# Dense Graph Convolutional Neural Networks on 3D Meshes for 3D Object
Segmentation and Classification

Wenming Tang  Guoping Qiu<EMAIL_ADDRESS>(G. Qiu). College of
Electronics and Information Engineering, Shenzhen University, Shenzhen, China
Guangdong Key Laboratory of Intelligent Information Processing, Shenzhen
University, Shenzhen, China  Shenzhen Institute of Artificial Intelligence and
Robotics for Society, Shenzhen, China  School of Computer Science, The
University of Nottingham, UK

###### Abstract

This paper presents new designs of graph convolutional neural networks (GCNs)
on 3D meshes for 3D object segmentation and classification. We use the faces
of the mesh as basic processing units and represent a 3D mesh as a graph where
each node corresponds to a face. To enhance the descriptive power of the
graph, we introduce a 1-ring face neighbourhood structure to derive novel
multi-dimensional spatial and structure features to represent the graph nodes.
Based on this new graph representation, we then design a densely connected
graph convolutional block which aggregates local and regional features as the
key construction component to build effective and efficient practical GCN
models for 3D object classification and segmentation. We will present
experimental results to show that our new technique outperforms state of the
art where our models are shown to have the smallest number of parameters and
consietently achieve the highest accuracies across a number of benchmark
datasets. We will also present ablation studies to demonstrate the soundness
of our design principles and the effectiveness of our practical models.

###### keywords:

3D Meshes, Graph Neural Network, Densely Connected, Classification,
Segmentation

††journal: Journal of Image and Vision Computing

##  1 Introduction

The increase in computing power and the emergence of various high-efficiency
deep learning architectures have rapidly advanced many areas of computer
vision. However, most of the deep learning architectures are designed for 1D
(Language) and 2D (Image) data. The main reason is that deep learning requires
a large amount of training data while the cost of acquiring 3D data is much
higher than that of 1D and 2D, and the representation of 3D data is much more
complicated. Therefore, the application of deep learning in 3D is not as
efficient in 2D [1]. Fortunately, with the development of 3D sensor
technology, the cost of 3D data acquisition is getting lower and lower, which
makes the application of deep learning increasingly popular in the 3D computer
vision community.

In 3D computer vision, 3D mesh is a very effective form for representing 3D
objects and occupies an important position in 3D computer vision and computer
graphics. Relying on the combination of geometrical structures of vertices,
edges and faces, 3D mesh has the advantages of containing rich details,
representing specific geometric features, and occupying a small memory
footprint. Therefore, 3D mesh has obvious advantages over point clouds and
voxels in describing 3D shapes. Among the many types of 3D mesh, 3D watertight
mesh is the most common one. Watertight means that the mesh on all of the
surfaces is complete, the lines of the mesh create valid elements, and the
mesh connects to adjacent surfaces around the perimeter so that the volume is
fully enclosed. Its geometric characteristics have not only become a research
hotspot, but also have been widely used in games, 3D animation, 3D printing
and scanning, industrial manufacturing, etc.

Classification and Segmentation are the two fundamental tasks in computer
vision. Deep learning has achieved great success in 2D image processing, such
as classification and segmentation [2, 3]. However, in 3D computer vision, the
irregularities of the data and the complexity of the 3D geometric structure
make the classification and segmentation tasks more challenging. Nevertheless,
researchers have attempted various methods, such as converting 3D data into
image data. For example, by using multi-view images to express 3D shapes [4],
thereby obtaining data that can be directly input to the Convolutional Neural
Networks(CNN), or by designing a unique multi-layer perception network
(PointNet) to implement 3D point cloud classification and segmentation tasks
[5]. PointNet is the first work where the original 3D data is directly input
into the neural network without conversion. Since PointNet only relies on 1D
convolution of 3D point cloud, it is difficult to capture the point cloud
neighborhood information. In a subsequent work, the authors proposed an
improved version of PointNet (PointNet++ [6]). The addition of point cloud
neighborhood information improves the multi-layer perceptron network’s ability
to perceive the local area of the point cloud, which enhanced the network’s
classification and segmentation capabilities. Continuing from this technical
route, Jiang et al. [7] proposed a new network named PointSIFT. They brought
SIFT to 3D point clouds which improved the performance of the network in 3D
point cloud learning. The neighborhood information of 3D data is not fixed and
unique as in image, so the results obtained by different neighborhood
constructions can vary greatly. Therefore, when applying the deep learning
framework to 3D data, enhancing the perception of local (neighborhood)
information is an effective method to improve network performance.

Meanwhile, deep learning on 3D mesh has made great progress, and some
excellent work has appeared the literature [8, 9, 10, 11]. Yutong et al. [8]
designed a MeshNet network by using mesh data. The MeshNet takes the face of
the mesh as a unit and uses the index of the face’s three neighborhood faces
and its own geometric characteristics to achieve mesh classification and
retrieval. MeshNet only uses the index of the neighborhood face, but other
geometric features of the neighborhood face deserve further study. Hanocka et
al. [11] proposed a CNN network based on the edge of the mesh and named it
MeshCNN. Based on the geometric characteristics of two faces on each edge, the
pooling of the mesh was designed through the collapse of the edge. The unique
CNN network achieved good performances in 3D mesh classification and
segmentation. MeshCNN feeds 3D mesh data in the non-Euclidean space to a CNN
network implemented by 2D convolution.

However, it’s well known that a 3D mesh is a graph with a natural topology
composed of vertices and edges. Therefore, we can directly utilize its
topological structure and geometric characteristics without having to convert
the graph structure to adapt to a deep learning framework. In this way, we can
use the original geometric characteristics of the 3D mesh, which is more
intuitive and general. 3D mesh’s geometric topology and non-Euclidean spatial
characteristics make it well suited for graph convolutional neural networks
(GCNs) [12]. GCNs are designed to solve non-Euclidean spatial data problems
and realise the conversion between the spatial domain and the spectral domain
through the Laplacian operator, which establishes associations between graph
nodes. Better exploiting the relations between the graph nodes is one of the
methods that researchers can use to improve the performance of deep learning
in 3D data.

In this paper, we present densely connected graph convolutional networks on 3D
meshes (MDC-GCNs). Aiming to make 3D mesh data as convenient as image data to
enter the deep learning framework, we convert 3D meshes into graph structure
data. We also introduce a novel set of features for each node to enhance the
expressive power of the nodes. The new features include the space coordinates
of the vertex $P$, the normal of the vertex $N_{v}$, the Gaussian curvature of
the vertex $GC$, the normal of the face $N_{f}$, and the angle between the
faces $\theta$. MDC-GCN is a deep architecture that utilises the densely
connected mechanisms of the GCN to integrate local and non-local features.
Hence, this network has better learning capabilities for 3D mesh than the
normal GCNs. The main contributions of our paper are as follows:

  * 1.

A novel graph data structure for representing 3D mesh data. We first convert
the 3D mesh to a graph where each face on the mesh has a corresponding node on
the graph. A novel 1-ring neighbourhood structure of the face (see Fig. 1) is
constructed to derive a multi-dimensional feature vector to represent the
node. Each node’s feature vector includes the geometric characteristics of the
face and descriptions of its physical connection with its 1-ring neighbours.

  * 2.

A novel graph convolutuional neural network architecture on 3D meshes for
achieving state of the art results in 3D object segmentation and
classification. Using the new graph representation of 3D mesh, we construct
densely connected graph convolutional neural networks for implementating 3D
object segmentation and classification. We present extensive experimental
results to show that our new method outperformed state of the art.

##  2 Related work

In this section, we will briefly review three related topics: densely
connected convolutional networks, graph convolutional networks, and graph
convolutional networks application in 3D mesh.

###  2.1 Densely Connected Convolutional Networks

In recent two decades, deep learning has played a pivotal role in computer
vision. In different applications, researchers have designed different
networks. As the complexity of the task increases, the network design becomes
deeper and deeper, which will bring about the vanishing gradient problem. To
solve this issue, researchers have proposed many solutions, such as: ResNets
[13], Highway Networks [14], Fractal Nets [15], and Stochastic depth networks
[16]. Gao et al. [17] designed a dense connection network (DenseNets) in which
each layer in the dense block receives feature maps from all previous layers
and passes its output to all subsequent layers. The feature maps received from
other layers are merged through cascade. Since the network uses a special
connection method of dense connections, the number of layers is reduced.
Besides, the reuse of feature maps reduces the network parameters which in
turn alleviates the vanishing gradient problem. This idea of efficiently using
feature maps between layers was widely used by subsequent researchers.

###  2.2 Graph Convolutional Networks

Graph convolutional networks fill the gaps in the development of deep learning
that CNNs cannot directly operate non-Euclidean spatial data such as social
networks [18], protein-protein interaction networks [19], and knowledge graphs
[20]. Similar to the convolution kernel operation of CNN, GCN is also updated
by the fusion of information between node neighborhoods. Through the
application of convolution operation to the Laplacian matrix of a graph, a
connection is established between the spatial domain and the spectral domain
[21]. This convolution operation is actually a special form of Laplacian
smoothing. Therefore, too many layers will cause over smoothing , hence, a GCN
network usually has a depth of 3 or 4 layers. As the complexity of the task
increases, a basic GCN network cannot meet the requirement. Li et al. [22]
introduced a deep GCNs by employing the concepts of residual, dense
connections and dilated convolutions of CNNs. They designed a 56-layer GCN
network that achieved state-of-the-art results. Guo et al. [22] designed a
graph-to-sequence learning Densely Connected Graph Convolutional Networks (DC-
GCNs). This form of GCN combines the local and non-local features of nodes to
achieve better graph structure representation in natural language processing
(NLP).

###  2.3 Graph Convolutional Networks application in 3D mesh

GCN has also made great progress in the field of 3D mesh. Choi et al. [23]
proposed a GCN network that can extract features from 2D human pose images and
generate a 3D human pose mesh in a coarse-to-fine manner. Xin et al. [24]
proposed a view-based GCN network which uses CNN to extract features from the
multi-view picture of the 3D mesh and then constructs a graph structure to
accomplish the classification and retrieval tasks by combining CNN and GCN.
The above-mentioned work use CNN directly or indirectly to extract the
features of the image and generate correlation with GCN. Researchers have
developed solutions that exploit the inherent geometric characteristics of 3D
surfaces to design different GCNs to accomplish different tasks [9, 25, 26,
27, 28]. Ilya Kostrikov et al. [9] designed a GCN network called surface
networks by leveraging extrinsic differential geometry properties of 3D
surfaces to enhance modelling power. In particular, they used the Dirac
operator, whose spectrum detects principal curvature directions instead of the
classical Laplace operator, which directly measures mean curvature. Miguel et
al. [25] designed a Graph CNN based on graph signal processing theory for 3D
point cloud and 3D mesh classification. Wu et al. [26] used the 3D mesh face
as a unit, and used the geometric features of the face to construct a graph
structure through node association. They designed a GCN network through the
layer-wise propagation rule to achieve 3D shape co-segmentation, and verify
the effectiveness of the algorithm through the COSEG dataset benchmark. Nitika
et al. [27] designed a novel graph convolution operator to establish the
connection between filter weights and arbitrary connected graph neighborhoods
which achieved shape representations without relying on shape descriptors.
Litany et al. [27] designed a variational autoencoder network based on graph
convolution operators to complete the deformable shape completion task. The
innovation is that it can handle local information of any 3D shape without
providing training data for local information and can be applied to any 3D
shaped data. Milano et al. [12] designed a primal-dual framework GCN named PD-
MeshNet. PD-MeshNet takes the edges and faces of the 3D meshes as the features
of the graph nodes, and then adds an attention mechanism to achieve
classification and segmentation.

##  3 Method overview

The standard GCN network only has 2-3 layers, which is difficult to meet the
needs of complex tasks such as 3D object segmentation and classification.
Therefore, we need to improve the design of CGN networks. Firstly, we improve
the descriptive ability of the graph data by improving the local features of
each node, which is achieved through utilizing 1-ring neighborhood spatial and
structural features of the mesh face. Secondly, we improve the aggregation
ability of the local and non-local features of the GCN network by designing
the Densely Connected (DC) GCN block and use it as the key component for
constructing effective and efficient GCN application models.

Figure 1:  The 1-ring neighbourhood structure of a face on a 3D mesh (top) and
the relation between a 3D mesh and its corresponding graph (bottom). Each face
in the mesh has a corresponding node in the graph, each edge on the graph
represents a connection between a face and one of its 1-ring neighbours.

(a) 1-ring neighborhood of a vertex (b) 1-ring neighborhood of a face

Figure 2:  3D triangle meshes.

###  3.1 1-ring neighborhood and node feature vector

It’s well known that in a 3D triangle mesh, each face contains at most three
adjacent faces, and the 3D triangle mesh is a non-Euclidean space. The 3D mesh
structure data $\mathcal{M}=\\{V,~{}\mathcal{E},~{}F\\}$ of vertices, edges
and faces has a clear topology. Therefore, we can convert the 3D triangle mesh
into graph data
$\mathcal{M}=\\{V,~{}\mathcal{E},~{}F\\}~{}\Rightarrow~{}\mathcal{G}=\\{N,~{}E\\}$.
A face $\\{F\\}$ on the mesh $\mathcal{M}$ has a corresponding node $\\{N\\}$
on the graph $\\{\mathcal{G}\\}$. An edge $\\{E\\}$ on the graph represents
the corresponding edge $\\{\mathcal{E}\\}$ between a face and one of its
1-ring neighbours on the mesh $\mathcal{M}$. In order to enhance the
perception of the local area of the graph node, we use the spatial and
structural geometric features ${F}^{(\mathcal{M})}_{faces}$ of the 1-ring
neighborhood of the mesh face as the feature ${F}^{(\mathcal{G})}_{nodes}$ of
the graph node, as shown in Eq. 1:

${F}^{(\mathcal{G})}_{nodes}={F}^{(\mathcal{M})}_{faces}=\\{\\{P\\},\\{N_{v}\\},\\{GC\\},\\{N_{f}\\},\\{\theta\\}\\}$
(1)

where 1-ring neighborhood spatial and structural features
${F}^{(\mathcal{M})}_{faces}$ (57-dimensional vectors) include spatial
features: the vertex spatial position ($\\{P\\}$), and structural features:
the vertex normal ($\\{N_{v}\\}$), Gaussian curvature ($\\{GC\\}$), the face
normal ($\\{N_{f}\\}$), angles between the face and its 1-ring neighbourhood
faces ($\\{\theta\\}$), as shown in Fig. 1. The definition of Gaussian
curvature on a triangle mesh is via a vertex’s angular deficit [29]:

$gc(v_{i}^{\prime})=(2\pi-\sum_{j\in N(i)}\theta_{j})/A_{N(i)}$ (2)

where $N(i)$ are the triangles incident on vertex $i$ and $\theta_{j}$ is the
angle at vertex $i$ in triangle $j$, $A_{N(i)}$ is the sum of areas of $N(i)$,
as shown in Fig. 2 (a). Gaussian curvature is the most important intrinsic
geometric quantity in surface theory, and it reflects the degree of curvature
of a curved surface. So we introduce it to describe the local features of the
mesh. The angle $\theta_{i}$ in the angle set $\\{\theta\\}$ is defined as the
angle between the normals of the two faces ( Eq. 3), as shown in Fig. 2 (b).

$\theta_{i}=\arccos(\frac{\vec{n}_{f_{i}}\cdot\vec{n}_{f_{j}}}{\|\vec{n}_{f_{i}}\|\cdot\|\vec{n}_{f_{j}}\|})$
(3)

The normal $\vec{n}_{v_{i}}$ in the normal set $\\{N_{v}\\}$ is as Eq. 4 [30]:

$\vec{n}_{v_{i}}=\sum_{j\in F_{v}(i)}A_{j}\vec{n}_{f_{j}},$ (4)

where $A_{j}$ is the corresponding face area and $\vec{n}_{f_{j}}$ is the
value of the $j$ face normal, $F_{v}(i)$ is the number of faces in the $i$-th
vertex-ring. The introduction of angles $\\{\theta\\}$, vertex normals
$\\{N_{v}\\}$ and face normals $\\{N_{f}\\}$ is to enhance the local
structural features of the graph nodes, and the vertex space coordinates are
used to describe the spatial features of the graph model.

Using 1-ring neighborhood of the faces instead of the vertices as graph nodes
has the following advantages: A triangular face contains three graph, and a
face contains more geometric features than a vertex. The 1-ring neighborhood
of a triangular face covers a larger area than a vertex, thus enabling nodes
to enhance their perception of local areas.

###  3.2 Densely Connected Convolutional Block

Graph convolution can directly operate on non-Euclidean space data, similar to
image convolution operations on image data. The GCNs convolution operation is
obtained by using the weighted average of its neighboring nodes. Kipf et al
[18] proposed a GCN where the node feature convolution operation in the graph
$G=\\{N,\varepsilon\\}$ is defined as:

$H_{i}^{(l+1)}=\sigma(\sum_{j\in{\tilde{N}(i)}}H_{j}^{(l)}W^{(l)}+b^{(l)})$
(5)

where $H_{i}^{(l+1)}$ represents the feature of the $i$-th node in the
$(l+1)$-th layer, $\sigma$ is a non-linear activation function, $W^{(l)}$ is
the weight matrix, $b^{(l)}$ is the bias vector, $\tilde{N}(i)$ is the
neighborhood set of node i $\tilde{N}(i)=\\{j\in N|(i,j)\in\varepsilon\\}$.

Similar to the traditional convolutional neural network of computer vision, as
the number of network layers increases, the gradient disappears. The increase
in the number of GCN layers will make the effect worse [18]. If there is no
mechanism to restrict the convolutional form of the GCN, then, the best result
is 2 layers [31]. However, shallow networks cannot effectively capture non-
local features, and have poor applicability to larger graph structures.
Bastingset et al. [32] introduced the resnet mechanism [13] into GCNs, as
shown in Eq. 6, each node feature is updated according to Eq. 5, and then
combined with the updated features of the previous layer.

$H_{i}^{(l+1)}=\sigma(\sum_{j\in{\tilde{N}(i)}}H_{j}^{(l)}W^{(l)}+b^{(l)})+H_{i}^{(l)}$
(6)

Subsequently, researchers have introduced the cyclic layers into the
convolutional layer to deepen the network (up to 6 layers) [33]. In order to
design a deeper GCN, Guo et al [33] introduced the densenet [34] propagation
mechanism into GCNs. As shown in Eq. 7, the feature of node $i$ in the
$(l+1)$-th layer is not only $H^{(l)}$, but also the set of node features of
all previous layers {$H^{(l)}$, $H^{(l-1)}$, …, $H^{(1)}$}.

$H_{j}^{(l)}=LA(H_{j}^{(l)},~{}H_{j}^{(l-1)},~{}...,~{}H_{j}^{(1)})$ (7)

Figure 3:  The dense connected block of GCN

where the LA function can be concatenation, maxpooling or LSTM-attention
operations [35]. The advantage of this design is that the number of network
layers can be increased deeper, local and non-local features can be better
combined, and it is more conducive to improving the description ability of
GCNs for graphs.

In 3D mesh classification and segmentation tasks, it is necessary to consider
the fusion of local features and non-local features, so we designed a densely
connected (DC) GCN block, as shown in Fig. 3. For DC block specifically, this
module is composed of five ordinary GCN layers, and the connection method is
realized as Eq. 8 and 5.

$H_{j}^{(l)}=[H_{j}^{(l-1)};~{}H_{j}^{(l-2)};~{}...;~{}H_{j}^{(1)}]$ (8)

where $[~{}]$ is a concatenation of the node feature produced in layers
$1,~{}2,~{}...,~{}(l-1)$. In order to improve the parameter efficiency similar
to DenseNets [34], and we define the feature dimension of each layer of the
output graph node in the DC block to be equal, as show in Eq. 9,

$DIM(H_{i(out)}^{(1)})=DIM(H_{i(out)}^{(2)})=...=DIM(H_{i(out)}^{(n)})=\tau$
(9)

where $H_{i(out)}^{(1)}$ represents the output feature of the $i$-th node of
the first layer, $DIM(X)$ represents the dimensionality of the vector $X$, and
$\tau$ can be freely set according to different tasks by the users. Therefor
the node feature growth rate could be defined as follow:

$DIM(H_{i(input)}^{(l)})=(n-1)*DIM(H_{i(out)}^{(l-1)})+DIM(H_{i(out)}^{(1)}).$
(10)

In order to effectively use the parameters and prevent the feature dimension
of GCNs from being too wide, $n$ cannot be set too large. Our experiments
empirically set $n~{}=~{}5$. As shown in Fig. 3, the dimension of the input
feature of the 5-th layer of the GCN in the DC block is
$DIM(H_{i(input)}^{(5)})=(5-1)*\tau+\tau=5\tau$. It should be noted that the
dimension of the DC input is $\tau$ and that of the output is $6\tau$.

###  3.3 MDC-GCN for 3D Mesh Classification

The proposed MDC-GCN for 3D mesh classification is composed of a DC block, a
GCN layer, a mean graph nodes layer, and a linear layer as illustrated in Fig.
4 . An ordinary GCN layer is used as the terminal layer between the DC block
and the input. Through this layer, we can get the features of the input graph
node defined by the DC block. After the feature extraction operation of the
graph convolution of a DC block, the feature average value of the graph node
is obtained through a layer of graph mean nodes, and finally the
classification category is output through the linear layer. The parameter
design of the MDC-GCN for 3D mesh classification is shown in Table 1.

With the help of the multi-dimensional features of 1-ring structure of a
single node and the design of the DC block that aggregates local and non-local
features, MDC-GCN only uses 8 GCN layers to complete the 3D mesh
classification task. The experiments and ablation experiments in the following
section verify that the modular design of MDC-GCN allows users to easily
change the parameters, which is very simple and efficient.

Figure 4:  MDC-GCN for 3D mesh classification. Table 1: MDC-GCN classification network configuration. Parameters | GCN layer |
---|---|---
$(in=\textbf{input},~{}out=1024)$ | 1 |
$(in=1024,~{}out=1024)$ | 2 |  }5*[DC block]
$(in=2048,~{}out=1024)$ | 3 |
$(in=3072,~{}out=1024)$ | 4 |
$(in=4096,~{}out=1024)$ | 5 |
$(in=5120,~{}out=1024)$ | 6 |
$(graph~{}mean~{}nodes)$ | 7 |
$(in=6144,~{}out=\textbf{output})$ | 8 |

###  3.4 MDC-GCN for 3D Mesh Segmentation

In the deep learning literature of 2D computer vision, the segmentation
network is often much more complicated than the classification network. The 3D
mesh segmentation task can be regarded as a multi-classification task, and its
complexity is far greater than that of 3D mesh classification, so more
parameters are needed to meet the needs of the segmentation task. Different
from the design of many deep GCN networks, the efficient local feature
description of the graph data and the design of the DC block enable MDC-GCN to
complete the 3D mesh segmentation with very few GCN layers. The design of 2 DC
blocks instead of more or fewer blocks is to achieve balance between
performance and parameters size. See Table 7.

The proposed MDC-GCN for 3D mesh segmentation is composed of 2 DC blocks and 3
GCN layers as illustrated in Fig. 5 . Different from the classification task,
we introduce two DC blocks and use a GCN as the intermediate connection layer
for feature extraction. Finally, a layer of GCN is used as the output layer,
and the dimension of its output is the segmentation category. The parameter
design of the MDC-GCN for 3D mesh segmentation is shown in Table 2.

Figure 5:  MDC-GCN for 3D mesh segmentation. Table 2: MDC-GCN segmentation network configuration. Parameters | GCN layer |
---|---|---
$(in=\textbf{input},~{}out=1024)$ | 1 |
$(in=1024,~{}out=1024)$ | 2 |  }5*[DC block 1]
$(in=2048,~{}out=1024)$ | 3 |
$(in=3072,~{}out=1024)$ | 4 |
$(in=4096,~{}out=1024)$ | 5 |
$(in=5120,~{}out=1024)$ | 6 |
$(in=6144,~{}out=1024)$ | 7 |
$(in=1024,~{}out=1024)$ | 8 |  }5*[DC block 2]
$(in=2048,~{}out=1024)$ | 9 |
$(in=3072,~{}out=1024)$ | 10 |
$(in=4096,~{}out=1024)$ | 11 |
$(in=5120,~{}out=1024)$ | 12 |
$(in=6144,~{}out=\textbf{output})$ | 13 |

###  3.5 Optimizer, loss function and implementation platform

The classification target variables of MDC-GCNs are discrete so we use cross
entropy to define loss function. Segmentation can also be seen as a multi-
classification task. Hence, the loss function of MDC-GCNs can be described as:

$loss(x,class)=-\log\left(\frac{\exp(x[class])}{\sum_{j}\exp(x[j])}\right)=-x[class]+\log\left(\sum_{j}\exp(x[j])\right)$
(11)

where $x$, $class$, $j$ are predicted vector, ground truth label and the
$j$-th value of $x$, respectively. In the selection of MDC-GCN optimizer, we
use the Adam optimizer proposed by Kingma et al [36].

We use the open source efficient graph neural network framework (Deep Graph
Library, DGL) proposed by Want et al [36] to implement the MDC-GCNs. Our
experimental platform: windows 10, NVIDIA GTX 1080 graphics card, Intel Xeon
quad-core 3.7 Ghz CPU, and 96 G RAM.

##  4 Experiments

We have conducted comparative analysis of multiple sets of experiments of
applying MDC-GCNs to the problems of segmenting 3D shapes and classifying 3D
objects represented in 3D meshes. We will first explain the datasets used in
the experiments and then present experimental results of 3D mesh segmentation
and classification respectively.

###  4.1 Datasets and Data Preprocessing

We use the open source datasets from [11] to test the MDC-GCNs’ performances
in 3D mesh segmentation and classification. As mentioned earlier, we use the
triangular faces of the 3D mesh as the basic processing units, which is
similar to the pixels of the image. In order to train MDC-GCNs more
effectively, we resize the meshes in the dataset to a mesh with a fixed number
of faces. As in meshCNN, we don’t need to fix the number of input mesh faces
and can input meshes of any resolution. Unlike meshCNN, our smallest unit is a
triangle face instead of the edge of a triangle face. We use triangular faces
as the unit, and use the spatial and structural features of its 1-ring
neighborhood to convert the mesh into the input graph structure data of MDC-
GCNs, as shown in Fig. 1. It is worth noting that MDC-GCNs do not rely on data
augmentation.

###  4.2 Classification

In the SHEREC dataset, there are 30 different types of rigid and non-rigid
watertight meshes. Fig. 6 shows four types of mesh. Each type contains 20
meshes of the same type but with different shapes. Each mesh contains 500
faces, and each face is used as a graph node. Each node contains
$57$-dimensional feature vectors. If the input batchsize is $B$, then the
input data dimension is $(B\times 500\times 57)$. Similar to GWCNN [37] and
meshCNN [11], we divide $80\%$ (16) and $50\%$ (10) of each category into two
training sets. We also follow the random division mechanism of meshCNN. Our
result is the average value obtained from $5$ randomly generated training
sets. The learning rate $lr=0.0003$, $droupout=0.3$, and the optimizer is
Adam. Comparison with methods in the literature is shown in Table 3.

Table 3: Results on SHREC dataset Algorithm | Accuracy (Split 16) | Accuracy (Split 10)
---|---|---
Our | 99.7$\%$ | 99.2$\%$
Milano et al. [12] | 99.7$\%$ | 99.1$\%$
Hanocka et al. [11] | 98.6$\%$ | 91.0$\%$
Ezuz et al. [37] | 96.6$\%$ | 90.3$\%$
Sinha et al. [38] | 96.6$\%$ | 88.6$\%$
Zhirong et al. [39] | 48.4$\%$ | 52.7$\%$
Bronstein et al. [40] | 70.8$\%$ | 62.6$\%$

Figure 6:  The four meshes in SHREC[40], from left to right are hand, lamp,
glasses, and notebook, respectively.

We have also conducted experiments on the Cube Engraving dataset. This dataset
comes from [41] was also used in meshCNN. Six meshes of the Cube Engraving
dataset are shown in Fig. 7. The special feature of this data set lies in a
model containing 500 faces, most of which are planes, and only a small part of
the small holes are the shapes that we need to recognize. Moreover, the
position of the shape in each cube is random, classification on this dataset
tests the 3D shape learning ability of MDC-GCNs to the extreme. The parameters
and optimizer are the same as the previous classification experiment (SHEREC).
Comparison with methods in the literature is shown in Table 4.

Table 4: Results on Cube Engraving dataset Algorithm | Accuracy
---|---
Our | 94.99$\%$
Milano et al. [12] | 94.39$\%$
Hanocka et al. [11] | 92.16$\%$
Charles et al. [6] | 64.26$\%$
Figure 7:  The four meshes in Cube Engraving dataset [41], from left to right
in the top row are tree, camel, and apple, respectively. From left to right in
the bottom row are key, bat, hammer, respectively.

(a) Telealiens (Ours) (b) Telealiens (Ground truth) (c) Vases (Ours) (d) Vases
(Ground truth)

(e) Chairs (Ours) (f) Chairs (Ground truth) (g) Human Body (Ours) (h) Human
Body (Ground truth)

Figure 8: Visualization of segmentation results with face labels. (a), (c),
(e), (f) are the prediction results of MDC-GCNs on the Telealiens, Vases,
Chairs, and Human Body data sets respectively (three randomly selected). (b),
(d), (f), (g) are the corresponding ground truth.  Table 5: Results on COSEG
segmentation

Algorithm | Vases Acc | Chairs Acc | Telealiens Acc
---|---|---|---
Our | 97.68$\%$ | 99.74$\%$ | 95.78$\%$
Milano et al. [12] | 97.83$\%$ | 98.21$\%$ | 99.03$\%$
Hanocka et al. [11] | 97.27$\%$ | 99.63$\%$ | 97.56$\%$
Charles et al. [5] | 91.5$\%$ | 70.2$\%$ | 54.4$\%$
Charles et al. [6] | 94.7$\%$ | 98.9$\%$ | 79.1$\%$
Yangyan et al. [42] | 96.37$\%$ | 99.31$\%$ | 97.40$\%$

Table 6: Results on Human Body Segmentation

Feature | Dimensions | Acc
---|---|---
Our | 57 | 94.65$\%$
Milano et al. [12] | 5 | 91.11$\%$
Hanocka et al. [11] | 5 | 92.30$\%$
Haim et al. [43] | 3 | 91.31$\%$
Maron et al. [44] | 26 | 88.00$\%$
Charles et al. [6] | 3 | 90.77$\%$
Yue et al. [45] | 3 | 89.72$\%$
Masci et al. [46] | 64 | 86.40$\%$
Poulenard and Ovsjanikov [47] | 64 | 89.47$\%$

Table 7: Comparison of MDC-GCN segmentation network parameters with PD-MeshNet
and MeshCNN.

Method | Parameters | Acc
---|---|---
Our (MDC-GCN) | 147,828 | 94.65$\%$
Milano et al. [12] (PD-MeshNet, 2020) | 173,728 | 91.11$\%$
Hanocka et al. [11] (MeshCNN, 2019) | 2,279,720 | 92.30$\%$

(a) Telealiens (Ours) (b) Telealiens (Ground truth) (c) Vases (Ours) (d) Vases
(Ground truth)

(e) Chairs (Ours) (f) Chairs (Ground truth) (g) Human Body (Ours) (h) Human
Body (Ground truth)

Figure 9: Visualization of segmentation results with edge labels. (a), (c),
(e), (f) are the prediction results of MDC-GCNs on the Telealiens, Vases,
Chairs, and Human Body data sets respectively (three randomly selected). (b),
(d), (f), (g) are the corresponding ground truth.

(a) Telealiens (b) Vases (c) Chairs (d) Human Body

(e) Telealiens (f) Vases (g) Chairs (h) Human Body

Figure 10: Visualization of the difference between segmentation result and
ground truth. (a), (b), (c), (d) are the difference between predicted results
and ground truth (face labels), (e), (f), (g), (h) are the corresponding edge
labels. Most errors made by the MDC-CGNs are located around the junctions
between two difference surfaces. Please see the zoomed detail. Table 8:
Classification results of the input graph node features ablation experiment on
SHREC and Cube Engraving

Feature | Dimensions | SHREC (16) Acc | SHREC (10) Acc | cubes Acc
---|---|---|---|---
$\\{\theta\\}$ | 3 | 97.5$\%$ | 95.0$\%$ | 72.53$\%$
$\\{GC\\}$ | 6 | 96.66$\%$ | 95.0$\%$ | 92.87$\%$
$\\{N_{f}\\}$ | 12 | 93.33$\%$ | 78.33$\%$ | 85.34$\%$
$\\{P\\}$ | 18 | 66.66$\%$ | 82.5$\%$ | 49.92$\%$
$\\{N_{v}\\}$ | 18 | 95.83$\%$ | 89.16$\%$ | 86.95$\%$
${\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 39 | 98.3$\%$ | 97.5$\%$ | 91.81$\%$
${\\{\\{P\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 39 | 98.3$\%$ | 96.67$\%$ | 94.70$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{\theta\\}\\}}$ | 45 | 97.5$\%$ | 97.5$\%$ | 94.08$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 51 | 98.33$\%$ | 95.83$\%$ | 92.87$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\}\\}}$ | 54 | 99.16$\%$ | 95.83$\%$ | 94.54$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 57 | 99.7$\%$ | 99.2$\%$ | 94.99$\%$

Table 9: Classification results of hidden layer output graph node features
ablation experiment on SHREC and Cube Engraving

Feature | Nodes | SHREC (16) Acc | SHREC (10) Acc | cubes Acc
---|---|---|---|---
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 8 | 64.16$\%$ | 59.16$\%$ | 70.10$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 16 | 78.33$\%$ | 74.16$\%$ | 82.24$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 32 | 92.5$\%$ | 76.67$\%$ | 89.2$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 64 | 94.1$\%$ | 94.1$\%$ | 92.71$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 128 | 96.6$\%$ | 95.0$\%$ | 93.32$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 256 | 98.3$\%$ | 93.33$\%$ | 93.93$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 512 | 97.5$\%$ | 96.67$\%$ | 94.08$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 1024 | 99.7$\%$ | 99.2$\%$ | 94.99$\%$

Table 10: Segmentation results of the input graph node features ablation
experiment on COSEG and Human Body

Feature | Dimensions | Vases Acc | Chairs Acc | Telealiens Acc | Human Acc
---|---|---|---|---|---
$\\{\theta\\}$ | 3 |  83.94$\%$ |  89.09$\%$ |  70.78$\%$ |  67.92$\%$
$\\{GC\\}$ | 6 |  88.16$\%$ |  91.72$\%$ |  84.24$\%$ |  85.57$\%$
$\\{N_{f}\\}$ | 12 |  93.69$\%$ |  98.97$\%$ |  88.56$\%$ |  90.92$\%$
$\\{P\\}$ | 18 |  95.11$\%$ |  85.60$\%$ |  57.25$\%$ |  68.87$\%$
$\\{N_{v}\\}$ | 18 |  93.79$\%$ |  99.30$\%$ |  92.61$\%$ |  92.61$\%$
${\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 39 |  94.47$\%$ |  99.64$\%$ |  94.67$\%$ |  94.50$\%$
${\\{\\{P\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 39 |  97.41$\%$ |  99.32$\%$ |  93.91$\%$ |  92.38$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{\theta\\}\\}}$ | 45 |  97.38$\%$ |  99.61$\%$ |  95.06$\%$ |  93.87$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 51 |  97.15$\%$ |  99.65$\%$ |  93.93$\%$ |  92.42$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\}\\}}$ | 54 | 97.49$\%$ |  99.71$\%$ |  94.95$\%$ |  94.27$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 57 |  97.68$\%$ | 99.74$\%$ | 95.78$\%$ | 94.65$\%$

Table 11: Segmentation results of hidden layer output graph nodes ablation
experiment on COSEG and Human Body

Feature | Nodes | Vases Acc | Chairs Acc | Telealiens Acc | Human Acc
---|---|---|---|---|---
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 8 | 90.07$\%$ | 96.00$\%$ | 88.39$\%$ | 82.27$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 16 | 94.33$\%$ | 98.70$\%$ | 89.65$\%$ | 84.03$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 32 | 95.49$\%$ | 98.96$\%$ | 90.61$\%$ | 87.52$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 64 | 96.43$\%$ | 98.45$\%$ | 91.82$\%$ | 89.73$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 128 | 96.80$\%$ | 98.59$\%$ | 92.88$\%$ | 91.24$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 256 | 96.97$\%$ | 98.56$\%$ | 93.34$\%$ | 92.66$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 512 | 97.12$\%$ | 99.17$\%$ | 94.00$\%$ | 93.54$\%$
${\\{\\{P\\},~{}\\{N_{v}\\},~{}\\{GC\\},~{}\\{N_{f}\\},~{}\\{\theta\\}\\}}$ | 1024 | 97.68$\%$ | 99.74$\%$ | 95.78$\%$ | 94.65$\%$

###  4.3 Segmentation

When applying MDC-GCNs to the task of segmentation, there is no need to make
major changes to the network, and only needs to cascade two DC-blocks through
the middle one-layer graph convolution layer, as shown in Fig. 5. Our network
does not need to perform image-like down-sampling and up-sampling operations
for the mesh structure, but only extracts features through the DC-block, and
then outputs node-level classification through graph convolution (different
from classification tasks, which are the entire graph structure
classification). We will verify the segmentation performances of MDC-GCNs on
the dataset COSEG [48] and Human Body Segmentation [44].

The three types of 3D meshes in COSEG are Telealiens, Chairs, and Vases. They
contain 200, 300 and 400 models of different shapes, respectively. Each model
of these three categories contains different numbers of face (Telealiens:
1500, Chairs: 1000, Vases: 1500). We follow the rule of dividing the training
and testing sets of MeshCNN, i.e., $85\%$ training and $15\%$ testing,
randomly generate 5 training and testing sets, and report the average results
of the 5 runs.

Human Body Segmentation contains 370 human body models used in [49], [50], and
[51] and 18 human body models from SHREC07 [52]. The humanoid models in the
dataset each contain 1500 faces. For the sake of fairness, we also randomly
divide 5 times according to the rules in the [44], and take the average value.
Since MeshCNN uses edge as the unit, in the segmentation comparison
experiment, we convert the face label into a label with edge as the unit for
comparison. Examples of the visual results are shown in Fig. 8 (using the face
as the label of the model) and Fig. 9 (using the edge as the label of the
model). Visualization of the errors made by the MDC-GCNs are shown in Fig. 10.
The quantitative results of COSEG and Human Body segmentation are shown in
Table 5 and 6. It should be noted that the quantitative results of all
segmentation comparison experiments use edge as the label.

##  5 Ablation studies

In order to better verify the 3D shape learning ability of MDC-GCNs, we set up
multiple sets of ablation experiments. The ablation experiment included the
influence of local feature of the 3D mesh and the dimension of hidden layer
output graph nodes on MDC-GCNs.

Table 8 and 9 respectively correspond to the effects of local features and
hidden layer output graph nodes on the classification performance of MDC-GCNs.
Table 10 and 11 respectively correspond to the effects of local features and
hidden layer output graph nodes on the segmentation performance of MDC-GCNs.
All experiments are to maintain the principle of single variable change. From
the results of these ablation experiments, the following conclusions can be
drawn: 1. It’s a better choice to select multi-dimensional geometric features
than a single geometric feature for the graph node description. 2\. Structural
features such as Gaussian curvature, vertex normals, and surface normals are
better than spatial features. 3\. The multi-dimensional geometric feature
fusion of spatial and structural features can make the network performance
better. 4\. It is better to use 1024 for the graph node dimension when the
performance and calculation cost are balanced.

In summary, the choice of 57-dimensional local features and 1024-dimensional
graph nodes achieves the best performance.

##  6 Conclusion

In this paper, we propose a novel densely connected graph convolutional
networks for 3D meshes. It achieved the classification and segmentation tasks
on 3D meshes without data augmentation. We used the face of the mesh as the
basic processing unit and introduced the 1-ring face neighbourhood structure
to derive multi-dimensional structural features for the graph node to enhance
the descriptive power of the graph. We then designed densely connected graph
convolutional blocks to aggregate local and non-local features to construct
effective and efficient GCNs for 3D object segmentation and classification.
Our network is simple and extremely easy to design, and can directly process
data in non-Euclidean spaces similar to 3D meshes.

##  7 Acknowledgments

## References

## References

  * [1] E. Ahmed, A. Saint, A. E. R. Shabayek, K. Cherenkova, R. Das, G. Gusev, D. Aouada, B. Ottersten, A survey on deep learning advances on different 3d data representations, arXiv preprint arXiv:1808.01462.
  * [2] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, L. Fei-Fei, Imagenet: A large-scale hierarchical image database, in: 2009 IEEE conference on computer vision and pattern recognition, Ieee, 2009, pp. 248–255.  doi:10.1109/CVPR.2009.5206848.
  * [3] T.-Y. Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, C. L. Zitnick, Microsoft coco: Common objects in context, in: European conference on computer vision, Springer, 2014, pp. 740–755.  doi:10.1007/978-3-319-10602-1_48.
  * [4] H. Su, S. Maji, E. Kalogerakis, E. Learned-Miller, Multi-view convolutional neural networks for 3d shape recognition, in: Proceedings of the IEEE international conference on computer vision, 2015, pp. 945–953.  doi:10.1109/ICCV.2015.114.
  * [5] C. R. Qi, H. Su, K. Mo, L. J. Guibas, Pointnet: Deep learning on point sets for 3d classification and segmentation, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp. 652–660.  doi:10.1109/CVPR.2017.16.
  * [6] C. R. Qi, L. Yi, H. Su, L. J. Guibas, Pointnet++: Deep hierarchical feature learning on point sets in a metric space, in: Advances in neural information processing systems, 2017, pp. 5099–5108.  doi:10.5555/3295222.3295263.
  * [7] M. Jiang, Y. Wu, T. Zhao, Z. Zhao, C. Lu, Pointsift: A sift-like network module for 3d point cloud semantic segmentation, arXiv preprint arXiv:1807.00652.
  * [8] Y. Feng, Y. Feng, H. You, X. Zhao, Y. Gao, Meshnet: Mesh neural network for 3d shape representation, AAAI 2019doi:10.1609/AAAI.V33I01.33018279.
  * [9] I. Kostrikov, Z. Jiang, D. Panozzo, D. Zorin, J. Bruna, Surface networks, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp. 2540–2548.
  * [10] S. Kumawat, S. Raman, Lp-3dcnn: Unveiling local phase in 3d convolutional neural networks, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE, 2019, pp. 4903–4912.  doi:10.1109/CVPR.2019.00504.
  * [11] R. Hanocka, A. Hertz, N. Fish, R. Giryes, S. Fleishman, D. Cohen-Or, Meshcnn: a network with an edge, ACM Transactions on Graphics (TOG) 38 (4) (2019) 90.  doi:10.1145/3306346.3322959.
  * [12] F. Milano, A. Loquercio, A. Rosinol, D. Scaramuzza, L. Carlone, Primal-dual mesh convolutional neural networks, in: Conference on Neural Information Processing Systems (NeurIPS), 2020.
URL https://github.com/MIT-SPARK/PD-MeshNet

  * [13] K. He, X. Zhang, S. Ren, J. Sun, Deep residual learning for image recognition, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2016, pp. 770–778.
  * [14] R. K. Srivastava, K. Greff, J. Schmidhuber, Training very deep networks, in: Advances in neural information processing systems, 2015, pp. 2377–2385.  doi:10.5555/2969442.2969505.
  * [15] G. Larsson, M. Maire, G. Shakhnarovich, Fractalnet: Ultra-deep neural networks without residuals, arXiv preprint arXiv:1605.07648.
  * [16] G. Huang, Y. Sun, Z. Liu, D. Sedra, K. Q. Weinberger, Deep networks with stochastic depth, in: European conference on computer vision, Springer, 2016, pp. 646–661.  doi:10.1007/978-3-319-46493-039.
  * [17] G. Huang, Z. Liu, L. Van Der Maaten, K. Q. Weinberger, Densely connected convolutional networks, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2017, pp. 4700–4708.  doi:10.1109/CVPR.2017.243.
  * [18] T. N. Kipf, M. Welling, Semi-supervised classification with graph convolutional networks, arXiv preprint arXiv:1609.02907.
  * [19] A. Fout, J. Byrd, B. Shariat, A. Ben-Hur, Protein interface prediction using graph convolutional networks, in: Advances in neural information processing systems, 2017, pp. 6530–6539.
  * [20] T. Hamaguchi, H. Oiwa, M. Shimbo, Y. Matsumoto, Knowledge transfer for out-of-knowledge-base entities: A graph neural network approach, arXiv preprint arXiv:1706.05674.
  * [21] J. Bruna, W. Zaremba, A. Szlam, Y. LeCun, Spectral networks and locally connected networks on graphs, arXiv preprint arXiv:1312.6203.
  * [22] G. Li, M. Muller, A. Thabet, B. Ghanem, Deepgcns: Can gcns go as deep as cnns?, in: Proceedings of the IEEE International Conference on Computer Vision, 2019, pp. 9267–9276.
  * [23] H. Choi, G. Moon, K. M. Lee, Pose2mesh: Graph convolutional network for 3d human pose and mesh recovery from a 2d human pose, arXiv preprint arXiv:2008.09047.
  * [24] X. Wei, R. Yu, J. Sun, View-gcn: View-based graph convolutional network for 3d shape analysis, in: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2020, pp. 1850–1859.
  * [25] M. Dominguez, F. P. Such, S. Sah, R. Ptucha, Towards 3d convolutional neural networks with meshes, in: 2017 IEEE International Conference on Image Processing (ICIP), IEEE, 2017, pp. 3929–3933.
  * [26] Z. Wu, M. Zeng, F. Qin, Y. Wang, J. Kosinka, Active 3-d shape cosegmentation with graph convolutional networks, IEEE computer graphics and applications 39 (2) (2019) 77–88.
  * [27] N. Verma, E. Boyer, J. Verbeek, Feastnet: Feature-steered graph convolutions for 3d shape analysis, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 2598–2606.
  * [28] O. Litany, A. Bronstein, M. Bronstein, A. Makadia, Deformable shape completion with graph convolutional autoencoders, in: Proceedings of the IEEE conference on computer vision and pattern recognition, 2018, pp. 1886–1895.
  * [29] M. Meyer, M. Desbrun, P. Schröder, A. H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, in: Visualization and mathematics III, Springer, 2003, pp. 35–57.  doi:10.1007/978-3-662-05105-4_2.
  * [30] H. Zhao, P. Xiao, An accurate vertex normal computation scheme, in: Computer Graphics International Conference, Springer, 2006, pp. 442–451.  doi:10.1007/11784203_38.
  * [31] Q. Li, Z. Han, X.-M. Wu, Deeper insights into graph convolutional networks for semi-supervised learning, in: Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 32, 2018.
  * [32] J. Bastings, I. Titov, W. Aziz, D. Marcheggiani, K. Sima’an, Graph convolutional encoders for syntax-aware neural machine translation, arXiv preprint arXiv:1704.04675.
  * [33] Z. Guo, Y. Zhang, Z. Teng, W. Lu, Densely connected graph convolutional networks for graph-to-sequence learning, Transactions of the Association for Computational Linguistics 7 (2019) 297–312.
  * [34] G. Huang, Z. Liu, G. Pleiss, L. Van Der Maaten, K. Weinberger, Convolutional networks with dense connectivity, IEEE transactions on pattern analysis and machine intelligence.
  * [35] K. Xu, C. Li, Y. Tian, T. Sonobe, K.-i. Kawarabayashi, S. Jegelka, Representation learning on graphs with jumping knowledge networks, in: International Conference on Machine Learning, PMLR, 2018, pp. 5453–5462.
  * [36] D. P. Kingma, J. Ba, Adam: A method for stochastic optimization, arXiv preprint arXiv:1412.6980.
  * [37] D. Ezuz, J. Solomon, V. G. Kim, M. Ben-Chen, Gwcnn: A metric alignment layer for deep shape analysis, in: Computer Graphics Forum, Vol. 36, Wiley Online Library, 2017, pp. 49–57.
  * [38] A. Sinha, J. Bai, K. Ramani, Deep learning 3d shape surfaces using geometry images, in: European Conference on Computer Vision, Springer, 2016, pp. 223–240.
  * [39] Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang, J. Xiao, 3d shapenets: A deep representation for volumetric shapes, in: Proceedings of the IEEE conference on computer vision and pattern recognition, IEEE, 2015, pp. 1912–1920.
  * [40] A. M. Bronstein, M. M. Bronstein, L. J. Guibas, M. Ovsjanikov, Shape google: Geometric words and expressions for invariant shape retrieval, ACM Transactions on Graphics (TOG) 30 (1) (2011) 1.
  * [41] L. J. Latecki, R. Lakamper, Shape similarity measure based on correspondence of visual parts, IEEE Transactions on Pattern Analysis and Machine Intelligence 22 (10) (2000) 1185–1190.  doi:10.1109/34.879802.
  * [42] Y. Li, R. Bu, M. Sun, B. Chen, Pointcnn, arXiv preprint arXiv:1801.07791.
  * [43] N. Haim, N. Segol, H. Ben-Hamu, H. Maron, Y. Lipman, Surface networks via general covers, in: Proceedings of the IEEE International Conference on Computer Vision, IEEE, 2019, pp. 632–641.  doi:10.1109/ICCV.2019.00072.
  * [44] H. Maron, M. Galun, N. Aigerman, M. Trope, N. Dym, E. Yumer, V. G. Kim, Y. Lipman, Convolutional neural networks on surfaces via seamless toric covers., ACM Trans. Graph. 36 (4) (2017) 71–1.
  * [45] Y. Wang, Y. Sun, Z. Liu, S. E. Sarma, M. M. Bronstein, J. M. Solomon, Dynamic graph cnn for learning on point cloudsv, ACM Transactions on Graphics (TOG) 38 (5) (2019) 146.
  * [46] J. Masci, D. Boscaini, M. Bronstein, P. Vandergheynst, Geodesic convolutional neural networks on riemannian manifolds, in: Proceedings of the IEEE international conference on computer vision workshops, IEEE, 2015, pp. 37–45.
  * [47] A. Poulenard, M. Ovsjanikov, Multi-directional geodesic neural networks via equivariant convolution, ACM Transactions on Graphics (TOG) 37 (6) (2019) 236\.
  * [48] Y. Wang, S. Asafi, O. Van Kaick, H. Zhang, D. Cohen-Or, B. Chen, Active co-analysis of a set of shapes, ACM Transactions on Graphics (TOG) 31 (6) (2012) 1–10.
  * [49] D. Anguelov, P. Srinivasan, D. Koller, S. Thrun, J. Rodgers, J. Davis, Scape: shape completion and animation of people, in: ACM SIGGRAPH 2005 Papers, 2005, pp. 408–416.
  * [50] D. Vlasic, I. Baran, W. Matusik, J. Popović, Articulated mesh animation from multi-view silhouettes, in: ACM SIGGRAPH 2008 papers, 2008, pp. 1–9.
  * [51] F. Bogo, J. Romero, M. Loper, M. J. Black, Faust: Dataset and evaluation for 3d mesh registration, in: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2014, pp. 3794–3801.
  * [52] D. Giorgi, S. Biasotti, L. Paraboschi, Shape retrieval contest 2007: Watertight models track, SHREC competition 8 (7).


# Black hole scalarizations induced by parity violations

Hao-Jie Lina,b<EMAIL_ADDRESS>Tao Zhua,b corresponding author:
<EMAIL_ADDRESS>Shao-Jun Zhanga,b<EMAIL_ADDRESS>Anzhong Wangc
<EMAIL_ADDRESS>a Institute for Theoretical Physics and Cosmology,
Zhejiang University of Technology, Hangzhou, 310032, China
b United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310032, China
c GCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316,
USA

###### Abstract

It is well-known that parity symmetry is broken in the weak interaction but
conserved for Einstein’s general relativity and Maxwell’s electromagnetic
theory. Nevertheless, parity symmetry could also be violated in the
gravitational/electromagnetic sectors if a fundamental scalar field couples to
the parity-violating gravitational/electromagnetic curvature terms. Such
parity-violating terms, which flip signs under reversed spatial directions,
can inevitably lead to a negative effective mass squared for the scalar field
perturbations near nonspherically symmetric black holes and thus are expected
to trigger tachyonic instability. As illustrative examples, we show that the
scalar field coupled to gravitational/electromagnetic Chern-Simons terms near
a Kerr-Newmann spacetime can develop tachyonic instabilities, leading to
equilibrium scalar field configurations in certain parameter regions of black
holes. This instability, which is an indication of the black hole
scalarization process, can occur in a broad class of nonspherically symmetric
black holes and parity-violating theories.

##  I Introduction

In recent years, the availability of detection data on black holes has
increased significantly a1 ; LIGOScientific:2021djp ; a2 ; a3 ; a4 , making
the study of black holes a topic of growing interest in the scientific
community. It is widely accepted that, as an indispensable prediction of the
theory of general relativity (GR), black holes can only be described in terms
of their mass, electric charge, and angular momentum. Any other charge is not
expected to exist according to the no-hair theorem Chrusciel:2012jk ;
Bekenstein:1998aw ; Bambi:2017khi . Remarkably, most of the current
observations, including the detection of gravitational waves Isi:2019aib ,
black hole images a2 ; a4 , and the stars orbiting the supermassive black hole
in our Galactic Center Qi:2020xzi , are in good agreement with hairless black
holes. Future precise observations, such as gravitational wave detection and
black hole images, can provide more accurate and details information about the
nature of the black hole spacetime in the regime of strong gravity.
Importantly, these precise observations can also provide a significant way to
probe or constrain possible extra hairs in black holes.

One type of such extra hairs, a fundamental scalar field, could survive around
black hole spacetime in many classes of extended scalar-tensor theories
through a tachyonic instability known as spontaneous scalarization. Hence, a
growth of scalar hair or spontaneous scalarization can render the new
fundamental scalar field detectable near black holes Maselli:2021men . Several
mechanisms that account for the spontaneous scalarization around black holes
have been proposed. In particular, matter-induced spontaneous scalarization
was proposed for compact neutron stars in scalar-tensor theories ns1 ; ns2 .
With nontrivial couplings of the scalar field to the spacetime curvature or
electromagnetic field, the curvature-induced and charge-induced spontaneous
scalarizations are also proven to be possible in the Einstein-Gauss-Bonnet
gravity esgb1 ; esgb2 ; esgb3 ; esgb4 ; esgb5 ; Brihaye:2019kvj  and Einstein-
scalar-Maxwell theories esm1 ; esm2 ; esm3 ; esm4 , respectively. More
recently, it has been shown that the spontaneous scalarization induced by the
spin of a black hole can also occur in various modified theories spin1 ; spin2
; spin3 ; spin4 ; spin5 ; Annulli:2022ivr ; Lai:2022spn . In addition, the
scalarization of black holes realized through the nonlinear accretion of
scalar fields has also been discovered and has attracted a lot of attention
recently nas1 ; nas2 ; nas3 ; nas4 ; nas5 .

A key ingredient to trigger the spontaneous scalarization around a black hole
is a threshold in both the coupling function, describing the interaction
between the scalar field and gravitational/electromagnetic fields, and the
black hole parameters, beyond which a tachyonic instability is induced by a
sufficiently large and negative effective mass squared of the scalar field
Doneva:2022ewd . Such a negative effective mass squared sensitively depends on
the sign of the coupling functions in both the curvature/charge-induced
scalarization esgb3 ; esm1 ; esm2 ; esm3  and the value of the spin of the
black hole in the spin-induced scalarization esgb5 ; esm3 ; esm4 ; spin1 ;
spin2 ; spin3 ; Lai:2022spn , for example.

In this paper, we present a study on a mechanism for the spontaneous
scalarization of nonspherically symmetric black holes, the parity-violation-
induced black hole scalarization. It is worth mentioning that the spontaneous
scalarization of black holes in a specific parity-violating theory—i.e., the
Chern-Simons modified gravity—has been extensively studied previously p1 ; p2
; Gao:2018acg ; nl2 ; mdcsg ; ecsg ; rnmdcsg . In these theories, the scalar
field coupled with the parity-violating gravitational/electromagnetic field
can inevitably lead to a negative effective mass squared for the scalar field
and thus can trigger tachyonic instability when it is sufficiently large.

To illustrate this mechanism, we analyze the behavior of the scalar field
which couples to two specific parity-violating terms: the gravitational and
electromagnetic Chern-Simons terms, in a fixed Kerr-Newmann background. Due to
the inherent complexity and nonlinearity of scalarization dynamics in rotating
black hole backgrounds, our study only focuses on the scalarization dynamics
within the framework of the “decoupling limit.” In this way, we numerically
evolve the nonlinear scalar field equation on the fixed Kerr-Newmann geometry,
disregarding the backreaction of the scalar field on the background spacetime
nl2 . This simplification allows us to gain valuable insights into the
scalarization phenomenon while bypassing the significant challenges associated
with capturing its complete generality and nonlinearity. Our result reveals
that the scalar field in a Kerr-Newmann spacetime can develop tachyonic
instabilities, eventually leading to the formation of an equilibrium scalar
field configuration.

We additionally check that the effective mass squared of the scalar field can
be inevitably negative in a broad class of nonspherically symmetric black
holes and parity-violating theories, which are expected to lead to tachyonic
instabilities in specific regions of the spacetime. Our results suggest that
the phenomenon of black hole scalarization potentially occurs for a broad
class of nonspherically symmetric black holes in numerous parity-violating
theories.

##  II The model

Let us describe our model by starting with the following scalar field action
on a nontrivial black hole spacetime,

$\displaystyle S_{\phi}=-\int
d^{4}x\sqrt{-g}\left(\frac{1}{2}\nabla_{\mu}\phi\nabla^{\mu}\phi+f(\phi)I(\psi;g_{\mu\nu})\right),$

where $\phi$ is the scalar field, $I(\psi;g_{\mu\nu})$ represents the source
term depending on $g_{\mu\nu}$ and matter fields $\psi$, and $f(\phi)$ is the
coupling function determining the coupling strength between the scalar field
$\phi$ and the spacetime metric $g_{\mu\nu}$ and other matter fields $\psi$.
By varying the above action with respect to the scalar field one obtains the
equation of motion of the scalar field on the black hole spacetime,

$\displaystyle\Box\phi-f^{\prime}(\phi)I=0,$ (2)

where $f^{\prime}(\phi)=df(\phi)/d\phi$. This equation allows for the
existence of scalar-free solutions which are also solutions of GR with
conditions $\phi=0$ and $f^{\prime}(0)=0$ Astefanesei:2020qxk ;
Herdeiro:2021ftk . And spontaneous scalarization occurs if the scalar-free
solution is unstable against scalar perturbations $\delta\phi$. To study how
the scalar-free background is affected by the small scalar field perturbation
$\delta\phi$, it is convenient to consider the linearized scalar field
equation, which is

$\displaystyle(\Box-\mu_{\rm eff}^{2})\delta\phi=0,$ (3)

where $\mu_{\rm eff}^{2}=f^{\prime\prime}(0)I(\psi;g_{\mu\nu})$ is the
effective mass squared of the scalar field. Once $\mu^{2}_{\rm eff}$ becomes
sufficiently negative, tachyonic instability occurs, causing the scalar-free
spacetime to be unstable under $\delta\phi$ in a certain region of the
parameter space, indicating an onset of the black hole scalarization process
Doneva:2022ewd .

One well-studied model of the spontaneous scalarization is realized in the
framework of the Einstein-scalar-Gauss-Bonent gravity with
$I(\psi;g_{\mu\nu})=R_{\rm GB}^{2}$, where $R_{\rm GB}^{2}$ denotes the Gauss-
Bonent scalar. For a Schwarzschild black hole, one has $R_{\rm
GB}^{2}=48M^{2}/r^{6}$, with $M$ being the mass of the black hole. It is
evident that effective mass $\mu_{\rm eff}$ is only allowed to be negative for
$f^{\prime\prime}(0)<0$, with which a tachyonic instability occurs when
$|f^{\prime\prime}(0)|$ is sufficiently large spin3 . Similar curvature-
induced scalarization occurs in Reissner-Nordström(RN) and Kerr black holes as
well with a negative $f^{\prime\prime}(0)$ esgb2 ; Brihaye:2019kvj ; spin3 ;
spin4 ; spin5 . For a Kerr black hole, $\mu_{\rm eff}^{2}$ can become negative
even for positive $f^{\prime\prime}(0)$ with a high spin and thus can trigger
the spin-induced scalarization spin3 ; spin4 ; spin5 . The black hole
scalarization with other models—for example,
$I(\psi,g_{\mu\nu})=F_{\mu\nu}F^{\mu\nu}$—have also been extensively studied
in the literature esm1 ; esm2 ; esm3 ; esm4 ; Lai:2022spn .

In this paper, we consider two choices of $I(\psi,g_{\mu\nu})$:

$\displaystyle I(\psi,g_{\mu\nu})=R\tilde{R}\;\;\;{\rm
and}\;\;\;I(\psi,g_{\mu\nu})=F\tilde{F},$ (4)

representing two parity-violating interactions between the scalar field and
the gravitational/electromagnetic field, where
$R\tilde{R}=\frac{1}{2}\epsilon^{\mu\nu\lambda\gamma}R^{\eta}_{~{}\xi\lambda\gamma}R^{\xi}_{~{}\eta\mu\nu}$
and
$F\tilde{F}=\frac{1}{2}\epsilon^{\mu\nu\lambda\gamma}F_{\mu\nu}F_{\lambda\gamma}$
with $\epsilon^{\mu\nu\lambda\gamma}$ being the totally antisymmetric Levi-
Civita tensor. Intriguingly, such couplings break the parity symmetry in the
gravitational/electromagnetic sectors, and thus
$I(\psi;g_{\mu\nu})\to-I(\psi;g_{\mu\nu})$ under parity transformations. When
$I(\psi;g_{\mu\nu})$ does not vanish for a nontrivial black hole background,
$I(\psi;g_{\mu\nu})$ has to be negative in a certain region of the spacetime,
which indicates that a negative effective mass squared for the scalar field
inevitably exists. When $\mu_{\rm
eff}^{2}=f^{\prime\prime}(0)I(\psi;g_{\mu\nu})$ is sufficiently large and
negative, exceeding a threshold, the tachyonic instability of the scalar field
occurs which triggers the process of the spontaneous scalarization.

##  III Scalarization process in Kerr-Newmann black holes

To illustrate the mechanisms of the parity-violation-induced scalarization
process, let us consider the scalarization of the Kerr-Newmann black hole as a
concrete example, in which both of the parity-violating terms $R\tilde{R}$ and
$F\tilde{F}$ do not vanish.

In the Boyer-Lindquist coordinates $\\{t,r,\theta,\varphi\\}$, the Kerr-
Newmann metric with charge $Q$, mass $M$, and spin $a$, is given by

$\displaystyle ds^{2}$ $\displaystyle\equiv$
$\displaystyle-\frac{\Delta-a^{2}\sin^{2}\theta}{\Sigma^{2}}dt^{2}-\frac{2a\sin^{2}\theta\left(r^{2}+a^{2}-\Delta\right)}{\Sigma^{2}}dtd\varphi$
(5) $\displaystyle+\frac{\left[\left(r^{2}+a^{2}\right)^{2}-\Delta
a^{2}\sin^{2}\theta\right]\sin^{2}\theta}{\Sigma^{2}}d\varphi^{2}$
$\displaystyle+\frac{\Sigma^{2}}{\Delta}dr^{2}+\Sigma^{2}d\theta^{2},$

where

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle r^{2}-2Mr+a^{2}+Q^{2},$
(6) $\displaystyle\Sigma^{2}$ $\displaystyle=$ $\displaystyle
r^{2}+a^{2}\cos^{2}\theta,$ (7) $\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{J}{M}.$ (8)

The corresponding vector potential is

$\displaystyle A_{\mu}dx^{\mu}=-\frac{Qr}{\Sigma^{2}}\left(dt-a\sin^{2}\theta
d\varphi\right).$ (9)

Therefore, the explicit form of the parity-violating terms $R\tilde{R}$ and
$F\tilde{F}$ in a Kerr-Newmann black hole, are given, respectively, by

$\displaystyle R\tilde{R}$ $\displaystyle=$
$\displaystyle\frac{96a\cos\theta\left[\left(3Mr-2Q^{2}\right)r-Ma^{2}\cos^{2}\theta\right]}{\left(r^{2}+a^{2}\cos^{2}\theta\right)^{6}}$
$\displaystyle\times\left[\left(Mr-Q^{2}\right)r^{2}-\left(3Mr-Q^{2}\right)a^{2}\cos^{2}\theta\right],$
$\displaystyle F\tilde{F}$ $\displaystyle=$
$\displaystyle\frac{8aQ^{2}r\cos\theta\left(r^{2}-a^{2}\cos\theta^{2}\right)}{\left(r^{2}+a^{2}\cos\theta^{2}\right)^{4}}.$
(11)

Intriguingly, both of the parity-violating terms contain a factor
$\cos\theta$, such that it always has a negative sign under the following
parity transformation:

$\displaystyle\theta\rightarrow\pi-\theta,$ (12) $\displaystyle
R\tilde{R}\to-R\tilde{R},$ (13) $\displaystyle
F\tilde{F}\rightarrow-F\tilde{F},$ (14)

which implies that regardless of the specific form of $f(\phi)$, the effective
mass square $\mu_{\rm eff}^{2}$ always has negative values in the interval
$\theta\in[0,\pi]$. Therefore, this nonminimal coupling between the parity-
violating terms and the scalar field will inevitably lead to tachyonic
instability as long as $|f^{\prime\prime}(0)|$ is sufficiently large,
resulting in the occurrence of the spontaneous scalarization process of Kerr-
Newman black holes at the linear level.

To further validate this assertion, we consider the specific choice,
$f(\phi)=\frac{\alpha}{2\beta}(1-e^{-\beta\phi^{2}})$ nl1 ; nl2 , where
$\alpha$ and $\beta$ are two constants. When $\beta\rightarrow 0$, the
coupling function $f(\phi)$ reduces to the quadratic form
$\frac{1}{2}\alpha\phi^{2}$, which is sufficient for studying spontaneous
scalarization resulting from linear tachyonic instability. As the instability
progresses and the scalar field grows, the significance of nonlinear terms
increases and eventually quenches the instability. The final state is an
equilibrium scalar field configuration.

To solve Eq. (2) for the time evolution of the scalar field, we utilize a
hyperboloidal foliation method described in Ref. Zhang:2020pko .

First, we transform the Bayer-Lindquist coordinates into the ingoing Kerr-
Schild coordinates $\\{\tilde{t},r,\theta,\tilde{\varphi}\\}$:

$\displaystyle d\tilde{t}$ $\displaystyle=$ $\displaystyle
dt+\frac{2Mr-Q^{2}}{\Delta}dr,$ (15) $\displaystyle d\tilde{\varphi}$
$\displaystyle=$ $\displaystyle d\varphi+\frac{a}{\Delta}dr.$ (16)

Considering the axisymmetry of the Kerr-Newmann spacetime, the scalar
perturbation can be decomposed as

$\displaystyle\phi(\tilde{t},r,\theta,\tilde{\varphi})=\sum_{m}\frac{\Psi_{m}(\tilde{t},r,\theta)}{r}e^{im\tilde{\varphi}},$
(17)

where $m$ is the azimuthal mode number and $\Psi_{m}$ is a new field
variable.111The introduction of a conformal compactification, as described in
Eqs. (20) and (21), necessitates the rescaling of the scalar field variable,
such as $\phi=r^{-1}\Psi$. This rescaling is crucial to mitigate the
singularity of the physical metric at the future null infinity
$\mathscr{I}^{+}$, which is also present in the wave equation te1 ; te2 .
Substituting the above expressions into Eq. (2), we obtain

$\displaystyle
A^{\tilde{t}\tilde{t}}\partial_{\tilde{t}}^{2}\Psi_{m}+A^{\tilde{t}r}\partial_{\tilde{t}}\partial_{r}\Psi_{m}+A^{rr}\partial_{r}^{2}\Psi_{m}+A^{\theta\theta}\partial_{\theta}^{2}\Psi_{m}$
$\displaystyle+B^{\tilde{t}}\partial_{\tilde{t}}\Psi_{m}+B^{r}\partial_{r}\Psi_{m}+B^{\theta}\partial_{\theta}\Psi_{m}+C\Psi_{m}=0,$

where

$\displaystyle A^{\tilde{t}\tilde{t}}=2Mr-Q^{2}+\Sigma^{2},$ $\displaystyle
A^{\tilde{tr}}=-4Mr+2Q^{2},$ $\displaystyle A^{rr}=-\Delta,$ $\displaystyle
A^{\theta\theta}=-1,$ $\displaystyle B^{\tilde{t}}=2M-\frac{2Q^{2}}{r},$
$\displaystyle B^{r}=\frac{2\left(a^{2}+Q^{2}-Mr\right)}{r}-2ima,$ (19)
$\displaystyle B^{\theta}=-\cot\theta,$ $\displaystyle
C=-\frac{2\left(a^{2}+Q^{2}-Mr\right)}{r^{2}}+\frac{2ima}{r}+\frac{m^{2}}{\sin^{2}\theta}$
$\displaystyle\quad\quad+\Sigma^{2}\alpha
e^{-\beta(\sum_{m}\Psi_{m}e^{im\tilde{\varphi}}/r)^{2}}I.$

Each azimuthal mode $m\neq 0$ has harmonic dependence on $\tilde{\varphi}$,
which causes Eq. (III) to be nonlinear. To eliminate the interference from
$\tilde{\varphi}$, we only consider the axisymmetric mode with $m=0$ to study
the time evolution of the scalar field perturbation. This particular choice
has an impact on the excited quasinormal modes in stable scenarios and the
characteristic times involved, but it does not alter the overall outcome of
the scalarization process Doneva:2022yqu .

The second step is to define the compactified horizon-penetrating,
hyperboloidal coordinates (HH coordinates)
$\\{\tau,\rho,\theta,\tilde{\varphi}\\}$. Specifically, we replace the ingoing
Kerr-Schild coordinates $\tilde{t}$ and $r$ with

$\displaystyle\tilde{t}$ $\displaystyle=$ $\displaystyle\tau+h(\rho),$ (20)
$\displaystyle r$ $\displaystyle=$ $\displaystyle\frac{\rho}{\Omega(\rho)},$
(21)

where

$\displaystyle h(\rho)$ $\displaystyle=$
$\displaystyle\frac{\rho}{\Omega}-\rho-4M\mathrm{ln}\Omega,$ (22)
$\displaystyle\Omega(\rho)$ $\displaystyle=$ $\displaystyle 1-\frac{\rho}{S}.$
(23)

By these coordinate transformations, the future null infinity
$\mathscr{I}^{+}$ is compactified at $\rho=S$, and the event horizon of the
Kerr-Newman black hole $r_{+}=M+\sqrt{M^{2}-a^{2}-Q^{2}}$ is located at

$\displaystyle\rho_{+}=\frac{S^{2}\left(M+\sqrt{M^{2}-a^{2}-Q^{2}}\right)+S\left(a^{2}+Q^{2}\right)}{a^{2}+2MS+Q^{2}+S^{2}}.$

Next, we define a boost function as

$\displaystyle H(\rho)=\frac{dh(\rho)}{dr},$ (25)

and then the partial derivatives for $\tilde{t}$ and $r$ can be rewritten as

$\displaystyle\partial_{\tilde{t}}=\partial_{\tau},~{}\partial_{r}=-H\partial_{\tau}+\frac{d\rho}{dr}\partial_{\rho}.$
(26)

Substituting the above expressions into Eq. (III), we obtain

$\displaystyle
A^{\tau\tau}\partial_{\tau}^{2}\Psi_{m}+A^{\tau\rho}\partial_{\tau}\partial_{\rho}\Psi_{m}+A^{\rho\rho}\partial_{\rho}^{2}\Psi_{m}+A^{\theta\theta}\partial_{\theta}^{2}\Psi_{m}$
$\displaystyle+B^{\tau}\partial_{\tau}\Psi_{m}+B^{\rho}\partial_{\rho}\Psi_{m}+B^{\theta}\partial_{\theta}\Psi_{m}+C\Psi_{m}=0~{},$

where

$\displaystyle A^{\tau\tau}=A^{\tilde{t}t}-HA^{\tilde{tr}}+H^{2}A^{rr},$
$\displaystyle A^{\rho\rho}=\left(\frac{d\rho}{dr}\right)^{2}A^{rr},$
$\displaystyle
A^{\tau\rho}=\frac{d\rho}{dr}\left(A^{\tilde{t}r}-2HA^{rr}\right),$ (28)
$\displaystyle B^{\tau}=B^{\tilde{t}}-HB^{r}-\frac{dH}{dr}A^{rr},$
$\displaystyle
B^{\rho}=\frac{d\rho}{dr}\left[B^{r}+\frac{d}{d\rho}\left(\frac{d\rho}{dr}\right)A^{rr}\right],$

By introducing a new auxiliary variable $\Pi_{m}=\partial_{\tau}\Psi_{m}$,
Eq.(III) is recast into a first-order form of coupled partial differential
equations:

$\displaystyle\partial_{\tau}\Psi_{m}$ $\displaystyle=$
$\displaystyle\Pi_{m},$ (29) $\displaystyle\partial_{\tau}\Pi_{m}$
$\displaystyle=$
$\displaystyle-\frac{1}{A^{\tau\tau}}\left(A^{\tau\rho}\partial_{\rho}\Pi_{m}+A^{\rho\rho}\partial_{\rho}^{2}\Psi_{m}+A^{\theta\theta}\partial_{\theta}^{2}\Psi_{m}\right.$
$\displaystyle\left.+B^{\tau}\Pi_{m}+B^{\rho}\partial_{\rho}\Psi_{m}+B^{\theta}\partial_{\theta}\Psi_{m}+C\Psi_{m}\right).$

In terms of numerical implementations, we employ a fourth-order finite
difference method for spatial grid discretizations, while the time evolution
is accomplished using a fourth-order Runge-Kutta integrator. The HH
coordinates automatically satisfy the ingoing/outgoing boundary condition at
the horizon/infinity, eliminating the need to handle the complex outer
boundary problem that can impact accuracy. However, at the angular poles
($\theta=0$ and $\pi$), we impose physical boundary conditions:
$\left.\Psi_{m}\right|_{\theta=0,\pi}=0$ for $m\neq 0$, and
$\left.\partial_{\theta}\Psi_{m}\right|_{\theta=0,\pi}=0$ for $m=0$ bd . For
the initial data, we consider a Gaussian distribution localized outside the
horizon at $\rho=\rho_{c}$. Specifically, we have

$\displaystyle\Psi_{lm}(\tau=0,\rho,\theta)\sim\mathbf{Y}_{lm}(\theta)e^{-\frac{\left(\rho-\rho_{c}\right)^{2}}{2\sigma}}$
(31) $\displaystyle\Pi_{m}(\tau=0,\rho,\theta)=0$ (32)

where $\mathbf{Y}_{lm}$ represents the $\theta$-dependent part of the
spherical harmonic function, and $\sigma$ represents the width of the Gaussian
distribution.

It is noteworthy to mention that the absence of spherical symmetry in rotating
black holes, leads to a phenomenon known as mode mixing. This phenomenon has
been extensively studied in Kerr and Kerr-Newmann black holes in different
theories of gravity, including GR mx1 ; mx2 ; mx3 , Chern-Simons modified
gravity ecsg ; Gao:2018acg , Gauss-Bonnet gravity spin3 ; spin5 ; esgb5 , and
Einstein-Maxwell-scalar gravity Lai:2022spn , which indicates that a pure
initial $l$ multipole will induce the presence of other $l^{\prime}$
multipoles with the same $m$ as it evolves. The evolution of different $m$
modes is decoupled in the presence of axisymmetry. Additionally, the
reflection symmetry results in the decoupling of even $l$ modes from odd $l$
modes. However, the evolution of a specific mode $(l,m)$ is coupled to that of
all the modes $(l+2k,m)$ with $k$ being an integer spin3 . Notably, the
$l=|m|$ mode plays a prominent role in the later stages of the evolution of
the scalar field, as indicated in Ref. spin5 . Among the dominant modes with
different values of $m$, the mode with $m=0$ exhibits the shortest growth
times, holding the greatest relevance to the instabilities, and has the
largest parameter region of the instability of the Kerr/Kerr-Newmann black
holes spin5 ; Gao:2018acg ; spin3 ; esgb5 . Moreover, the equations governing
the evolution of the scalar field perturbation with the chosen azimuthal mode,
$m=0$, are linear in nature. This will significantly simplify the analysis as
we avoid the complexities associated with solving nonlinear partial
differential equations. Therefore, we solely focus on the perturbations with
$l=m=0$ in the subsequent discussion. Additionally, we adopt the convention of
setting $M=1$ to express all quantities in units of $M$.

Our results, as presented below, demonstrate that the occurrence of
instability at the linear level is contingent upon the values of spin $a$,
charge $Q$, and $\alpha$. Figure 1 illustrates the time evolution of the
scalar field for a fixed set of parameters. It is evident that the scalar
field decays over time when $\alpha$ is below the scalarization threshold.
However, when $\alpha$ surpasses this threshold, the scalar field experiences
exponential growth. As anticipated, such growth is quenched by nonlinearity of
the scalar field, leading to an equilibrium scalar field configuration, i.e.,
the black hole scalarizes. Conversely, in the case of a decaying scalar field
perturbation, the effect of nonlinearity is largely insignificant. Figure 2
depicts the parameter space in which spontaneous scalarization arises due to
the linear tachyonic instability, where the extreme Kerr-Newmann black hole
provides an upper limit for the spin parameter for a fixed charge $Q$ (dashed
lines). In the case of zero spin, the Kerr-Newmann black hole degenerates to a
RN black hole with no parity-violating terms, leading to a decaying scalar
field perturbation. For $Q=0$, $F\tilde{F}$ vanishes, and only $R\tilde{R}$
remains, which has been extensively investigated in Refs. Gao:2018acg ; nl2 .
Additionally, as $Q$ increases, the parameter space for the linear stability
of the scalar field decreases.

Figure 1: Time evolutions of the scalar field perturbation due to $F\tilde{F}$
and $R\tilde{R}$ for $a=0.5$, $Q=0.8$ with various values of $\alpha$ and
$\beta$. Here we fix $\rho_{c}=6$ and $\sigma=0.2$. Observers are assumed to
be located at $\rho=6$ and $\theta=\pi/4$.

Figure 2: The parameter spaces of $a$ and $\alpha$ for $F\tilde{F}$ and
$R\tilde{R}$ at different $Q$ values ($0.8$, $0.5$, $0.25$, and $0.1$) in the
linear level. The solid lines represent the thresholds of the dominant $m=0$
mode for which scalarization occurs. The dashed lines represent the upper
limits for the spin parameter for the fixed charges $Q$ (extreme Kerr-Newmann
black holes). The filled areas indicate the stable regions, while the areas
between the dashed and solid lines of the same color in both panels indicate
instability.

##  IV Extensions to other nonspherically symmetric black holes

The scalarizations of the Kerr-Newmann black hole, with $I(\psi;g_{\mu\nu})$
being chosen to be the gravitational/electromagnetic Chern-Simons term,
represent two illustrative examples of the scalarized mechanism induced by the
parity violation. One remarkable property of such a mechanism is that it can
exist in a broad class of black hole backgrounds, provided that the
gravitational/electromagnetic Chern-Simons term does not vanish. Due to the
parity violations, $I(\psi;g_{\mu\nu})$ simply vanishes in the spherical
backgrounds and could be nontrivial only for nonspherically symmetric black
holes.

We now show the nonzero of the parity-violating gravitational/electromagnetic
curvature invariants, $R\tilde{R}$ and $F\tilde{F}$ for several specific
nonspherically symmetric black holes. We consider five different
nonspherically symmetric black hole solutions, including the Kerr, Kerr-
Newmann, RN-Melvin rnm1 ; rnm2 , Kerr-Melvin km1 , and Magnetized Kerr-Newmann
black holes rnm1 ; mkn1 ; km1 ; rnm2 . As shown in Table 1, $R\tilde{R}$ is
nonzero for all the five black holes, while $F\tilde{F}$ is nonzero for four
spacetimes, all except the Kerr black hole. Once $R\tilde{R}$ or $F\tilde{F}$
is nonzero, it must give rise to a negative effective mass squared in a
certain region of spacetime for the scalar perturbation no matter what the
sign of $f^{\prime\prime}(0)$ is. This is shown in Figure 3, in which we find
the $R\tilde{R}$ and $F\tilde{F}$ for all the nonspherically symmetric black
holes considered in this paper. It is evident that all the nonzero
$R\tilde{R}$ and $F\tilde{F}$ exhibit a negative sign in the region
$\theta\in[0,\pi]$ outside the horizon of the nonspherically symmetric black
holes. When this negative mass squared is sufficiently negative, the scalar
field will develop a tachyonic instability, resulting in the spontaneous
scalarization of the nonspherically symmetric black holes. Here we would like
to mention that the spontaneous scalarization of the Kerr and RN-Melvin black
holes with $I=R\tilde{R}$ has been previously investigated in Refs. p2 ;
Gao:2018acg ; nl2 ; mdcsg ; ecsg ; rnmdcsg .

Table 1: Summary of the nonzero of $R\tilde{R}$ and $F\tilde{F}$ for several specific nonshperical black holes. Here a “✓” represents the nonzero of $R\tilde{R}$ or $F\tilde{F}$.  | $I(\psi;g_{\mu\nu})$
---|---
Black holes | $R\tilde{R}$ | $F\tilde{F}$
Kerr | ✓ | …
Kerr-Newmann (KN) | ✓ | ✓
Reissner-Nordström-Melvin (RNM) | ✓ | ✓
Kerr-Melvin (KM) | ✓ | ✓
Magnetized Kerr-Newmann (MKN) | ✓ | ✓
Figure 3: The parity-violating terms of non-spherical black holes as a
function of $\theta$, where $B$ represents the magnetic field strength, $Q$
represents the charges of the KN and RNM black holes, and $q$ and $p$
represent the electric and magnetic charges of the MKN black holes,
respectively. We simply choose $r=4$. For other value of $r$, the result is
similar.

##  V Extensions to other parity-violating theories

The models studied in this paper can be easily extended to other types of
parity-violating theories—for example, the chiral-scalar tensor theory which
extends the Chern-Simons gravity by including parity-violating interactions
between the higher derivatives of the scalar field and the spacetime
curvatures Crisostomi:2017ugk . Similarly to the Chern-Simons gravity, the
chiral-scalar-tensor theory can naturally trigger the tachyonic instability of
the scalar field in non-spherical backgrounds and can exhibit more rich
phenomena of black hole scalarizations.

In the framework of the teleparallel gravity, the GR equivalent teleparallel
gravity can be modified by adding a parity-violating interaction between the
scalar field and the Nieh-Yan term in the gravitational action, i.e.,
$I(\psi;g_{\mu\nu})={\cal T}_{A\mu\nu}\tilde{\cal T}^{A\mu\nu}$ with ${\cal
T}_{A\mu\nu}$ being the torsion tensor two-form and $\tilde{\cal
T}^{A\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\lambda\gamma}{\cal
T}^{A}_{~{}~{}\lambda\gamma}$ being the dual torsion tensor two-form
Li:2020xjt ; Li:2021wij . Similarly, the parity-violating symmetric
teleparallel gravity can also be constructed by including an interaction
between the scalar field and the nonmetricity tensor $Q_{\mu\nu\lambda}$—i.e.,
$I(\psi;g_{\mu\nu})=\epsilon^{\mu\nu\lambda\gamma}Q_{\mu\nu}^{\xi}Q_{\lambda\gamma\xi}$
Li:2021mdp . More parity-violating gravities can be found in Refs.
Conroy:2019ibo ; Rao:2023doc ; Chen:2022wtz  and references therein.

In all these theories, $f^{\prime\prime}(0)I(\psi;g_{\mu\nu})$ can inevitably
become negative in a certain region of the nontrivial black hole backgrounds
and thus can naturally trigger tachyonic instabilities. Therefore, it is
interesting to explore such parity-violation-induced black hole scalarization
and the corresponding scalarized black hole solutions in such a broad class of
theories. We would like to address this issue in detail in our future works.

##  VI Summary and Discussion

The results presented here exhibit a possible black hole scalarization process
induced by the couplings between a scalar field and the parity-violating
spacetime/electromagnetic curvature terms, potentially extending the family of
known black hole scalarization mechanisms in the literature to a broader range
of modified theories of gravity.

For illustrative purposes, we only consider the dynamics in the “decoupling
limit” by numerically evolving the nonlinear scalar field equation on the
fixed Kerr-Newmann geometry. While a full scalarized black hole induced by
parity violation is still lacking, it is crucial to understand the full
scalarization dynamics by constructing the full scalarized black hole with
parity-violating interactions, which remains a challenging problem nl2 . The
main issue in this problem is that one has to solve the dynamical equation
that may contain higher time derivatives in some specific parity-violating
theories. For example, in the Chern-Simons modified gravity Alexander:2009tp ,
the dynamical equation contains the third-order time derivative which results
in the presence of ghost modes that generically render the theory
pathological. In this sense, the parity-violating theories that contain high-
order derivative terms can only be treated as effective field theories. This
is also the reason why we only consider the dynamics in the “decoupling
limit.” Recently, several parity-violating theories which are expected to be
healthy have been proposed—for example, the Nieh-Yan modified teleparallel
gravity Li:2020xjt ; Li:2021wij  and symmetric teleparallel gravity with
parity violations Li:2021mdp . These particular parity-violating theories do
not contain high-order derivative terms and do not have any ghost degree of
freedom. In addition, a ghost-free parity-violating scalar-tensor theory, as
an extension of the Chern-Simons modified gravity, has also been proposed
Crisostomi:2017ugk . It is interesting to explore the full dynamics of the
scalar field and seek any possible full scalarized black hole solutions in
these parity-violating theories. We expect to come back to this issue in our
future works.

## ACKNOWLEDGEMENTS

This work is supported by the National Key Research and Development Program of
China under Grant No. 2020YFC2201503, the Zhejiang Provincial Natural Science
Foundation of China under Grants No. LR21A050001 and No. LY20A050002, and the
National Natural Science Foundation of China under Grants No. 12275238, No.
11975203, and No. 12075207, and the US NSF Grant, No. PHY2308845.

## References

  * (1) B. P. Abbott et al. (LIGO Scientific and Virgo collaborations), Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116, 061102 (2016).
  * (2) R. Abbott et al. (LIGO Scientific, VIRGO and KAGRA collaborations), GWTC-3: Compact binary coalescences observed by LIGO and Virgo during the second part of the third observing run, arXiv:2111.03606.
  * (3) K. Akiyama et al. (Event Horizon Telescope collaboration), First M87 event horizon telescope results. I. The shadow of the supermassive black hole, Astrophys. J. Lett. 875, L1 (2019).
  * (4) K. Akiyama et al. (Event Horizon Telescope collaboration), First M87 event horizon telescope results. IV. Imaging the central supermassive black hole, Astrophys. J. Lett. 875, L4 (2019).
  * (5) K. Akiyama et al. (Event Horizon Telescope collaboration), First Sagittarius A* event horizon telescope results. I. The shadow of the supermassive black hole in the center of the Milky Way, Astrophys. J. Lett. 930, L12 (2022).
  * (6) J. D. Bekenstein, Black holes: Classical properties, thermodynamics and heuristic quantization, arXiv:gr-qc/9808028.
  * (7) P. T. Chrusciel, J. Lopes Costa and M. Heusler, Stationary black holes: Uniqueness and beyond, Living Rev. Relativity 15, 7 (2012).
  * (8) C. Bambi, Black Holes: A Laboratory for Testing Strong Gravity, (Springer,New York, 2017).
  * (9) M. Isi, M. Giesler, W. M. Farr, M. A. Scheel and S. A. Teukolsky, Testing the No-Hair Theorem with GW150914, Phys. Rev. Lett. 123, 111102 (2019).
  * (10) H. Qi, R. O’Shaughnessy and P. Brady, Testing the black hole no-hair theorem with Galactic Center stellar orbits, Phys. Rev. D 103, 084006 (2021).
  * (11) A. Maselli, N. Franchini, L. Gualtieri, T. P. Sotiriou, S. Barsanti and P. Pani, Detecting fundamental fields with LISA observations of gravitational waves from extreme mass-ratio inspirals, Nat. Astron. 6, 464 (2022).
  * (12) T. Damour and G. Esposito-Farese, Nonperturbative Strong Field Effects in Tensor-Scalar Theories of Gravitation, Phys. Rev. Lett. 70, 2220 (1993).
  * (13) T. Damour and G. Esposito-Farese, Tensor-scalar gravity and binary pulsar experiments, Phys. Rev. D 54, 1474 (1996).
  * (14) D. D. Doneva and S. S. Yazadjiev, New Gauss-Bonnet Black Holes with Curvature-Induced Scalarization in Extended Scalar-Tensor Theories, Phys. Rev. Lett. 120, 131103 (2018).
  * (15) D. D. Doneva, S. Kiorpelidi, P. G. Nedkova, E. Papantonopoulos and S. S. Yazadjiev, Charged Gauss-Bonnet black holes with curvature induced scalarization in the extended scalar-tensor theories, Phys. Rev. D 98, 104056 (2018).
  * (16) H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou and E. Berti, Spontaneous Scalarization of Black Holes and Compact Stars from a Gauss-Bonnet Coupling, Phys. Rev. Lett. 120, 131104 (2018).
  * (17) G. Antoniou, A. Bakopoulos and P. Kanti, Evasion of No-Hair Theorems and Novel Black-Hole Solutions in Gauss-Bonnet Theories, Phys. Rev. Lett. 120, 131102 (2018).
  * (18) M. Zhang and J. Jiang, Curvature induced scalarization of Kerr-Newman black hole spacetimes, Phys. Rev. D 107, 044002 (2023).
  * (19) Y. Brihaye and B. Hartmann, Spontaneous scalarization of charged black holes at the approach to extremality, Phys. Lett. B 792, 244 (2019).
  * (20) C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual and J. A. Font, Spontaneous Scalarization of Charged Black Holes, Phys. Rev. Lett. 121, 101102 (2018).
  * (21) P. G. S. Fernandes, C. A. R. Herdeiro, A. M. Pombo, E. Radu and N. Sanchis-Gual, Spontaneous scalarisation of charged black holes: coupling dependence and dynamical features, Classical Quantum Gravity 36, 134002 (2019).
  * (22) S. Hod, Spin-charge induced scalarization of Kerr-Newman black-hole spacetimes, J. High Energy Phys. 08, 272 (2022).
  * (23) M. Y. Lai, Y. S. Myung, R. H. Yue and D. C. Zou, Spin-charge induced spontaneous scalarization of Kerr-Newman black holes, Phys. Rev. D 106, 084043 (2022).
  * (24) M. Y. Lai, Y. S. Myung, R. H. Yue and D. C. Zou, Spin-induced scalarization of Kerr-Newman black holes in Einstein-Maxwell-scalar theory, Phys. Rev. D 106, 044045 (2022).
  * (25) M. Elley, H. O. Silva, H. Witek and N. Yunes, Spin-induced dynamical scalarization, descalarization, and stealthness in scalar-Gauss-Bonnet gravity during a black hole coalescence, Phys. Rev. D 106, 044018 (2022).
  * (26) E. Berti, L. G. Collodel, B. Kleihaus and J. Kunz, Spin-Induced Black-Hole Scalarization in Einstein-Scalar-Gauss-Bonnet Theory, Phys. Rev. Lett. 126, 011104 (2021).
  * (27) A. Dima, E. Barausse, N. Franchini and T. P. Sotiriou, Spin-Induced Black Hole Spontaneous Scalarization, Phys. Rev. Lett. 125, 231101 (2020).
  * (28) S. Hod, Onset of spontaneous scalarization in spinning Gauss-Bonnet black holes, Phys. Rev. D 102, 084060 (2020).
  * (29) D. D. Doneva, L. G. Collodel, C. J. Krüger and S. S. Yazadjiev, Black hole scalarization induced by the spin: 2 + 1 time evolution, Phys. Rev. D 102, 104027 (2020).
  * (30) L. Annulli, C. A. R. Herdeiro and E. Radu, Spin-induced scalarization and magnetic fields, Phys. Lett. B 832, 137227 (2022).
  * (31) Y. Liu, C. Y. Zhang, Q. Chen, Z. Cao, Y. Tian and B. Wang, The critical scalarization and descalarization of black holes in a generalized scalar-tensor theory, arXiv:2208.07548.
  * (32) C. Y. Zhang, Q. Chen, Y. Liu, W. K. Luo, Y. Tian and B. Wang, Critical Phenomena in Dynamical Scalarization of Charged Black Holes, Phys. Rev. Lett. 128, 161105 (2022).
  * (33) C. Y. Zhang, Q. Chen, Y. Liu, W. K. Luo, Y. Tian and B. Wang, Dynamical transitions in scalarization and descalarization through black hole accretion, Phys. Rev. D 106, L061501 (2022).
  * (34) D. D. Doneva and S. S. Yazadjiev, Beyond the spontaneous scalarization: New fully nonlinear mechanism for the formation of scalarized black holes and its dynamical development, Phys. Rev. D 105, L041502 (2022).
  * (35) J. L. Blázquez-Salcedo, D. D. Doneva, J. Kunz and S. S. Yazadjiev, Radial perturbations of scalar-Gauss-Bonnet black holes beyond spontaneous scalarization, Phys. Rev. D 105, 124005 (2022).
  * (36) D. D. Doneva, F. M. Ramazanoğlu, H. O. Silva, T. P. Sotiriou and S. S. Yazadjiev, Scalarization, arXiv:2211.01766.
  * (37) Y. S. Myung and D. C. Zou, Black holes in Gauss-Bonnet and Chern-Simons-scalar theory, Int. J. Mod. Phys. D 28, 1950114 (2019).
  * (38) Y. S. Myung and D. C. Zou, Onset of rotating scalarized black holes in Einstein-Chern-Simons-Scalar theory, Phys. Lett. B 814, 136081 (2021)
  * (39) D. D. Doneva and S. S. Yazadjiev, Spontaneously scalarized black holes in dynamical Chern-Simons gravity: dynamics and equilibrium solutions, Phys. Rev. D 103, 083007 (2021).
  * (40) Y. X. Gao, Y. Huang and D. J. Liu, Scalar perturbations on the background of Kerr black holes in the quadratic dynamical Chern-Simons gravity, Phys. Rev. D 99, 044020 (2019).
  * (41) Y. S. Myung and D. C. Zou, Onset of rotating scalarized black holes in Einstein-Chern-Simons-scalar theory, Phys. Lett. B 814, 136081 (2021).
  * (42) S. J. Zhang, Massive scalar field perturbation on Kerr black holes in dynamical Chern–Simons gravity, Eur. Phys. J. C 81, 441 (2021).
  * (43) S. J. Zhang, B. Wang, E. Papantonopoulos and A. Wang, Magnetic-induced spontaneous scalarization in dynamical Chern–Simons gravity, Eur. Phys. J. C 83, 97 (2023)
  * (44) D. Astefanesei, C. Herdeiro, J. Oliveira and E. Radu, Higher dimensional black hole scalarization, J. High Energy Phys. 09, 186 (2020).
  * (45) C. Herdeiro, E. Radu and D. H. Tchrakian, Embedding Gauss–Bonnet scalarization models in higher dimensional topological theories, Symmetry 13, 590 (2021).
  * (46) D. D. Doneva and S. S. Yazadjiev, Neutron star solutions with curvature induced scalarization in the extended Gauss-Bonnet scalar-tensor theories, J. Cosmol. Astropart. Phys. 04 (2018) 011.
  * (47) S. J. Zhang, B. Wang, A. Wang and J. F. Saavedra, Object picture of scalar field perturbation on Kerr black hole in scalar-Einstein-Gauss-Bonnet theory, Phys. Rev. D 102, 124056 (2020).
  * (48) D. D. Doneva, L. G. Collodel and S. S. Yazadjiev, Spontaneous nonlinear scalarization of Kerr black holes, Phys. Rev. D 106, 104027 (2022).
  * (49) I. Racz and G. Z. Toth, Numerical investigation of the late-time Kerr tails, Classical Quantum Gravity 28, 195003 (2011).
  * (50) E. Harms, S. Bernuzzi and B. Brügmann, Numerical solution of the 2 + 1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays, Classical Quantum Gravity 30, 115013 (2013).
  * (51) S. R. Dolan, L. Barack and B. Wardell, Self force via $m$-mode regularization and 2+1D evolution: II. Scalar-field implementation on Kerr spacetime, Phys. Rev. D 84, 084001 (2011).
  * (52) A. Zenginoğlu, G. Khanna and L. M. Burko, Intermediate behavior of Kerr tails, Gen. Relativ. Gravit. 46, 1672 (2014).
  * (53) L. M. Burko and G. Khanna, Mode coupling mechanism for late-time Kerr tails, Phys. Rev. D 89, 044037 (2014).
  * (54) I. Thuestad, G. Khanna and R. H. Price, Scalar fields in black hole spacetimes, Phys. Rev. D 96, 024020 (2017).
  * (55) F. J. Ernst, Black holes in a magnetic universe, J. Math. Phys. (N.Y.) 17, 54 (1976).
  * (56) G. W. Gibbons, A. H. Mujtaba and C. N. Pope, Ergoregions in magnetised black hole spacetimes, Classical Quantum Gravity 30, 125008 (2013).
  * (57) A. N. Aliev and D. V. Galtsov, Exact solutions for magnetized black holes, Astrophys. Space Sci. 155, 181 (1989).
  * (58) F. J. Ernst and W. J. Wild, Kerr black holes in a magnetic universe, J. Math. Phys. (N.Y.) 17, 182 (1976).
  * (59) M. Crisostomi, K. Noui, C. Charmousis and D. Langlois, Beyond Lovelock gravity: Higher derivative metric theories, Phys. Rev. D 97, 044034 (2018).
  * (60) M. Li, H. Rao and D. Zhao, A simple parity violating gravity model without ghost instability, J. Cosmol. Astropart. Phys. 11 (2020) 023.
  * (61) M. Li, H. Rao and Y. Tong, Revisiting a parity violating gravity model without ghost instability: Local Lorentz covariance, Phys. Rev. D 104, 084077 (2021).
  * (62) M. Li and D. Zhao, A simple parity violating model in the symmetric teleparallel gravity and its cosmological perturbations, Phys. Lett. B 827, 136968 (2022).
  * (63) A. Conroy and T. Koivisto, Parity-violating gravity and GW170817 in non-Riemannian cosmology, J. Cosmol. Astropart. Phys. 12 (2019) 016.
  * (64) H. Rao and D. Zhao, Parity violating scalar-tensor model in teleparallel gravity and its cosmological application, arXiv:2304.07138.
  * (65) Z. Chen, Y. Yu and X. Gao, Polarized gravitational waves in the parity violating scalar-nonmetricity theory, J. Cosmol. Astropart. Phys. 06 (2023) 001.
  * (66) S. Alexander and N. Yunes, Chern-Simons modified general relativity, Phys. Rep. 480, 1(2009).