data in a neighborhood are in a strong correlation assumption) and is not
suitable for non-Euclidean data. However, the graph-structured data does not
satisfy the assumption, and the pooling technology brings the key invariance
and increases the receptive field of the whole graph, is very important for
GNNs.

Many researchers paid attention to the GNN pooling technologies, which are
divided into Global Pooling [34, 1], Topology-based Pooling [5, 4], and
Hierarchical Pooling [31, 7]. Global pooling is a simple and effective
technology, which is a feature-based technology differing from the topology-
based pooling technology. Global pooling methods use manual summation or
learning methods to pool all the representations of nodes in each layer.
Although global pooling technology does not explicitly consider the topology
of the input graph, the previous graph convolutional filter has already
considered the topology and implicitly influenced the later pooling. Hence,
this paper focuses on the global pooling methods because of its computational
efficiency and universality.

Figure 1: We show the main difference between sorting pooling and the proposed
GP. Different graphs containing different-number nodes are marked in different
colors. The numbers in different nodes represent the sorting indices. For
example, the amplitude of the node indexed by 0 is larger than that of indexed
by 1. Meanwhile, the color contrast among different nodes in a particular
graph means the similarity among different nodes. For the above Sorting
Pooling, the maintaining nodes are indexed by 0, 1, and 2 (we assume there are
three nodes being maintained). In the below Geometric Pooling, the replaceable
nodes are dropped and the nodes carrying unique information of the node are
maintained.

However, the existing global pooling methods dropped much useful information
due to their decision criteria and the activated function, such as tanh, used
in the GNNs. For example, Sorting Pooling [34] maintains the large-value units
and drops small-value units by sorting the units. The effectiveness of Sorting
Pooling is depended on the assumption that the dropped units contribute little
to the final decision. However, due to the activation function, tanh, used in
Sorting Pooling, the sorting methods not only drop the small-value units but
also the negative-value ones. The negative-value units with large absolute
values actually carry considerable information, representing the node and
contributing much to the final decision. And unfortunately, tanh cannot be
replaced by the positive-value activation functions (such as ReLU, ReLU6)
because this leads to a performance drop.

To explore the number of dropped negative-value units and the small-value
units, which respectively contribute much and little to the final decision, we
counted their number in Fig. 2a. Obviously, the “useless” units accounted for
a small proportion of all dropped units, and “useful” units accounted for a
larger proportion. The dropped information problem cannot be addressed by a
simple absolute sorting pooling because when the number of dropped units is
fixed (Seeing [34]), the number of “useless” units is not enough and the
useful units are also dropped.

Seeing this, we proposed a novel pooling decision criterion - Geometric
Pooling (GP) to maintain more useful information, whether positive-value or
negative-value units. Different from the global pooling methods, which drop
the units with relatively less contribution, the proposed GP dropped the
relatively most replaceable units. Particularly, we used a metric such as
Euclidean distance, inner product, cosine similarity [22], and Mahalanobis
distance [16], to measure the similarity between different nodes. We used
Euclidean distance because of its simplicity and low computation cost. In the
graph classification task, we dropped the most replaceable node features. The
main difference between sorting pooling and GP is visualized in Fig. 1. By
exploring the effectiveness, we found that the proposed GP can be viewed as a
sort of regularization method, entropy reduction, such as Label Smoothing
[18], Mixup[33], and Cutmix [32]. They alleviate the over-confidence problem,
which leads to the over-fitting problem. Dropping the replaceable units
maximizes the compressing information rate and improves the representation of
the node. We stated the regularization view in Section III-D.

To verify the effectiveness of GP, we used the backbone of DGCNN [34], and
used GP to replace the original pooling technology. The evaluated experiments
are conducted on the TUdataset [17]. The experimental results show that GP
outperforms the SOTA graph pooling methods.

The main contributions of this paper are as follows.

  1. 1.

We analyzed the units dropped by the existing global pooling technologies and
found that many negative-value units with useful information are dropped.
Seeing this, this paper proposed a novel pooling decision criterion -
Geometric Pooling (GP) to drop the most replaceable units, maintaining more
useful information.

  2. 2.

By exploring the effectiveness, this paper viewed the proposed GP as a
regularization method, which smooths the distribution of the outputs and
maximizes the compressing information rate, addressing the overfitting
problem.

  3. 3.

Comprehensive evaluated experiments show that GP outperforms the existing deep
and kernel-based pooing technologies.

##  II Related Works

Graph global pooling technologies. Sorting Pooling [34] maintains the large-
value units and drops the small-value units. This is similar to the norm-based
pruning technologies, which determine the importance of the units by their
contributions to the final decision. Furthermore, SAGPooling [15] uses self-
attention [23] to mask the features pooled by Sorting Pooling. Different from
the Sorting Pooling, gPooling [8] not only uses the sorting indices to select
the node features but also the corresponding adjacency matrix. According to
the selected adjacency matrix, the topological links between the selected
nodes are strengthened, and the unimportant links are dropped.

Entropy reduction. Entropy reduction appear in many regularization methods.
The typical one is label smoothing [18]. Label smoothing improves the
robustness of the trained network by using soft targets that are a weighted
average of the hard targets and the uniform distribution over labels. This is
equivalent to the effectiveness of the method which encourages the output
distribution away from the distribution of the hard targets. Following label
smoothing, many researchers proposed many different modified version, such as
Mixup [33], Manifold Mixup [24], CutMix [32], to strengthen the entropy
reduction according to different tasks.

Zhang et. al proposed Mixup [33] that used the linear combination of different
sample of different classes to augment the training samples, and label
smoothing is also used to assign the label to the augmented training sample.
Manifold Mixup [24] is modified by Mixup, which randomly augments the feature
by the linear combination of different features. CutMix [32] is a
regularization method which fuses the advantages of Cutout and Mixup.

The above methods encourage the output distribution far away from the
distribution of the hard targets. This alleviates the over-confidence problem
of the model output, which results in the over-fitting problem. The proposed
GP is based on an argument that the similar features works repetitively for
the final decision, and we can drop some of them. This also alleviates the
over-confidence problem.

##  III Methods

In this section, we first set up a typical GNNs model. Then the number
distribution of the activated units is analyzed in the Sec. III-B. According
to the conclusion of the analysis, we stated the proposed Geometric Pooling
(GP) in the Sec. III-C. Finally, we give the effectiveness of GP from the
entropy reduction view in Section III-D.

###  III-A Set Up

A graph neural network consists of four parts: 1) graph convolution layers
extract nodes’ local substructure features and define a consistent node
ordering; 2) a pooling technology is used to unify the size of nodes and
impose invariance on the network; 3) an activation layer, such as tanh, is
used to give the non-linearity to the previous layers; 4) One or several
linear layers are used to inference the class of the input graph.

$\mathbf{X}=\\{\mathbf{x}_{i}\\}_{i=0}^{N}$ denotes the input set and
$\mathbf{x}_{i}\in\mathcal{R}^{n\times d}$, i.e., the input $\mathbf{x}_{i}$
consists of $n$ nodes and each node is a d-dimensional vector.
$\mathbf{Y}=\\{\mathbf{y}_{i}\\}_{i=0}^{N}$ denotes the predict set. In the
graph classification task, $\mathbf{A}\in\mathcal{R}$ is the adjacent matrix.
$\mathbf{A}$ is a symmetric 0/1 matrix, and the graph has no self-loops.

Graph convolutional layers. Graph convolutional layers aim at learning a
mapping
$\mathit{f}:(\mathbf{H}^{(l)},\widetilde{\mathbf{A}})\rightarrow\mathbf{H}^{(l+1)}$,
where $\mathbf{H}^{(l)}$ is the activation vector in the $l_{th}$ layer and
$\mathbf{H}^{(0)}=\mathbf{X}$. For example, we formulated the graph
convolution layers defined in [34]:

$\mathbf{H}^{(l+1)}=\mathit{\sigma}(\widetilde{\mathbf{D}}^{-1}\widetilde{\mathbf{A}}\mathbf{H}^{(l)}\mathbf{W}^{(l)}),$
(1)

where $\widetilde{\mathbf{A}}=\mathbf{A}+\mathbf{I}$ is the adjacency matrix
with added self-loops, $\widetilde{\mathbf{D}}$ is its diagonal degree matrix
with $\widetilde{\mathbf{D}}_{ii}=\sum_{j}\widetilde{\mathbf{A}}$,
$\mathbf{W}^{(l)}\in\mathcal{R}^{d_{l}\times d_{l+1}}$ is the trainable graph
convolution parameters of the $l_{th}$ layers, and $\mathit{\sigma}$ is a
nonlinear activation function. In Eq. 1, $\mathbf{H}^{(l)}\mathbf{W}^{(l)}$ is
a linear transformation for the input $\mathbf{H}^{(l)}$ to extract the key
information. Then $\widetilde{\mathbf{A}}\mathbf{H}^{(l)}\mathbf{W}^{(l)}$
used self-looped adjacency matrix to propagate node information to neighboring
nodes along with the topological structure. Finally,
$\widetilde{\mathbf{D}}^{-1}$ is used to keep a fixed feature scale after
graph convolution.

Pooling. Global pooling operator of GNNs can be viewed as a selected operator,
which selects the representative node features and impose invariance on the
model. The global pooling operator considers the importance of each node
feature.

Activation function. A typical activation function is the hyperbolic tangent
function (tanh):

$z=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}.$

###  III-B Analysis of the Number Distribution

We analyzed two sorts of sorting pooling: DGCNN [34] and Filter Norm Sorting
[15]. DGCNN concatenates the features of different graph convolutional layers
to represent the whole graph. In DGCNN, the contributions of the units are
determined by the values of the units in the last graph convolutional layer,
i.e., DGCNN drops the node features with small-value units in the last graph
convolutional layer. However, the contribution of a particular node ought to
be jointly determined by all features of this node, not only the units in the
last layer. This dropped a lot of representative node features (useful
information).

(a) sorting pooling

(b) geometric pooling

Figure 2: The counting histograms of sorting pooling and GP technologies. In
(a), the units are dropped by sorting pooling; in (b), the units are dropped
by geometric distance between the node features.

We counted the values of dropped units in Fig. 2. We counted the dropped unit
on D&D dataset, with setting $k$ to 20. As shown in Fig. 2 (a), sorting
pooling drops a large number of negative-value units but with large absolute
values. Differently, the proposed GP drops the node features sharing large
similarity with other node features. Obviously, in Fig. 2 (b), the dropped
units are concentrated around zero. In addition, the other dropped units
distribute uniformly, i.e., replaceable units are dropped. Consequently, more
useful information are maintained.

###  III-C Geometric Pooling

To get rid of the constraints in the analysis of Sec. III-B, we proposed a
novel pooling decision criterion - Geometric Pooling (GP). GP used the
similarity between different node features to judge the importance of the
node. The whole pooling process is formulated as follows.

We use the feature concatenation method in DGCNN [34] as an example, i.e., the
input features of a graph are the concatenation of the output different graph
convolution layers, denoted as $\mathbf{H}^{0:L}\in\mathcal{R}^{n\times
d^{\prime}}$, where $n$ is the node number of the graph,
$d^{\prime}=\sum_{l=0}^{L}d_{l}$, and $L$ is layer number of the whole
network. $\mathbf{H}^{0:L}$ consists of
$\mathbf{H}^{0:L}_{j}\in\mathcal{R}^{d^{\prime}}(j=1,\ldots,n)$, which denotes
the concatenated feature of each node. Then, the similarity (distance) between
different node features is computed as follows.

$\mathbf{S}_{j}=\sum_{i=1}^{n}\|\mathbf{H}^{0:L}_{i}-\mathbf{H}^{0:L}_{j}\|,$
(2)

where $\mathbf{S}=\\{\mathbf{S}_{j}\\}_{j=1}^{n}$ denotes the similarity
vector of all nodes, and $\|\cdot\|$ denotes the metric used to measure the
similarity.

According to $\mathbf{S}$, we sorted the $\\{\mathbf{S}_{j}\\}_{j=1}^{n}$ in
Eq. 2 to get the node features with least similarity:

$\mathbf{Idx}=Topk(\\{\mathbf{S}_{j}\\}_{j=1}^{n}).$ (3)

In Eq. 3, $Topk$ is a operator to select the index of the top-k minimal
similarity and the selected indices are saved in $\mathbf{Idx}$. $k$ is a
hyper-parameter, which means the number of maintaining node features, to make
sure the node number of graphs be same. By $\mathbf{Idx}$, we maintain the
node features with the index in $\mathbf{Idx}$. The whole pooling process is
stated in Alg. 1.

Algorithm 1 Geometric Pooling

1:k - the number of maintaining nodes

2:Input: $\mathbf{H}^{0:L}=\\{\mathbf{H}^{0:L}_{j}\\}_{j=1}^{n}$

3:for $1\leq j\leq n$ do

4: Compute $\mathbf{S}_{j}$ by Eq. 2

5:end for

6:Sort $\mathbf{S}=\\{\mathbf{S}_{j}\\}_{j=1}^{n}$

7:Compute $\mathbf{Idx}$ by Eq. 3

8:Select the node feature with index in $\mathbf{Idx}$

9:Output: $\\{\mathbf{H}^{0:L}_{k}\\}_{k\in\mathbf{Idx}}$

###  III-D Entropy Reduction: distribution drag

The typical convolutional neural networks gradually strengthen the influence
of the training samples, i.e., the distribution of output logits is forced far
away from the initial uniform distribution. We visualized this process in Fig.
3.

Figure 3: We explain the effectiveness of GP from the output distribution drag
view. Typically, common cross entropy loss singly encourage the output
distribution close to the distribution of the ground truth in one-hot manner.
This results to an over-fitting problem. By GP, the output distribution is
encouraged to be close to the uniform distribution, i.e., it is punished to be
far away from the ground truth.

To state the problem, we first give the formulation of the typical loss
function - Cross Entropy Loss:

$\mathcal{L}_{CE}=\sum_{\mathbf{x}_{i}}\mathbf{y}_{i}\log\frac{1}{\mathbf{q}(\mathbf{x}_{i})}=-\sum_{\mathbf{x}_{i}}\mathbf{y}_{i}\log\mathbf{q}(\mathbf{x}_{i}),$
(4)

where $\mathbf{x}_{i}$ is the training sample; $\mathbf{p}(\mathbf{x}_{i})$
denotes the output of the model; $\mathbf{y}_{i}$ is the label of
$\mathbf{x}_{i}$. When $\mathcal{L}_{CE}$ decreases, the distribution of
$\mathbf{y}_{i}$ is pulled to be close to $\mathbf{p}(\mathbf{x}_{i})$, which
is in a one-hot manner. The one-hot label is an extreme probability
formulation that is far away from the uniform distribution.

Although the one-hot label is the target of the task, a lot of regularization
research shows that just pulling the distribution close to the one-hot
distribution results in an over-fitting problem. For example, Miller [18]
proposed that the effectiveness of Label Smoothing results from the
distribution drag. Particularly, the one-hot label is modified to a
probability form:

$[0,0,1,0,0]\rightarrow[0.05,0.05,0.8,0.05,0.05].$

Obviously, $\mathbf{q}(\mathbf{x}_{i})$ is enforced to be close to the uniform
distribution. Label Smoothing effectively strengthens the robustness by
alleviating the too-high confidence problem at the training stage, i.e., the
over-fitting problem [12].

Many similar units filtered by the convolutional kernels jointly output the
final decision of the too-high confidence [11]. Hence dropping the units of
high similarity alleviates the too-high confidence problem. Except for the
pooling operator, the regularization effectiveness of the proposed GP can be
viewed as a penalty term:

$\mathbf{L}=\mathbf{L}_{CE}+\mathbf{L}_{ana}=-\sum_{\mathbf{x}_{i}}\left[\mathbf{y}_{i}\log\mathbf{q}(\mathbf{x}_{i})+\lambda\cdot
D_{KL}(\mathbf{U}_{[0,C]}\|\mathbf{q}(\mathbf{x}_{i}))\right],$ (5)

where $D_{KL}(\cdot\|\cdot)$ is a Kullback-Leibler divergence; $\lambda$ is
the scaling factor; $C$ is the class number of the task. We give the
$D_{KL}(\mathbf{U}_{[0,C]}\|\mathbf{q}(\mathbf{x}_{i}))$ as follows.

$\begin{split}D_{KL}(\mathbf{U}_{[0,C]}\|\mathbf{q}(\mathbf{x}_{i}))&=-\mathbf{U}_{[0,C]}\log\frac{\mathbf{q}(\mathbf{x}_{i})}{\mathbf{U}_{[0,C]}}\\\
&=\frac{1}{C}\log\frac{1}{C}-\frac{1}{C}\log\mathbf{q}(\mathbf{x}_{i})\end{split},$
(6)

where $\frac{1}{C}\log\frac{1}{C}$ is the negative entropy of the uniform
distribution $\mathbf{U}_{[0,C]}$ and
$-\frac{1}{C}\log\mathbf{q}(\mathbf{x}_{i})$ is the cross entropy between
$\mathbf{U}_{[0,C]}$ and $\mathbf{q}(\mathbf{x}_{i})$.
$\frac{1}{C}\log\frac{1}{C}$ is a constant in a particular task. When
$-\frac{1}{C}\log\mathbf{q}(\mathbf{x}_{i})$ decreases, the output is enforced
to be close to $\mathbf{U}_{[0,C]}$, and hence the over-fitting problem is
alleviated. To clearly show the distribution drag, we visualized the
distribution variation of the vanilla training in the above row and the
proposed GP in the below row.

##  IV Experiments

To show the effectiveness of the proposed GP, we embedded GP in different
methods to replace the original pooling technologies on the typical Graph
classification dataset - TUdataset.

###  IV-A Set Up

Datasets. TUdataset consists of too many sub-datasets. Hence we chose MUTAG,
PTC, NCI1, PROTEINS, D&D, COLLAB, IMDB-B, and IMDB-M to evaluate the
performance. Most chosen sub-datasets contain graphs of two categories, except
for COLLAB and IMDB-M, which contain graphs of three categories. The detailed
information of the sub-datasets is given in Table I.

TABLE I: The dataset description. Dataset | MUTAG | PTC | NCI1 | PROTEINS
---|---|---|---|---
#Graphs | 188 | 344 | 4,110 | 1,113
#Classes | 2 | 2 | 2 | 2
#Node Attr. | 8 | 19 | 38 | 5
Dataset | D&D | COLLAB | IMDB-B | IMDB-M
#Graphs | 1,178 | 5,000 | 1,000 | 1,500
#Classes | 2 | 3 | 2 | 3
#Node Attr. | 90 | 1 | 1 | 1

Comparison methods. We compare the proposed GP with sort pooling in DGCNN
[34], SAGPooling [15], DiffPooling [31], and GRAPHSAGE [10] to show GP’s
effectiveness. Also, several kernel-based algorithms are compared, including
GRAPHLET [21], Shortest-Path [2], Weisfeiler-Lehman kernel (WL) [20], and
Weisfeiler-Lehman Optimal Assignment Kernel (WL-OA) [13].

Model Configuration. We used five Graph Convolutional layers to extract
different-level features of the graph and a GP layer is used to obtain the
salient features and make each graph share the same node number and the same
node feature number. The hyper-parameter $k$ in Eq. 3 is varying for different
datasets. We set the $k$ such that 60% graphs have nodes more than $k$. For
fair evaluation, we used a single network structure on all datasets, and ran
GP using exactly the same folds as used in graph kernels in all the 100 runs
of each dataset.

###  IV-B Comparison with deep approaches

As given in Table II, the results of the compared deep approaches show the
effectiveness of the proposed GP and its variation – GP-mixed. GP-mixed first
used sort pooling [34] to drop some units contributing little to the final
decision And then GP is used to drop the replaceable units.

TABLE II: The performance comparison among different deep approaches. The top two test classification accuracy on each dataset is in bold. Methods | Datasets
---|---
MUTAG | PTC | NCI1 | PROTEINS
DGCNN | 85.83 | 58.59 | 74.44 | 75.54
GRAPHSAGE | – | – | – | 70.48
SAGPooling | – | – | 74.06 | 70.04
DiffPooing | – | – | – | 76.25
GP | 86.10 | 61.30 | 75.80 | 77.10
GP-mixed | 86.44 | 61.86 | 76.29 | 77.64
| D&D | COLLAB | IMDB-B | IMDB-M
DGCNN | 79.37 | 73.76 | 71.50 | 46.47
GRAPHSAGE | 75.42 | 68.25 | 68.80 | 47.6
SAGPooling | 76.19 | 69.47 | – | –
DiffPooing | 79.64 | 75.48 | – | –
GP | 80.21 | 74.63 | 72.20 | 47.89
GP-mixed | 80.28 | 75.50 | 72.68 | 48.32

In Table II, we give several state-of-the-art GNNs pooling performance. For
example, on D&D, the proposed GP outperforms DGCNN and SAGPooling by 0.84% and
4.02%, respectively. On COLLAB, GP outperforms DGCNN by 0.87%. Although GP
underperforms DiffPooling, GP-mixed slightly outperforms DiffPooling. In
general, the proposed GP outperforms the SOTA deep approaches by 0.7% $\sim$
5.05%.

###  IV-C Comparison with kernel-based approaches

Kernel-based approaches are also compared. As shown in Table III, the proposed
GP and its variant outperform the kernel-based approaches on most evaluation
datasets. For example, GP-mixed outperforms the compared kernel-based
approaches by 1.21% $\sim$ 4.73% on PROTEINS. Unfortunately, on COLLAB, GP and
GP-mixed underperform WL and WL-OA.

TABLE III: The performance comparison between GP and kernel-based approaches. The top two test classification accuracy on each dataset is in bold. Methods | Datasets
---|---
MUTAG | PTC | NCI1 | PROTEINS
GRAPHLET | 85.7 | 54.7 | 76.20 | 72.91
SHORTEST-PATH | 83.15 | – | – | 76.43
WL | 84.11 | 57.97 | 84.46 | 73.76
WL-OA | 84.50 | – | – | 75.26
GP | 86.10 | 61.30 | 75.80 | 77.10
GP-mixed | 86.44 | 61.86 | 76.29 | 77.64
| D&D | COLLAB | IMDB-B | IMDB-M
GRAPHLET | 74.85 | 64.66 | – | –
SHORTEST-PATH | 78.86 | 59.10 | – | –
WL | 74.02 | 78.61 | 71.26 | 46.52
WL-OA | 79.04 | 80.74 | – | –
GP | 80.21 | 74.63 | 72.20 | 47.89
GP-mixed | 80.28 | 75.50 | 72.68 | 48.32

###  IV-D Comparison on the computation cost

To show the simplification of the proposed GP, we also compute the computation
cost of the compared deep approaches.

Figure 4: Parameter comparison. We give the parameters of different deep
approaches on two datasets: D&D and PROTEINS. For example, let GP/DGCNN (D)
denote the parameter amount of the proposed GP and DGCNN on D&D dataset. Also,
SAGPooling (P) denotes the parameter amount of SAGPooling on PROTEINS dataset.
The numbers on the colored bars mean the exact parameter number.

As shown in Fig. 4, the red bars, which mean the parameter amount of the
proposed GP, are far less than that of SAGPooling and DiffPooling approaches.
On D&D, the parameter amount of GP accounts for 39.44% of SAGPooling and
24.55% of DiffPooling. This shows the computation efficiency of the proposed
GP.

##  V Conclusion

In the work, we proposed a novel graph pooling technology based on the
geometric distance between different node features, called Geometric Pooling
(GP). The proposed GP maintains more useful information, which is dropped by
other global graph pooling technologies. In addition, the replaceable feature
drop brings the entropy reduction, which encourages the output distribution
away from the hard targets. This alleviates the over-confidence problem and
hence address the over-fitting problem.

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