# TeachAugment: Data Augmentation Optimization Using Teacher Knowledge

Teppei Suzuki
Denso IT Laboratory, Inc.

###### Abstract

Optimization of image transformation functions for the purpose of data
augmentation has been intensively studied. In particular, adversarial data
augmentation strategies, which search augmentation maximizing task loss, show
significant improvement in the model generalization for many tasks. However,
the existing methods require careful parameter tuning to avoid excessively
strong deformations that take away image features critical for acquiring
generalization. In this paper, we propose a data augmentation optimization
method based on the adversarial strategy called TeachAugment, which can
produce informative transformed images to the model without requiring careful
tuning by leveraging a teacher model. Specifically, the augmentation is
searched so that augmented images are adversarial for the target model and
recognizable for the teacher model. We also propose data augmentation using
neural networks, which simplifies the search space design and allows for
updating of the data augmentation using the gradient method. We show that
TeachAugment outperforms existing methods in experiments of image
classification, semantic segmentation, and unsupervised representation
learning tasks.000Code: https://github.com/DensoITLab/TeachAugment

##  1 Introduction

Data augmentation is an important technique used to improve model
generalization. To automatically search efficient augmentation strategies for
model generalization, AutoAugment[9] has been proposed. Searched data
augmentation policies lead to significant generalization improvements.
However, AutoAugment requires thousands of GPU hours to search for efficient
data augmentation.

Recent studies[30, 17, 22] have demonstrated methods leading to dramatic
reductions in search costs with AutoAugment, meaning that computational costs
are no longer a problem. In particular, online data augmentation optimization
frameworks[61, 28, 51, 18] that alternately update augmentation policies and a
target network have not only reduced the computational costs but also
simplified the data augmentation search pipeline by unifying the search and
training processes.

Figure 1: Concept of TeachAugment. Adversarial data augmentation, a baseline
method, transforms data to increase loss for the target model $f_{\theta}$.
The augmented data are often meaningless (e.g., black-out and noise images) or
difficult to recognize without any constraints. TeachAugment, the proposed
method, transforms data so that they are adversarial for the target model but
they are recognizable for the teacher model $f_{\hat{\theta}}$. As a result,
the augmented images will be more informative than the adversarial data
augmentation.

Many online optimization methods are based on an adversarial strategy that
searches augmentation maximizing task loss for the target model, which is
empirically known to improve model generalization[61, 38, 50, 43, 28].
However, the adversarial data augmentation is unstable without any constraint
because maximizing the loss can be achieved by collapsing the inherent
meanings of images, as shown in Fig. 1. To avoid the collapse, previous
methods regularize augmentation based on prior knowledge and/or restrict the
search range of the magnitude parameters of functions in the search space, but
there are many tuned parameters that will annoy practitioners.

To alleviate the parameter tuning problem, we propose an online data
augmentation optimization method using teacher knowledge, called TeachAugment.
TeachAugment is also based on the adversarial data augmentation strategy, but
it searches augmentation in the range where the transformed image can be
recognizable for a teacher model, as shown in Fig. 1. Unlike previous
adversarial data augmentation methods [61, 51, 38, 50], thanks to the teacher
model, TeachAugment does not require priors and hyperparameters to avoid the
excessively strong augmentation that collapses the inherent meanings of
images. As a result, TeachAugment does not require parameter tuning to ensure
that the transformed images are recognizable.

Moreover, we propose data augmentation using neural networks that represent
two functions, geometric augmentation and color augmentation. Our augmentation
model applies only two transformations to data but they can represent most
functions included in the search space of AutoAugment and their composite
functions. The use of the neural networks has two advantages compared to
conventional augmentation functions: (i) we can update the augmentation
parameters using the gradient method with backpropagation, and (ii) we can
reduce the number of functions in the search space from tens of functions to
two functions. In particular, because of the latter advantage, practitioners
only need to consider the range of the magnitude parameters for the two
functions when they adjust the size of the search space for better
convergence.

Contribution. Our contribution is summarized as follows: (1) We propose an
online data augmentation optimization framework based on the adversarial
strategy using teacher knowledge called TeachAugment. TeachAugment makes the
adversarial augmentation more informative without careful parameter tuning by
leveraging the teacher model to avoid collapsing the inherent meaning of
images. (2) We also propose data augmentation using neural networks. The
proposed augmentation simplifies the search space design and enables updating
of its parameters by the gradient method in TeachAugment. (3) We show that
TeachAugment outperforms previous methods, including online data
augmentation[28, 61] and state-of-the-art augmentation strategies[10, 40] in
classification, semantic segmentation, and unsupervised representation
learning tasks without adjusting the hyperparameters and size of the search
space for each task.

##  2 Related work

As conventional data augmentation for image data, geometric and color
transformations are widely used in deep learning. Besides, advanced data
augmentations[54, 63, 21, 23, 60, 12, 58] have been recently developed and
they have improved accuracy on image recognition tasks. Data augmentation not
only enhances the image recognition accuracy, but also plays an important role
in recent unsupervised representation learning[19, 6, 15, 5, 7] and semi-
supervised learning[38, 50, 47]. While the data augmentation usually improves
model generalization, it sometimes hurts performance or induces unexpected
biases. Thus, one needs to manually find the effective augmentation policies
based on domain knowledge to enhance model generalization.

Cubuk et al.[9] proposed a method to automatically search for effective data
augmentation, called AutoAugment. AutoAugment outperforms hand-designed
augmentation strategies and shows state-of-the-art performances on various
benchmarks. Data augmentation search has become a research trend, and many
methods have been proposed[30, 17, 29, 61, 53, 45, 28, 51, 31, 40, 10, 33, 18,
55].

We roughly categorize them into two types: a proxy task based method and a
proxy task free method. The proxy task based methods[30, 17, 29, 61, 53]
search data augmentation strategies on proxy tasks that use subsets of the
training data and/or small models to reduce computational costs. Thus, policy
searches using the proxy task based method might be sub-optimal. The proxy
task free methods [28, 33, 40, 61, 10] directly search data augmentation
strategies on the target network with all training data. Thus, policies
obtained with this method are potentially optimal.

In proxy task free methods, several approaches, such as RandAugment [10] and
TrivialAugment [40], randomize the parameters searched and reduce the size of
the search space. Other methods, such as Adversarial AutoAugment [61] and
PointAugment [28], update augmentation policies in an online manner, meaning
that they alternately update a target network and augmentation policies. The
online optimization methods simplify data augmentation optimization frameworks
by unifying the search and training processes. However, these methods are
fraught with minor problems. For example, PointAugment [28] unjustly binds the
difficulty of augmented images, Adversarial AutoAugment [61] manually
restricts the search space to guarantee convergence, and OnlineAugment [51]
has many hyperparamters for regularization. These problems stem from reliance
on the adversarial strategy that searches augmentation maximizing task loss.
In other words, these problems are induced to ensure that the transformed
images are recognizable.

In this work, we focus on proxy task free methods updating the policies in an
online manner for two reasons: (i) they can directly search data augmentation
strategies on the target network with all training data, and (ii) they unify
the search and training processes, simplifying the framework.

##  3 Data augmentation optimization using teacher knowledge

Figure 2: Overview of training procedure for TeachAugment. Our method
alternately updates the target model $f_{\theta}$ and the augmentation model
$a_{\phi}$ (i.e., step 1 and step 2 are repeated).

###  3.1 Preliminaries

Let $x\sim\mathcal{X}$ and $a_{\phi}$ be an image sampled from dataset
$\mathcal{X}$ and an augmentation function parameterized by $\phi$,
respectively. In conventional data augmentation, $\phi$ corresponds to the
magnitude of augmentation, while in this work, it corresponds to the neural
network’s parameters. In general training procedures using data augmentation,
the mini-batch samples are transformed by $a_{\phi}$ and fed into a target
network $f_{\theta}$ (i.e., $f_{\theta}(a_{\phi}(x))$). Then, the parameters
of the target network are updated to minimize task loss $L$ in the stochastic
gradient descent. This training procedure is represented as
$\min_{\theta}\mathbb{E}_{x\sim\mathcal{X}}L(f_{\theta}(a_{\phi}(x))$. Note
that we omitted the target label because we consider not only supervised
learning but also unsupervised representation learning in this work.

In addition, adversarial data augmentation searches the parameters $\phi$
maximizing the loss. The objective is defined as
$\max_{\phi}\min_{\theta}\mathbb{E}_{x\sim\mathcal{X}}L(f_{\theta}(a_{\phi}(x))$.
This objective is sometimes solved by alternately updating $\phi$ and
$\theta$[61, 28]. The adversarial data augmentation is empirically known to
improve model generalization[61, 43]. However, it does not work without some
regularization or restriction in the size of the search space because the
maximization with respect to $\phi$ can be achieved by collapsing the inherent
meanings of $x$. Thus, instead of using regularization based on prior
knowledge, we utilized a teacher model to avoid the collapse.

###  3.2 TeachAugment

Let $f_{\hat{\theta}}$ be the teacher model, which allows for any model as
long as it is different from the target model $f_{\theta}$.111In most of our
experiments, we define the teacher model as the same model as the target model
but with different parameters. Thus, the same symbol $f$ is used for the
teacher model and the target model although it may be a little complicated. In
this work, we suggest two types of the teacher model, a pretrained teacher and
an EMA teacher whose weights are updated as an exponential moving average of
the target model’s weights. We provide detailed definitions and evaluate the
effect of teacher model choices in Sec. 5.2.

The proposed objective is defined as follows:

$\displaystyle\max_{\phi}\min_{\theta}\mathbb{E}_{x\sim\mathcal{X}}\left[L(f_{\theta}(a_{\phi}(x)))-L(f_{\hat{\theta}}(a_{\phi}(x)))\right].$
(1)

For the target model, this objective has the same properties as the
adversarial data augmentation, but the augmentation function needs to minimize
the loss for the teacher model, in addition to maximizing the loss for the
target model. The augmentation that is obtained in this manner avoids
collapsing the inherent meanings of images because the loss for the teacher
model will explode when the transformed images are unrecognizable. In other
words, the introduced teacher loss requires the augmentation function to
transform images so that they are adversarial for the target model in the
range where they are recognizable for the teacher model.

As shown in Fig. 2, the objective is solved by alternately updating the
augmentation function and the target model in the stochastic gradient descent,
similar to previous methods [18, 28, 61]. We first update the target network
for $n_{\text{inner}}$ steps, and then update the augmentation function. A
pseudo-code can be found in the appendix. Note that the augmentation function
is updated by the gradient method because our proposed augmentation using the
neural networks introduced in Sec. 4 is differentiable with respect to $\phi$.
We refer to the augmentation strategy following Eq. 1 as TeachAugment.

Unlike previous methods[28, 51], TeachAugment does not regularize augmentation
functions based on the domain knowledge, such as cycle consistency and
smoothness[51]. It also does not bind the difficulty of the transformed images
as in [28] to ensure that the transformed images are recognizable.

###  3.3 Improvement techniques

The training procedure for TeachAugment is similar to that for generative
adversarial networks (GANs)[14] and actor-critic methods in reinforcement
learning[49, 26], which alternately update two networks. Practitioners in both
fields have amassed a large number of strategies to mitigate instabilities and
improve training[44]. TeachAugment also benefits from three techniques used in
both fields, experience replay, non-saturating loss, and label smoothing.
Moreover, we introduced color regularization to mitigate inconsistency between
color distributions of images before and after data augmentation. The
techniques introduced here are also applicable for other online methods[28,
61, 51].

Non-saturating loss. For classification tasks, the loss function $L$ is
usually defined as the cross-entropy loss,
$L(f_{\theta}(a_{\phi}(x))=\sum_{k=1}^{K}-y_{k}\log
f_{\theta}(a_{\phi}(x))_{k}$, where $y\in\\{0,1\\}^{K}$ and $K$ denote the
one-hot ground truth label and the number of classes, respectively. In this
case, the gradient of the first term in Eq. (1) often saturates in the
maximization problem with respect to $\phi$ when the target model’s
predictions are very confident. Thus, we use
$\sum_{k=1}^{K}y_{k}\log(1-f_{\theta}(a_{\phi}(x))_{k})$ when updating the
augmentation model rather than $\sum_{k=1}^{K}-y_{k}\log
f_{\theta}(a_{\phi}(x))_{k}$. This technique has been used in GANs[14].

The non-saturating loss is a key factor for TeachAugment; it improves error
rates of WideResNet-28-10[59] on CIFAR-100[27] from 18.7% to 17.4% (a
baseline’s error rate is 18.4%). Thus, we basically use the non-saturating
loss for updating augmentation models in our experiments.

Experience replay. In reinforcement learning, experience replay[32, 39] stores
actions chosen by the actor in the past and reuses them to update the critic.
We apply this technique to our method by storing the augmentation models and
prioritizing them in a manner similar to prioritized experience replay[46].
Then, the target network is updated using the augmentation model randomly
sampled from the buffer following their priorities.

Let $p_{i}$ be a priority of the $i$-th stored augmentation model. We compute
the priority as $p_{i}=\gamma^{S-i}$, where $S$ denotes the number of
augmentation models stored in the buffer, $\gamma$ denotes a decay rate, and
we set $\gamma$ to 0.99. In our experiments, we stored the augmentation model
every $n_{\text{buffer}}$ epochs. For image classification on CIFAR-10,
CIFAR-100, and semantic segmentation, $n_{\text{buffer}}$ was set to 10, and
for other tasks and datasets, it was set to 1.

Label smoothing. Label smoothing is a technique that replaces the one-hot
labels with $\hat{y}_{k}=(1-\epsilon)y_{k}+\epsilon/K$, where
$\epsilon\in[0,1)$ is a smoothing parameter. For our method, the label
smoothing prevents gradients from exploding when the target model’s
predictions are very confident under the non-saturating loss. In particular,
such a situation tends to occur for easy tasks or strong target models. To
mitigate exploding gradients, we used the smoothed labels for the first term
in Eq. (1) when updating the augmentation model. Note that for a fair
comparison, we did not apply it when updating the target model.

Color regularization. In practice, the color augmentation model tends to
transform the pixel colors outside the color distribution of the training
data. As a result, the augmented images will be out-of-distribution data,
which may hurt the recognition accuracy for the in-distribution samples. To
align the color distributions before and after augmentation, we regularized
the color augmentation model by introducing sliced Wasserstein distance (SWD)
[3] between pixel colors before and after the augmentation. SWD is a variant
of Wasserstein distance that represents a distance between two distributions.

We define the color regularization term as follows:

$\displaystyle
L_{\text{color}}(\\{x^{b}\\}_{b}^{B},\\{\tilde{x}^{b}\\}_{b}^{B})=\sum_{i}\mathrm{SWD}(\\{x_{i}^{b}\\}_{b}^{B},\\{\tilde{x}_{i}^{b}\\}_{b}^{B}),$
(2)

where $\\{x_{i}^{b}\\}_{b}^{B}$ denotes a set of $i$-th pixels of images in a
mini-batch with a batch size of $B$, and $\\{\tilde{x}^{b}\\}_{b}^{B}$ denotes
color-augmented images defined in Eq. (4). Because costs of computing SWD at
each pixel position depends linearly on the image resolution, we can compute
SWD for high-resolution images handled in semantic segmentation with low
computational resources. Then, in the stochastic gradient descent, the
gradient with respect to $\phi$ in each iteration is represented as follows:

$\displaystyle\frac{\partial}{\partial\phi}\frac{1}{B}\sum_{b}^{B}\left[L(f_{\theta}(a_{\phi}(x^{b})))-L(f_{\tilde{\theta}}(a_{\phi}(x^{b})))\right]$
$\displaystyle-\lambda
L_{\text{color}}(\\{x^{b}\\}_{b}^{B},\\{\tilde{x}^{b}\\}_{b}^{B}),$ (3)

where $\lambda$ is a hyperparameter controlling the effect of the
regularization that was set to 10 in our experiments.

##  4 Data augmentation using neural networks

We propose data augmentation using neural networks parameterized by $\phi$,
which consists of two models, a color augmentation model $c_{\phi_{c}}$ and a
geometric augmentation model $g_{\phi_{g}}$. Thus, $a_{\phi}$ is defined as
the composite function of $c_{\phi_{c}}$ and $g_{\phi_{g}}$,
$a_{\phi}=g_{\phi_{g}}\circ c_{\phi_{c}}$, and the parameter $\phi$ is a set
of $\phi_{c}$ and $\phi_{g}$, $\phi=\\{\phi_{c},\phi_{g}\\}$.

Figure 3: Data augmentation procedure. Our data augmentation consists of the
color augmentation $c_{\phi_{c}}$ and the geometric augmentation
$g_{\phi_{g}}$. Each augmentation is applied to an input image $x$ with
probabilities $p_{c}$ and $p_{g}$.

Our augmentation model enables updating of its parameters by the gradient
method and construction of the search space with only two functions.

The augmentation procedure is illustrated in Fig. 3. Given an image
$x\in\mathbb{R}^{M\times 3}$, where $M$ denotes the number of pixels and 3
corresponds to the RGB color channels, the color augmentation is applied with
probability of $p_{c}\in(0,1)$, and then the geometric augmentation is applied
with probability of $p_{g}\in(0,1)$.

The color augmentation is defined as follows:

$\displaystyle\tilde{x}_{i}=t(\alpha_{i}\odot x_{i}+\beta_{i}),\
(\alpha_{i},\beta_{i})=c_{\phi_{c}}(x_{i},z,c),$ (4)

where $\alpha_{i},\ \beta_{i}\in\mathbb{R}^{3}$ denote scale and shift
parameters; $\odot$ denotes the element-wise multiplication between vectors;
$t(\cdot)$ denotes the triangle wave, $t(x)=\arccos(\cos(x\pi))/\pi$, which
ensures $\tilde{x}_{i}\in[0,1]$; $z\sim\mathcal{N}(0,I_{N})$, where
$\mathcal{N}(0,I_{N})$ denotes $N$-dimensional unit Gaussian distribution, and
$c$ is an optional context vector. In our experiments, we used a one-hot
ground truth label as $c$ for image classification and omitted it for other
tasks. The color augmentation model can transform the input image into any
image in principle when the augmentation model is sufficiently large because
of the universal approximation theorem.

The geometric augmentation is defined as follows:

$\displaystyle\hat{x}=\mathrm{Affine}(\tilde{x},A+I),\ A=g_{\phi_{g}}(z,c),$
(5)

where $\mathrm{Affine}(\tilde{x},A+I)$ denotes an affine transformation of
$\tilde{x}$ with a parameter $A+I$; $I\in\mathbb{R}^{2\times 3}$ denotes a
matrix where $\forall i,I_{ii}=1$ and 0 otherwise, which makes the affine
transformation the identity mapping $\mathrm{Affine}(x,I)=x$;
$A\in\mathbb{R}^{2\times 3}$ denotes the residual parameter, and $c$ and $z$
are the same vectors used in Eq. (4).

The geometric augmentation model can be defined by transformations other than
the affine transformation, similar to [24]. However, the affine transformation
can represent all geometric transformations in the search space of AutoAugment
and their composite functions. Thus, we considered only the affine
transformation in this work.

In addition to $\phi_{c}$ and $\phi_{g}$, we also learned the probabilities
$p_{c}$ and $p_{g}$ by using the gradient method. However, the decision
process of applying the augmentation is non-differentiable with respect to
$p_{c}$ and $p_{g}$. To make it differentiable with respect to the
probabilities, we used the same approach as in previous works[17, 29]. A
detailed pipeline can be found in the appendix.

##  5 Experiments

We evaluate our method with three tasks. For an ablation study and comparison
with existing automatic data augmentation search methods, we evaluate our
method on image classification tasks. We train WideResNet-40-2 (WRN-40-2)[59],
WideResNet-28-10 (WRN-28-10)[59], Shake-Shake (26 2$\times$96d)[13], and
PyramidNet[16] with ShakeDrop regularization[57] on CIFAR-10 and
CIFAR-100[27], and train ResNet-50 [20] on ImageNet[11]. The training and
evaluation protocols are the same settings as in previous work[30].

In addition to the above experiments, we examine our method with semantic
segmentation and unsupervised representation learning. For semantic
segmentation, we train FCN-32s [34], PSPNet [62], and Deeplabv3 [4] on
Cityscapes [8]. The training protocol is the same as [62]. For unsupervised
representation learning, we pretrain ResNet-50 [20] on ImageNet [11] using
SimSiam [7] with various data augmentation and then evaluate the model
following the linear evaluation protocol [7].

More details can be found in the appendix.

###  5.1 Implementation details

We construct the geometric augmentation model using a multi-layer perceptron
(MLP) and the color augmentation model using two MLPs that receive an RGB
vector and a noise vector as inputs and then add up the outputs. We apply the
sigmoid function to the outputs of each augmentation model and normalize the
outputs in a range of $A\in(-0.25,0.25)^{2\times 3}$, $\alpha\in(0.6,1.4)$,
and $\beta\in(-0.5,0.5)$. The dimension of the noise vector $z$ is set to 128.
For stochasticity, Dropout [48] is applied after the linear layers, except to
the output layer. We initialize the weights of the output layer as zero to
make the augmentation the identity mapping in the initial state. We use AdamW
optimizer [35] to train the augmentation model. All the hyperparameters of
AdamW (e.g., learning rate and weight decay) are set to the PyTorch default
parameters [42]. More details can be found in the appendix.

###  5.2 Ablation study

Effect of teacher model choices. We first investigate the effect of teacher
model choices. We trained WRN-40-2 with two types of teacher models: a teacher
that was pretrained with the baseline augmentation, and an EMA teacher that is
exponential moving average of the target model (i.e.,
$\hat{\theta}\leftarrow\xi\hat{\theta}+(1-\xi)\theta$). We set the decay rate
$\xi$ to 0.999, following [52]. For the pretrained teacher, we pretrained
WRN-40-2, which is the same model as the target model, and WRN-28-10, which is
a stronger model than the target model.

The results are shown in Tab. 1.

Teacher | WRN-40-2 | WRN-28-10 | EMA
---|---|---|---
CIFAR-10 | 4.4 | 3.7 | 3.7
CIFAR-100 | 21.3 | 21.6 | 20.3

Table 1: Effect of teacher models. We report the error rates (%) of
WideResNet-40-2 trained with various teacher models. WRN-40-2 and WRN-28-10
are pretrained with the baseline augmentation. EMA is an exponential moving
average of the target model.

For both datasets, the EMA teacher achieves lower error rates than the others.
The augmentation model may cause overfitting to the teacher model when the
pretrained teacher is used because they are not updated during training. The
EMA teacher would prevent overfitting by updating parameters at the same time
as the target model during training, bringing more improvement than the
pretrained teachers.

Interestingly, we found the stronger teacher was not a better teacher. In
fact, WRN-28-10 demonstrates error rates comparable to the EMA teacher in
CIFAR-10, but in CIFAR-100, it was slightly worse than WRN-40-2.

In the rest of the experiments, we used the EMA teacher for TeachAugment
because the EMA teacher not only has lower error rates but also eliminates the
architecture choices of the teacher model.

Effect of stabilizing techniques. We next investigate the effect of
stabilizing techniques introduced in Sec. 3.3. We trained WRN-28-10 on
CIFAR-10 and CIFAR-100, with the exception of each technique.

The results are shown in Tab. 2.

Dataset | CIFAR-10 | CIFAR-100
---|---|---
All | 2.5 | 16.8
w/o Color Reg. | 3.0 | 16.5
w/o Experience Replay | 2.7 | 17.3
w/o Label Smoothing | 3.0 | 17.3
w/o all techniques | 3.0 | 17.4

Table 2: Error rates (%) of WideResNet-28-10 on CIFAR-10 and CIFAR-100 with
various technique choices.

Except for the color regularization on CIFAR-100, all of the techniques
contributes to the improvement of error rates.

We visualize the effect of the color regularization in Fig. 4.

|  |
---|---|---

(a) CIFAR-10

(b) CIFAR-100

Figure 4: Augmented images and corresponding color distributions on CIFAR-10
and CIFAR-100. From left to right, original images, augmented images from the
model trained without color regularization, and augmented images from the
model trained with color regularization. We randomly picked 128 images and
plotted their pixel colors (i.e., each point corresponds to a pixel).

The color distributions are aligned by the color regularization. In
particular, the augmented images without regularization lose brightness in
many pixels with CIFAR-100. However, alignment of the color distribution
improve the error rates only for CIFAR-10. This would be due to the color
diversity of CIFAR-10 being narrower than CIFAR-100, which can be seen from
the color distributions in Fig. 4. In other words, in the case of training
without color regularization, the color augmentation produces out-of-
distribution colors, and the obtained images hurt the recognition accuracy of
the target model for the in-distribution samples. The transformed image from
CIFAR-10 tends to be this kind of an out-of-distribution sample due to the low
diversity of colors.

We evaluate label smoothing with various smoothing parameter values, and the
results are shown in Fig. 5. For the easy tasks of CIFAR-10, label smoothing
with the large $\epsilon$ works well. As already described in Sec. 3.3, the
gradient of the non-saturating loss tends to be large for easy tasks or strong
models, meaning that the model’s predictions tend to be very confident. Thus,
a larger $\epsilon$ avoids exploding gradients and stabilizes training for
CIFAR-10.

Figure 5: Effect of $\epsilon$ on label smoothing. The upper figure shows
error rates on CIFAR-100, and the lower figure shows the error rates on
CIFAR-10. Stars indicate $\epsilon$ achieving minimum error rates.

Evaluation of the proposed objective function. We evaluate the effectiveness
of the proposed objective function, eq. (1). As baseline methods, we replace
the objective function in our framework with the loss of adversarial
AutoAugment (Adv. AA) [61] and PointAugment (PA) [28]. For a fair comparison,
all of the models were trained with the same protocol and the difference
between them was the objective function. The detailed setting is described in
the appendix.

The results are shown in Tab. 3.

Objective | Baseline | Adv. AA [61] | PA [28] | Ours
---|---|---|---|---
CIFAR-10 | 3.1 | 3.6 | 2.9 | 2.5
CIFAR-100 | 18.4 | 19.4 | 17.5 | 16.8

Table 3: Error rates (%) of WideResNet-28-10 trained with various online
optimization frameworks. All results are obtained under the same augmentation
models and training protocol, except for the objective function.

The error rates of Adv. AA degrades from the baseline because the augmentation
model produces the unrecognizable images that confuse the target model. For
Adv. AA, one needs to carefully tune the size of the search space to guarantee
the convergence[61] but we do not adjust it. In addition, for such
instability, the proposed augmentation model would be unsuitable because of
its property, i.e., the color augmentation model can transform the input image
into any image. In other words, the proposed augmentation would work only for
the methods that guarantee transformed images to be recognizable. Our method
achieves better error rates than PointAugment. PointAugment controls the upper
bound of the loss for the augmented data with a dynamic parameter, but our
method has no such restriction. As a result, our method provides more diverse
transformations and better improvements in the error rates than PointAugment.
The augmented images are visualized in the appendix.

Evaluation of the proposed augmentation model. We compare the proposed
augmentation model to a differentiable data augmentation pipeline (DDA)
proposed in [17] and augmentation models used in OnlineAugment (OA)[51]. The
results are shown in Tab. 4.

Augmentation | Baseline | DDA | OA | Ours
---|---|---|---|---
CIFAR-10 | 3.1 | 2.8 | 16.7 | 2.5
CIFAR-100 | 18.4 | 18.0 | 26.2 | 16.8
Table 4: Error rates (%) of WideResNet-28-10 with DDA [17], the augmentation
models of OnlineAugment [51] (OA) and the proposed augmentation on CIFAR-10
and CIFAR-100. Each augmentation is optimized by the TeachAugment strategy.

The improvement of DDA from baseline is less than that of ours. DDA uses some
techniques (e.g., relaxation[37, 25] and straight-through estimator[1]) to
make some data augmentation functions differentiable, but these techniques
induce inaccurate gradients and the vanishing gradient problem that would make
it difficult to learn effective augmentation in the TeachAugment framework. OA
is much worse than baseline. OnlineAugment regularizes augmentation models
based on prior knowledge, such as cycle consistency and smoothness, instead of
bounding the search space. However, TeachAugment does not impose strong
regularization. The lack of regularization would be unsuitable for
augmentation with the unbounded search space. We note that the proposed
augmentation is better than DDA and OA for TeachAugment, but as seen in Tab.
3, it does not work in all cases.

###  5.3 Image classification

We compare our method to several previous methods: AutoAugment (AA) [9],
PBA[22], Fast AA[30], Faster AA[17], DADA[29], RandAugment (RandAug)[10],
OnlineAugment (OnlineAug)[51], and Adv. AA[61].

For CIFAR-10 and CIFAR-100, we used the smoothing parameter $\epsilon$
achieving the best error rate in the previous section. We set $\epsilon$ to
0.1 for training on ImageNet, following the tendency in Fig. 5 (i.e., the
smaller $\epsilon$ tends to be better for difficult tasks).

We show the comparison results in Tab. 5.

| CIFAR-10 | CIFAR-100 | ImageNet
---|---|---|---
| WRN-28-10 | Shake-Shake | PyramidNet | WRN-28-10 | Shake-Shake | PyramidNet | ResNet-50
Baseline | 3.9 | 2.9 | 2.7 | 18.8 | 17.1 | 14.0 | 23.7/6.9
Cutout[12] | 3.1 | 2.6 | 2.3 | 18.4 | 16.0 | 12.2 | -
AA[9] | 2.6 | 2.0 | 1.5 | 17.1 | 14.3 | 10.7 | 22.4/6.4
PBA[22] | 2.6 | 2.0 | 1.5 | 16.7 | 15.3 | 10.9 | -
RandAug[10] | 2.7 | 2.0 | 1.5 | 16.7 | - | - | 22.4/6.2
Fast AA[30] | 2.7 | 2.0 | 1.7 | 17.3 | 14.6 | 11.7 | 22.4/6.3
Faster AA[17] | 2.6 | 2.0 | - | 17.3 | 15.0 | - | 23.5/6.8
DADA[29] | 2.7 | 2.0 | 1.7 | 17.5 | 15.3 | 11.2 | 22.5/6.5
OnlineAug[51] | 2.4 | - | - | 16.6 | - | - | 22.4/-
Adv. AA† [61] | (1.9) | (1.9) | (1.4) | (15.5) | (14.1) | (10.4) | (20.6/5.5)
Ours | 2.5 | 2.0 | 1.5 | 16.8 | 14.5 | 11.8 | 22.2/6.3

Table 5: Test set top-1 error rates (%) on CIFAR-10 and CIFAR-100 and
validation set top-1/top-5 error rates on ImageNet. Please note that Adv. AA†
uses multiple augmented samples per mini-batch.

Our method achieves error rates comparable to other method except for Adv. AA,
which uses multiple augmented samples per mini-batch. In particular, our
method achieves the lowest top-1 error rate on ImageNet among the methods
without a multiple augmentation strategy.

###  5.4 Semantic segmentation

We also evaluate our method on Cityscapes[8]. We implement widely used models
with a ResNet-101 backbone, FCN-32s[34], PSPNet[62], and Deeplabv3[4]. As
baseline methods, we adopted RandAugment[10] and TrivialAugment[40]. These
methods also does not need careful parameter tuning because they have few or
no hyperparameters.

We show the results in Tab. 6.

Model | Baseline | RA[10] | TA[40] | Ours
---|---|---|---|---
FCN-32 | 71.7 | 71.0 | 71.3 | 72.1
PSPNet | 77.7 | 78.8 | 77.5 | 78.8
Deeplabv3 | 78.4 | 79.3 | 78.9 | 79.4

Table 6: Validation set mIoU (%) on Cityscapes. RA and TA denote
RandAugment[10] and TrivialAugment[40], respectively.

Our method achieves the best mIoU for each model. RandAugment hurts the mIoU
for FCN-32s, and TrivialAugment does not improve the mIoU for any of the
models. In fact, it has been reported that TrivialAugment does not work well
for tasks other than image classification[40]. We suspect that the search
spaces of RandAugment and TrivialAugment are not suitable for semantic
segmentation tasks and model capacities. However, our method improves mIoU for
all conditions without adjusting parameters from the classification tasks.

###  5.5 Unsupervised representation learning

Finally, we evaluate our method with unsupervised representation learning
tasks using SimSiam[7]. To generate two different views of the same image, we
used two augmentation models, $a_{\phi_{1}}$ and $a_{\phi_{2}}$. The detailed
settings can be found in the appendix.

The results are shown in Tab. 7.

#Epochs | 100 | 200 | 400
---|---|---|---
Baseline [7] | 68.1 | 70.0 | 70.8
RandAugment [10] | 68.0 | 70.0 | 70.7
TrivialAugment [40] | 62.7 | 68.7 | 71.3
Ours | 68.2 | 70.2 | 71.0

Table 7: ImageNet linear evaluation accuracy (%) for various pretraining
epochs. We set batch size to 256 for pretraining.

TrivialAugment causes underfitting for 100 and 200 epoch training due to the
diversity of augmentation induced by the strong randomness. RandAugment does
not contribute to the improvement of accuracy. Our proposed method
consistently improves the accuracy for all training schedules, although the
improvements are slight. This would be because the online optimization
frameworks have aspects similar to curriculum learning [2]. In other words,
our method adjusts the augmentation magnitude according to the learning
progress of the target model.

##  6 Conclusion

We proposed an online data augmentation optimization method called
TeachAugment that introduces a teacher model into the adversarial data
augmentation and makes it more informative without the need for careful
parameter tuning. We also proposed neural network based augmentation that
simplified the search space design and enabled updating of the data
augmentation with the gradient method. In experiments, our method outperformed
existing data augmentation search frameworks, including state-of-the-art
methods, on image classification, semantic segmentation, and unsupervised
representation learning tasks without adjusting the hyperparameters for each
task.

Limitation. The proposed color augmentation cannot represent transformation
using global information of a target image, such as Equalize and AutoContrast,
because of the lack of global information in the input. Such transformations
may be realized using the color histogram as the context vector, although at
the expense of computational cost, especially for high-resolution images.
Moreover, we only focused on geometric and color augmentation, but there are
many advanced augmentation that are not categorized with them, for example,
cutout[12] and mixup[60]. Considering such augmentations will be future work.

The training time for TeachAugment is approximately three times longer than
the conventional training procedure because updating the augmentation model
requires the forward and backward computation of the target model and the
teacher model, in addition to updating the target network. However, the
training time of TeachAugment is almost the same as other online methods[28,
51] and is a realistic time in comparison to AutoAugment [9]. We believe that
it is a negligible problem under the recent advances of computational power.

##  Appendix A Training setups

###  A.1 Image classification

We summarize the hyperparamters for training in Tab. 8.

| WideResNet-40-2 | WideResNet-28-10 | Shake-Shake (26 2$\times$96d) | PyramidNet | ResNet-50
---|---|---|---|---|---
Learning rate | 0.1 | 0.1 | 0.01 | 0.05 | 0.05
Weight decay | 2e-4 | 5e-4 | 1e-3 | 5e-5 | 1e-4
Epochs | 200 | 200 | 1,800 | 1,800 | 270
Batch size | 128 | 128 | 128 | 64 | 128
Learning rate decay | cosine | cosine | cosine | cosine | step
Table 8: Hyperparameters for classification tasks. We set parameters following
[30]. Note that we show the batch size per GPU and the learning rate was
multiplied by the number of GPUs (e.g., if two GPUs are used for training, the
learning rate is doubled). We used a single GPU for WideResNet-40-2 and
WideResNet-28-10, and four GPUs for the other models. We decayed the learning
rate for ResNet-50 by 10-fold at epochs 90, 180, and 240.

Each model was trained with Nesterov’s accelerated gradient method[41] in the
stochastic gradient descent. The cross entropy loss between the model
prediction and the ground truth label was used as the loss function. We
gradually warmed up the learning rate for five epochs until it reached the
predefined learning rate shown in Tab. 8. As baseline augmentation, we used
random horizontal flipping and random cropping with a crop size of 32 for
CIFAR-10 and CIFAR-100 and 224 for ImageNet. In addition, we also used Cutout
with a crop size of 16 for CIFAR-10 and CIFAR-100, and random color distortion
for ImageNet.222The implementation of the baseline augmentation and models are
based on https://github.com/kakaobrain/fast-autoaugment.

###  A.2 Semantic segmentation

We trained all models using the stochastic gradient descent with momentum of
0.9 for 300 epochs. The cross entropy loss between the model prediction and
the ground truth label was used as the loss function. We set the initial
learning rate to 5e-3 and decayed it with poly learning rate decay where the
initial learning rate was multiplied by
$\left(1-\frac{iter}{max\\_iter}\right)^{0.9}$. The coefficient of the
auxiliary loss used in [62] was set to 0.4. As baseline augmentation, we used
random horizontal flipping and random cropping with a crop size of 1024. For
TeachAugment, the smoothing parameter $\epsilon$ in label smoothing was set to
0.1. For RandAugment, the number of transformations $n$ and the magnitude $m$,
were set to 1 and 5, which were tuned for FCN-32s using grid search.

###  A.3 Unsupervised representation learning

We evaluated each method following the linear evaluation setting[7].333The
experimental code is based on https://github.com/facebookresearch/simsiam. We
modified the objective of our method for SimSiam as follows:

$\displaystyle\max_{\phi_{1},\phi_{2}}\min_{\theta}\mathbb{E}_{x\sim\mathcal{X}}[$
$\displaystyle L(f_{\theta}(a_{\phi_{1}}(x)),f_{\theta}(a_{\phi_{2}}(x)))$
$\displaystyle-L(f_{\theta}(a_{\phi_{1}}(x)),f_{\theta}(a_{\phi_{2}}(x)))],$
(6)

where $L$ denotes the cosine distance. Because the non-saturating loss and
label smoothing cannot be applied to the cosine distance, we omitted them in
this experiment.

Note that our method with the EMA teacher cannot be simply applied to other
methods, such as BYOL[15] and MoCo[19, 6], because they already integrate the
EMA teacher into their training frameworks. It will be future work to
investigate combinations of such methods.

As pretraining, we trained ResNet-50 for 100, 200, and 400 epochs with a batch
size of 256. The momentum SGD was employed as the optimizer. The learning rate
and the momentum were set to 0.05 and 0.9, respectively. After pretraining, we
trained a linear classifier for 90 epochs with a batch size of 4,096. We set
the hyperparameters for each of the methods to the same as the parameters for
ImageNet classification. As baseline augmentation, we used the same
augmentation as in SimSiam[7], namely, random cropping, random horizontal
flipping, color jittering, and Gaussian blur. For more details and SimSiam’s
hyperparameters, please refer to [7].

##  Appendix B Pseudo-Code of TeachAugment

We show the pseudo-code of TeachAugment in Algorithm 1.

Algorithm 1 Training procedure for TeachAugment

0: A target model $f_{\theta}$, a teacher model $f_{\hat{\theta}}$, dataset
$\mathcal{X}$, the number of inner iterations $n_{\text{inner}}$, learning
rate $\eta_{\theta}$ and $\eta_{\phi}$, and decay rate for the EMA teacher
$\xi$

1: while $\theta$ has not converged do

2: for $i=0,\cdots,n_{\text{inner}}$ do

3: if $f_{\hat{\theta}}$ is the EMA teacher then

4: Update teacher weights,
$\hat{\theta}\leftarrow\xi\hat{\theta}+(1-\xi)\theta$

5: end if

6: Randomly sample a mini-batch, $\\{x^{b}\\}^{B}_{b}\sim\mathcal{X}$

7: Compute loss for the target
model,$L_{\theta}=\sum_{b}L(f_{\theta}(a_{\phi}(x^{b})))$

8: Update $\theta$ by the gradient
descent,$\theta\leftarrow\theta-\eta_{\theta}\partial
L_{\theta}/\partial\theta$

9: end for

10: Randomly sample a mini-batch, $\\{\bar{x}^{b}\\}^{B}_{b}\sim\mathcal{X}$

11: Compute loss for the augmentation
model,$L_{\phi}=\sum_{b}(L(f_{\theta}(a_{\phi}(\bar{x}^{b})))-L(f_{\hat{\theta}}(a_{\phi}(\bar{x}^{b}))))$

12: Update $\phi$ by the gradient ascent,
$\phi\leftarrow\phi+\eta_{\phi}\partial L_{\phi}/\partial\phi$

13: end while

##  Appendix C Details of augmentation model

For the geometric augmentation, we used a three-layer perceptron. The
dimension of the noise vector was 128 and the number of units in hidden layers
was 512. As a non-linear activation function, we used leaky ReLU[36] with the
negative slope of 0.2. The output, $A^{\mathrm{unnorm}}$, was normalized
through the sigmoid function:

$\displaystyle
A=\lambda_{g_{\text{scale}}}(\mathrm{sigmoid}(A^{\mathrm{unnorm}})-0.5),$ (7)

where $\lambda_{g_{\text{scale}}}$ controls the search range of $A$, and we
set it to 0.5 (i.e., $A\in(-0.25,0.25)^{2\times 3}$).

For the color augmentation, we used two three-layer perceptrons that receive
an RGB vector and a noise vector as inputs and add up their outputs. The
number of units in hidden layers was 128 and 512, respectively. As the non-
linear activation, leaky ReLU was used. We illustrate the computational scheme
of the color augmentation model in Fig. 6.

Figure 6: Illustration of the color augmentation model. $\sigma$ denotes the
sigmoid function.

The former model outputs 3-dimensional scale and shift parameters:

$\displaystyle(\alpha_{i}^{\text{RGB}},\beta_{i}^{\text{RGB}})\in\mathbb{R}^{3\times
2},$ (8)

and the latter outputs scalar scale and shift parameters:

$\displaystyle(\alpha_{i}^{\text{Noise}},\beta_{i}^{\text{Noise}})\in\mathbb{R}^{1\times
2}.$ (9)

The scale and shift parameters from the noise vector control the global
brightness of images. Then, we add up these scale and shift parameters:

$\displaystyle(\alpha_{i}^{\mathrm{unnorm}})_{j}=(\alpha_{i}^{\text{RGB}})_{j}+\alpha_{i}^{\text{Noise}},$
(10)
$\displaystyle(\beta_{i}^{\mathrm{unnorm}})_{j}=(\beta^{\text{RGB}}_{i})_{j}+\beta^{\text{Noise}}_{i},$
(11)

where $(\alpha_{i})_{j}$ denotes the $j$-th element of
$\alpha_{i}\in\mathbb{R}^{3}$. We normalized the scale and shift parameters,
$(\alpha_{i}^{\mathrm{unnorm}},\beta_{i}^{\mathrm{unnorm}})$, using the
sigmoid function:

$\displaystyle\alpha_{i}=\lambda_{c_{\text{scale}}}(\mathrm{sigmoid}(\alpha_{i}^{\mathrm{unnorm}})-0.5)+1,$
(12)
$\displaystyle\beta_{i}=\lambda_{c_{\text{scale}}}(\mathrm{sigmoid}(\beta_{i}^{\mathrm{unnorm}})-0.5),$
(13)

where $\lambda_{c_{\text{scale}}}$ controls the search range of $\alpha_{i}$
and $\beta_{i}$, and we set it to 0.8, namely, $\alpha_{i}\in(0.6,1.4)$ and
$\beta_{i}\in(-0.4,0.4)$.

We adopted AdamW [35] as the optimizer for the augmentation model. The
learning rate and the weight decay were set to 1e-3 and 1e-2, which are the
default parameters in PyTorch[42]. Dropout[48] was applied after the linear
layers except for the output layer with the drop ratio of 0.8.

##  Appendix D Learning pipeline of probabilities $p_{c}$ and $p_{g}$

We made the decision process of applying the augmentation differentiable using
weights sampled from the relaxed Bernoulli distribution defined in [37].444The
relaxed Bernoulli distribution is referred to as the BinConcrete distribution
in [37]. A sample from the relaxed Bernoulli, $w\sim\mathrm{ReBern}(p,\tau)$,
is obtained as follows:

$\displaystyle w=\mathrm{sigmoid}((L+\log p)/\tau),\
L\sim\mathrm{Logistic}(0,1),$ (14)

where $\tau$ and $p$ denote the temperature parameter and a probability that
corresponds to $p_{c}$ and $p_{g}$ in our case and $L$ is a sample from the
Logistic distribution, which is obtained by $L=\log(U)-\log(1-U),\
U\sim\mathrm{Uniform}(0,1)$. We set $\tau$ to 0.05 following [17]. We note
that the sampling procedure, Eq. 14, is differentiable with respect to the
probability $p$.

We compute weighted sum of the parameters generated by augmentation models and
parameters that make the augmentation the identity mapping:

$\displaystyle\hat{\alpha}_{i}=w_{c}\alpha_{i}+(1-w_{c})\cdot 1,$ (15)
$\displaystyle\hat{\beta}_{i}=w_{c}\beta_{i}+(1-w_{c})\cdot 0,$ (16)
$\displaystyle\hat{A}=w_{g}A+(1-w_{g})I,$ (17)

where $w_{c}\sim\mathrm{ReBern}(p_{c},\tau)$ and
$w_{g}\sim\mathrm{ReBern}(p_{g},\tau)$. We use
$(\hat{\alpha}_{i},\hat{\beta}_{i})$ and $\hat{A}$ to transform images,
instead of $(\alpha_{i},\beta_{i})$ and $A$:

$\displaystyle\tilde{x}_{i}=t(\hat{\alpha}_{i}\odot x_{i}+\hat{\beta}_{i}),$
(18) $\displaystyle\hat{x}=\mathrm{Affine}(\tilde{x},\hat{A}+I).$ (19)

Because the sampling procedure from $\mathrm{ReBern}(p,\tau)$ is
differentiable with respect to $p$, we can also update the probabilities using
the gradient method.

##  Appendix E Additional results

###  E.1 Relation between consistency regularization and TeachAugment

TeachAugment can be viewed as a method that minimizes the distance between
predictions of the target model and the teacher model. We show it
qualitatively.

We assume that data augmentation $a_{\phi}$ can transform an input data point
to any point in the input space, and the teacher model is fixed during
training. Then, solving
$\max_{\phi}L(f_{\theta}(a_{\phi}(x)))-L(f_{\hat{\theta}}(a_{\phi}(x)))$
corresponds to searching the data augmentation maximizing the distance between
the predictions under the constraint of
$L(f_{\theta}(a_{\phi}(x)))>L(f_{\hat{\theta}}(a_{\phi}(x)))$. Then, the
target model updates its decision boundary for minimizing the loss for the
augmented data. This procedure is illustrated in Fig. 7. If the decision
boundary of the target model completely corresponds to that of the teacher
model, the data augmentation reaches the stationary point because the
objective $L(f_{\theta}(a_{\phi}(x)))-L(f_{\hat{\theta}}(a_{\phi}(x)))$ is 0
for all data points.

Figure 7: Illustration of the training process in TeachAugment. The data
augmentation transforms a data point to the point so that
$L(f_{\theta}(a_{\phi}(x)))>L(f_{\hat{\theta}}(a_{\phi}(x)))$. Then, the
target model is updated to minimize loss for the augmented point. By repeating
this process, the decision boundary of the target model is close to that of
the teacher model.

However, the target model would not match the teacher model in practice,
because above analysis lacks certain points: (1) The data augmentation cannot
transform an input data point to any point in the input. (2) We used the non-
saturating loss for the image classification tasks instead of cross entropy.
Thus, $L(f_{\theta}(a_{\phi}(x)))-L(f_{\hat{\theta}}(a_{\phi}(x)))\neq 0$ if
the decision boundary of the target model is the same as that of the teacher
model. (3) TeachAugment does not explicitly minimize the distance between the
predictions. In particular, (3) is a more critical point compared to (1) and
(2) because they may be solved by adopting a model large enough to satisfy the
requirement as the augmentation model, and using an original loss function
instead of the non-saturating loss. Because of the lack of explicit costs for
the consistency, TeachAugment does not ensure that the target model converges
with the teacher model.

However, we believe that the above analysis gives some insights, and comparing
TeachAugment to consistency regularization is important. Thus, we trained
model to minimize the following costs and compared the results to
TeachAugment:

$\displaystyle\min_{\theta}L(f_{\theta}(x))+D(f_{\theta}(x),f_{\hat{\theta}}(x)),$
(20)

where $D(\cdot,\cdot)$ denotes the distance function. We used the mean squared
error and Kullback–Leibler divergence as $D(\cdot,\cdot)$. These functions are
widely used in the consistency regularization and the knowledge distillation.
We refer to the former as MSE Consistency and the later as KLD Consistency.
Note that we use random horizontal flipping, random cropping, and cutout[12]
as the data augmentation but omitted them in Eq. (20) because they were not
optimized. As the teacher model, we used the same EMA teacher in TeachAugment.

The results are shown in Tab. 9.

Dataset | CIFAR-10 | CIFAR-100
---|---|---
Baseline | 3.1 | 18.4
KLD consistency | 3.1 | 18.2
MSE consistency | 3.2 | 18.5
TeachAugment | 2.5 | 16.8
Table 9: Comparison with the consistency regularization. We report the error
rates of WideResNet-28-10. For training with KLD and MSE consistency, we used
random horizontal flipping, random cropping, and cutout[12] as data
augmentation.

The consistency methods do not improve the error rates from the baseline.
Thus, TeachAugment has different properties than the consistency
regularization, and it works well in supervised learning.

###  E.2 Qualitative analysis

We show augmented images obtained by the augmentation model trained with
various objective functions: TeachAugment, Adversarial AutoAugment (Adv.
AA)[61], and PointAugment[28]. All methods used the same proposed augmentation
model as data augmentation.

The objective of Adv. AA is as follows:

$\displaystyle\max_{\phi}\min_{\theta}L(f_{\theta}(a_{\phi}(x))).$ (21)

Also, the objective of PointAugment is as follows:

$\displaystyle\min_{\theta}L(f_{\theta}(a_{\phi}(x))),$ (22)
$\displaystyle\min_{\phi}L(f_{\theta}(a_{\phi}(x)))$ $\displaystyle+|1-\exp($
$\displaystyle L(f_{\theta}(a_{\phi}(x)))-\rho L(f_{\theta}(x)))|,$ (23)

where $\rho$ is a dynamic parameter defined as $\rho=\exp(y^{T}\cdot
f_{\theta}(a_{\phi}(x)))$. Note that Adv. AA updates augmentation functions
using the REINFORCE algorithm[56] because many functions in the search space
of AutoAugment are non-differentiable. However, in our experiments, because
our proposed augmentation is differentiable, we updated augmentation with the
gradient descent rather than the REINFORCE algorithm.

The augmented images of CIFAR-10 and CIFAR-100 are shown in Figs. 8 and 9. The
images augmented by Adv. AA obviously collapse, so the meaningful information
is lost. The augmented images obtained by PointAugment and the proposed method
are recognizable, but the proposed method transforms images more strongly than
PointAugment so that the proposed method distorts the aspect ratio for the
geometric augmentation. PointAugment binds the difficulty of the augmented
images through a dynamic parameter $\rho\leq\exp(1)$, but the proposed method
requires only that the augmented images are recognizable for the teacher
model. As a result, the proposed method allows stronger augmentation than
PointAugment.

|  |  |
---|---|---|---
(a) Original Image | (b) Adv. AA | (c) PointAugment | (d) Ours
Figure 8: Augmented images obtained with CIFAR-10 using various methods.. |  |  |
---|---|---|---
(a) Original Image | (b) Adv. AA | (c) PointAugment | (d) Ours
Figure 9: Augmented images obtained with CIFAR-100 using various methods..

###  E.3 Example of augmented images

We show example results of augmentation for ImageNet and Cityscapes in Figs.
10 and 11. As can see from Fig. 11, the augmentation obtained for FCN-32s is
obviously different from the others. Because the output stride of FCN-32s
(i.e., the ratio of input image spatial resolution to final output resolution)
is lower than that of the others, FCN-32s will have different properties than
PSPNet and Deeplav3. We believe that the difference leads the different
augmentation of these models. Moreover, it also leads the degradation of mIoU
for FCN-32s in RandAugment and TrivialAugment.

|
---|---
(a) Original images | (b) Augmented images
Figure 10: Augmentations obtained with ImageNet. |  |  |
---|---|---|---
(a) Original images | (b) FCN-32s | (c) PSPNet | (d) Deeplabv3
Figure 11: Augmentations obtained with Cityscapes.

## References

  * [1] Yoshua Bengio, Nicholas Léonard, and Aaron Courville.  Estimating or propagating gradients through stochastic neurons for conditional computation.  arXiv preprint arXiv:1308.3432, 2013.
  * [2] Yoshua Bengio, Jérôme Louradour, Ronan Collobert, and Jason Weston.  Curriculum learning.  In Proceedings of the 26th annual international conference on machine learning, pages 41–48, 2009.
  * [3] Nicolas Bonneel, Julien Rabin, Gabriel Peyré, and Hanspeter Pfister.  Sliced and radon wasserstein barycenters of measures.  Journal of Mathematical Imaging and Vision, 51(1):22–45, 2015.
  * [4] Liang-Chieh Chen, George Papandreou, Florian Schroff, and Hartwig Adam.  Rethinking atrous convolution for semantic image segmentation.  arXiv preprint arXiv:1706.05587, 2017.
  * [5] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton.  A simple framework for contrastive learning of visual representations.  In International conference on machine learning, pages 1597–1607. PMLR, 2020.
  * [6] Xinlei Chen, Haoqi Fan, Ross Girshick, and Kaiming He.  Improved baselines with momentum contrastive learning.  arXiv preprint arXiv:2003.04297, 2020.
  * [7] Xinlei Chen and Kaiming He.  Exploring simple siamese representation learning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 15750–15758, 2021.
  * [8] Marius Cordts, Mohamed Omran, Sebastian Ramos, Timo Rehfeld, Markus Enzweiler, Rodrigo Benenson, Uwe Franke, Stefan Roth, and Bernt Schiele.  The cityscapes dataset for semantic urban scene understanding.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3213–3223, 2016.
  * [9] Ekin D Cubuk, Barret Zoph, Dandelion Mane, Vijay Vasudevan, and Quoc V Le.  Autoaugment: Learning augmentation strategies from data.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 113–123, 2019.
  * [10] Ekin D Cubuk, Barret Zoph, Jonathon Shlens, and Quoc V Le.  Randaugment: Practical automated data augmentation with a reduced search space.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops, pages 702–703, 2020.
  * [11] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
  * [12] Terrance DeVries and Graham W Taylor.  Improved regularization of convolutional neural networks with cutout.  arXiv preprint arXiv:1708.04552, 2017.
  * [13] Xavier Gastaldi.  Shake-shake regularization.  arXiv preprint arXiv:1705.07485, 2017.
  * [14] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.  Generative adversarial nets.  Advances in neural information processing systems, 27, 2014.
  * [15] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Avila Pires, Zhaohan Daniel Guo, Mohammad Gheshlaghi Azar, et al.  Bootstrap your own latent: A new approach to self-supervised learning.  arXiv preprint arXiv:2006.07733, 2020.
  * [16] Dongyoon Han, Jiwhan Kim, and Junmo Kim.  Deep pyramidal residual networks.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 5927–5935, 2017.
  * [17] Ryuichiro Hataya, Jan Zdenek, Kazuki Yoshizoe, and Hideki Nakayama.  Faster autoaugment: Learning augmentation strategies using backpropagation.  In European Conference on Computer Vision, pages 1–16. Springer, 2020.
  * [18] Ryuichiro Hataya, Jan Zdenek, Kazuki Yoshizoe, and Hideki Nakayama.  Meta approach to data augmentation optimization.  arXiv preprint arXiv:2006.07965, 2020.
  * [19] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick.  Momentum contrast for unsupervised visual representation learning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9729–9738, 2020.
  * [20] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.  Deep residual learning for image recognition.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
  * [21] Dan Hendrycks, Norman Mu, Ekin D Cubuk, Barret Zoph, Justin Gilmer, and Balaji Lakshminarayanan.  Augmix: A simple data processing method to improve robustness and uncertainty.  arXiv preprint arXiv:1912.02781, 2019.
  * [22] Daniel Ho, Eric Liang, Xi Chen, Ion Stoica, and Pieter Abbeel.  Population based augmentation: Efficient learning of augmentation policy schedules.  In International Conference on Machine Learning, pages 2731–2741. PMLR, 2019.
  * [23] Hiroshi Inoue.  Data augmentation by pairing samples for images classification.  arXiv preprint arXiv:1801.02929, 2018.
  * [24] Max Jaderberg, Karen Simonyan, Andrew Zisserman, et al.  Spatial transformer networks.  Advances in neural information processing systems, 28:2017–2025, 2015.
  * [25] Eric Jang, Shixiang Gu, and Ben Poole.  Categorical reparameterization with gumbel-softmax.  arXiv preprint arXiv:1611.01144, 2016.
  * [26] Vijay R Konda and John N Tsitsiklis.  Actor-critic algorithms.  In Advances in neural information processing systems, pages 1008–1014, 2000.
  * [27] Alex Krizhevsky, Geoffrey Hinton, et al.  Learning multiple layers of features from tiny images.  2009\.
  * [28] Ruihui Li, Xianzhi Li, Pheng-Ann Heng, and Chi-Wing Fu.  Pointaugment: an auto-augmentation framework for point cloud classification.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 6378–6387, 2020.
  * [29] Yonggang Li, Guosheng Hu, Yongtao Wang, Timothy Hospedales, Neil M Robertson, and Yongxin Yang.  Dada: Differentiable automatic data augmentation.  arXiv preprint arXiv:2003.03780, 2020.
  * [30] Sungbin Lim, Ildoo Kim, Taesup Kim, Chiheon Kim, and Sungwoong Kim.  Fast autoaugment.  Advances in Neural Information Processing Systems, 32:6665–6675, 2019.
  * [31] Chen Lin, Minghao Guo, Chuming Li, Xin Yuan, Wei Wu, Junjie Yan, Dahua Lin, and Wanli Ouyang.  Online hyper-parameter learning for auto-augmentation strategy.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6579–6588, 2019.
  * [32] Long-Ji Lin.  Reinforcement learning for robots using neural networks.  Carnegie Mellon University, 1992.
  * [33] Tom Ching LingChen, Ava Khonsari, Amirreza Lashkari, Mina Rafi Nazari, Jaspreet Singh Sambee, and Mario A Nascimento.  Uniformaugment: A search-free probabilistic data augmentation approach.  arXiv preprint arXiv:2003.14348, 2020.
  * [34] Jonathan Long, Evan Shelhamer, and Trevor Darrell.  Fully convolutional networks for semantic segmentation.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3431–3440, 2015.
  * [35] Ilya Loshchilov and Frank Hutter.  Decoupled weight decay regularization.  arXiv preprint arXiv:1711.05101, 2017.
  * [36] Andrew L Maas, Awni Y Hannun, Andrew Y Ng, et al.  Rectifier nonlinearities improve neural network acoustic models.  In Proc. icml, volume 30, page 3. Citeseer, 2013.
  * [37] Chris J Maddison, Andriy Mnih, and Yee Whye Teh.  The concrete distribution: A continuous relaxation of discrete random variables.  arXiv preprint arXiv:1611.00712, 2016.
  * [38] Takeru Miyato, Shin-ichi Maeda, Masanori Koyama, and Shin Ishii.  Virtual adversarial training: a regularization method for supervised and semi-supervised learning.  IEEE transactions on pattern analysis and machine intelligence, 41(8):1979–1993, 2018.
  * [39] Volodymyr Mnih, Koray Kavukcuoglu, David Silver, Alex Graves, Ioannis Antonoglou, Daan Wierstra, and Martin Riedmiller.  Playing atari with deep reinforcement learning.  arXiv preprint arXiv:1312.5602, 2013.
  * [40] Samuel G Müller and Frank Hutter.  Trivialaugment: Tuning-free yet state-of-the-art data augmentation.  arXiv preprint arXiv:2103.10158, 2021.
  * [41] Yurii E Nesterov.  A method for solving the convex programming problem with convergence rate o (1/k^ 2).  In Dokl. akad. nauk Sssr, volume 269, pages 543–547, 1983.
  * [42] Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala.  Pytorch: An imperative style, high-performance deep learning library.  In Advances in Neural Information Processing Systems 32, pages 8024–8035. Curran Associates, Inc., 2019.
  * [43] Xi Peng, Zhiqiang Tang, Fei Yang, Rogerio S Feris, and Dimitris Metaxas.  Jointly optimize data augmentation and network training: Adversarial data augmentation in human pose estimation.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 2226–2234, 2018.
  * [44] David Pfau and Oriol Vinyals.  Connecting generative adversarial networks and actor-critic methods.  arXiv preprint arXiv:1610.01945, 2016.
  * [45] Colorado J Reed, Sean Metzger, Aravind Srinivas, Trevor Darrell, and Kurt Keutzer.  Selfaugment: Automatic augmentation policies for self-supervised learning.  In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2674–2683, 2021.
  * [46] Tom Schaul, John Quan, Ioannis Antonoglou, and David Silver.  Prioritized experience replay.  arXiv preprint arXiv:1511.05952, 2015.
  * [47] Kihyuk Sohn, David Berthelot, Chun-Liang Li, Zizhao Zhang, Nicholas Carlini, Ekin D Cubuk, Alex Kurakin, Han Zhang, and Colin Raffel.  Fixmatch: Simplifying semi-supervised learning with consistency and confidence.  arXiv preprint arXiv:2001.07685, 2020.
  * [48] Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov.  Dropout: a simple way to prevent neural networks from overfitting.  The journal of machine learning research, 15(1):1929–1958, 2014\.
  * [49] Richard S Sutton, David A McAllester, Satinder P Singh, and Yishay Mansour.  Policy gradient methods for reinforcement learning with function approximation.  In Advances in neural information processing systems, pages 1057–1063, 2000.
  * [50] Teppei Suzuki and Ikuro Sato.  Adversarial transformations for semi-supervised learning.  arXiv preprint arXiv:1911.06181, 2019.
  * [51] Zhiqiang Tang, Yunhe Gao, Leonid Karlinsky, Prasanna Sattigeri, Rogerio Feris, and Dimitris Metaxas.  Onlineaugment: Online data augmentation with less domain knowledge.  In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part VII 16, pages 313–329. Springer, 2020.
  * [52] Antti Tarvainen and Harri Valpola.  Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results.  arXiv preprint arXiv:1703.01780, 2017.
  * [53] Keyu Tian, Chen Lin, Ming Sun, Luping Zhou, Junjie Yan, and Wanli Ouyang.  Improving auto-augment via augmentation-wise weight sharing.  arXiv preprint arXiv:2009.14737, 2020.
  * [54] Yuji Tokozume, Yoshitaka Ushiku, and Tatsuya Harada.  Between-class learning for image classification.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5486–5494, 2018.
  * [55] Longhui Wei, An Xiao, Lingxi Xie, Xiaopeng Zhang, Xin Chen, and Qi Tian.  Circumventing outliers of autoaugment with knowledge distillation.  In European Conference on Computer Vision, pages 608–625. Springer, 2020.
  * [56] Ronald J Williams.  Simple statistical gradient-following algorithms for connectionist reinforcement learning.  Machine learning, 8(3):229–256, 1992.
  * [57] Yoshihiro Yamada, Masakazu Iwamura, Takuya Akiba, and Koichi Kise.  Shakedrop regularization for deep residual learning.  IEEE Access, 7:186126–186136, 2019.
  * [58] Sangdoo Yun, Dongyoon Han, Seong Joon Oh, Sanghyuk Chun, Junsuk Choe, and Youngjoon Yoo.  Cutmix: Regularization strategy to train strong classifiers with localizable features.  In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6023–6032, 2019.
  * [59] Sergey Zagoruyko and Nikos Komodakis.  Wide residual networks.  arXiv preprint arXiv:1605.07146, 2016.
  * [60] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz.  mixup: Beyond empirical risk minimization.  arXiv preprint arXiv:1710.09412, 2017.
  * [61] Xinyu Zhang, Qiang Wang, Jian Zhang, and Zhao Zhong.  Adversarial autoaugment.  arXiv preprint arXiv:1912.11188, 2019.
  * [62] Hengshuang Zhao, Jianping Shi, Xiaojuan Qi, Xiaogang Wang, and Jiaya Jia.  Pyramid scene parsing network.  In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 2881–2890, 2017.
  * [63] Zhun Zhong, Liang Zheng, Guoliang Kang, Shaozi Li, and Yi Yang.  Random erasing data augmentation.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 13001–13008, 2020.


# Studying the $\bar{D}_{1}K$ molecule in the Bethe-Salpeter equation approach

Jing-Juan Qi 111e-mail<EMAIL_ADDRESS>Junior College, Zhejiang Wanli
University, Zhejiang 315101, China Zhen-Yang Wang 222Corresponding author,
e-mail<EMAIL_ADDRESS>Physics Department, Ningbo University,
Zhejiang 315211, China Zhu-Feng Zhang 333e-mail<EMAIL_ADDRESS>Physics Department, Ningbo University, Zhejiang 315211, China Xin-Heng Guo
444Corresponding author, e-mail<EMAIL_ADDRESS>College of Nuclear Science
and Technology, Beijing Normal University, Beijing 100875, China

###### Abstract

We interpret the $X_{1}(2900)$ as an $S$-wave $\bar{D}_{1}K$ molecular state
in the Bethe-Salpeter equation approach with the ladder and instantaneous
approximations for the kernel. By solving the Bethe-Salpeter equation
numerically with the kernel containing one-particle-exchange diagrams and
introducing three different form factors (monopole, dipole, and exponential
form factors) in the verties, we find the bound state exists. We also study
the decay width of the decay $X_{1}(2900)$ to $D^{-}K^{+}$.

###### pacs:

**********

##  I Introduction

Recently, two new open flavor states were observed by LHCb collaboration in
the $D^{-}K^{+}$ invariant mass distribution of $B^{+}\rightarrow
D^{+}D^{-}K^{+}$, and the parameters are determined to be Aaij:2020ypa

$\begin{split}X_{0}(2900):M&=2.866\pm 0.007\pm 0.002\ \mathrm{GeV},\\\
\Gamma&=57\pm 12\pm 4\ \mathrm{MeV},\\\ X_{1}(2900):M&=2.904\pm 0.005\pm
0.001\ \mathrm{GeV},\\\ \Gamma&=110\pm 11\pm 4\ \mathrm{MeV},\\\ \end{split}$

respectively. Since the resonances $X_{0}(2900)$ and $X_{1}(2900)$ are
observed in the $D^{-}K^{+}$ channel, they should be manifestly exotic and
have minimal quark content $\bar{c}du\bar{s}$. These two states are new fully
open flavor states after the discovery of $X(5568)$ ($su\bar{b}\bar{d}$),
which was reported by D0 collaboration in the $B_{s}\pi$ invariant mass
distribution in 2016. However, since then LHCb, CMS, CDF, and ATLAS
Collaborations have not found evidence for $X(5568)$ Aaij:2016iev ;
Sirunyan:2017ofq ; Aaltonen:2017voc ; Aaboud:2018hgx .

In the past decades, a growing number of good candidates of exotic states have
been observed, with lots of them containing $c\bar{c}$ or $b\bar{b}$ quarks
Guo:2017jvc ; Olsen:2017bmm . Thus, the discovery of $X_{0}(2900)$ and
$X_{1}(2900)$ have drawn a lot of attentions. The $X_{0}(2900)$ can be
interpreted as a $cs\bar{u}\bar{d}$ compact tetraquark both in the universal
quark mass picture Karliner:2020vsi  and in the quark model Wang:2020prk , but
not within an extended relativized quark model Lu:2020qmp . In Ref. He:2020jna
, the authors used the two-body chromomagnetic interactions to find that the
$X_{0}(2900)$ can be interpreted as a radial excited tetraquark and the
$X_{1}(2900)$ can be an orbitally excited tetraquark. It was also suggested
the $X_{0}(2900)$ can be interpreted as the $S$-wave $D^{\ast-}K^{\ast+}$
molecule state and the $X_{1}(2900)$ as the $P$-wave $\bar{c}\bar{s}ud$
compact tetraquark state Chen:2020aos . In the chiral constituent quark model,
it was shown show that no candidate of $X(2900)$ was founded in the
$IJ^{P}=00^{+}$ and $IJ^{P}=01^{+}$ $cs\bar{q}\bar{q}$ system, while there
were two states in the $P$-wave excited $cs\bar{q}\bar{q}$ system,
$D_{1}\bar{K}$ and $D_{J}\bar{K}$ , which could be candidates of $X(2900)$
Tan:2020cpu . From the QCD Sum Rules, the $X_{0}(2900)$ and $X_{1}(2900)$ were
studied in molecular and diquark-antidiquark tetraquark pictures,
respectively, and the results for masses are in good agreement with the
observed masses in the experiment Mutuk:2020igv . Investigations bases the
one-boson exchange model Liu:2020nil  and the phenomenological Lagrangian
approach Huang:2020ptc , showed that the $X_{0}(2900)$ can be a
$D^{\ast}\bar{K}^{\ast}$ molecule, but the $X_{1}(2900)$ can not. In Ref.
Xiao:2020ltm , the decay width for $X_{0}(2900)\rightarrow\bar{D}K$ process
was found to be in agreement with the experimental data with the $S$-wave
$\bar{D}^{\ast}K^{\ast}$ scenario for $X_{0}(2900)$ in the effective
lagrangian approach. The study in Ref. Dong:2020rgs  showed that the
$X_{1}(2900)$ as a $\bar{D}_{1}K$ is disfavored within the meson exchange
model. In Ref. He:2020btl , in the quasipotential Bethe-Salpeter (BS) equation
approach, the authors supported the assignment of $X_{0}(2900)$ as a
$D^{\ast}\bar{K}^{\ast}$ molecular state and $X_{1}(2900)$ as a $\bar{D_{1}}K$
virtual state.

Considering the mass of $X_{1}(2900)$ is about 10 MeV below the $\bar{D}_{1}K$
threshold, it is natural to explore the existence of the $S$-wave
$\bar{D}_{1}K$ molecule. In this work, we will focus on the $X_{1}(2900)$ in
the BS equation approach, investigating whether the $X_{1}(2900)$ can be an
$S$-wave $\bar{D}_{1}K$ bound state. We will also to study the decay width of
$X_{1}(2900)\rightarrow D^{-}K^{+}$.

In the rest of the manuscript we will proceed as follows. In Sec. II, we will
establish the BS equation for the bound state of an axial-vector meson
($\bar{D}_{1}$) and a pseudoscalar meson ($K$). Then we will discuss the
interaction kernel of the BS equation and calculate numerical results of the
Lorentz scalar functions in the normalized BS wave function in Sec. III. In
Sec. IV, the decay width of the $X_{1}(2900)$ to $D^{-}K^{+}$ final state will
be calculated. In Sec. V, we will present a summary of our results.

##  II The BS formalism for $\bar{D}_{1}K$ system

For the molecule composed of an axial-vector meson ($\bar{D}_{1}$) and a
pseudoscalar meson ($K$), its BS wave function is defined as

$\chi^{\mu}\left(x_{1},x_{2},P\right)=\langle
0|T\bar{D}_{1}^{\mu}(x_{1})K(x_{2})|P\rangle,$ (1)

where $\bar{D}_{1}(x_{1})$ and $K(x_{2})$ are the field operators of the
axial-vector meson $\bar{D}_{1}$ and the pseudoscalar meson $K$ at space
coordinates $x_{1}$ and $x_{2}$, respectively, $P=Mv$ is the total momentum of
bound state and $v$ is its velocity. Let $m_{\bar{D}_{1}}$ and $m_{K}$ be the
masses of the $\bar{D}_{1}$ and $K$ mesons, respectively, $p$ be the relative
momentum of the two constituents, and define
$\lambda_{1}$=$m_{\bar{D}_{1}}/(m_{\bar{D}_{1}}+m_{K})$,
$\lambda_{2}$=$m_{K}/(m_{\bar{D}_{1}}+m_{K})$. The BS wave function in
momentum space is defined as

$\chi^{\mu}_{P}(x_{1},x_{2},P)=e^{-iPX}\int\frac{d^{4}p}{(2\pi)^{4}}e^{-ipx}\chi^{\mu}_{P}(p),$
(2)

where $X=\lambda_{1}x_{1}+\lambda_{2}x_{2}$ is the coordinate of the center of
mass and $x=x_{1}-x_{2}$. The momentum of the $\bar{D}_{1}$ meson is
$p_{1}=\lambda_{1}P+p$ and that of the $K$ meson is $p_{2}=\lambda_{2}P-p$.

It can be shown that the BS wave function of the $\bar{D}_{1}K$ system
satisfies the following BS equation:

$\chi^{\mu}_{P}(p)=S^{\mu\nu}_{\bar{D}_{1}}(p_{1})\int\frac{d^{4}q}{(2\pi)^{4}}K_{\nu\lambda}(P,p,q)\chi^{\lambda}_{P}(q)S_{K}(p_{2}),$
(3)

where $S^{\mu\nu}_{\bar{D}_{1}}(p_{1})$ and $S_{K}(p_{2})$ are the propagators
of $\bar{D}_{1}$ and $K$ mesons, respectively, and $K_{\nu\lambda}(P,p,q)$ is
the kernel, which is defined as the sum of all the two particle irreducible
diagrams with respect to $D_{1}$ and $K$ mesons. For convenience, in the
following we use the variables $p_{l}(=p\cdot v)$ and $p_{t}(=p-p_{l}v)$ as
the longitudinal and transverse projections of the relative momentum ($p$)
along the bound state momentum ($P$), respectively. Then, in the heavy quark
limit the propagator of $D_{1}$ is

$S^{\mu\nu}_{D_{1}}(\lambda_{1}P+p)=\frac{-i\left(g^{\mu\nu}-v^{\mu}v^{\nu}\right)}{2\omega_{1}\left(\lambda_{1}M+p_{l}-\omega_{1}+i\epsilon\right)},$
(4)

and the propagator of the $K$ meson is

$S_{K}(\lambda_{2}P-p)=\frac{i}{\left(\lambda_{2}M-p_{l}\right)^{2}-\omega_{2}^{2}+i\epsilon},$
(5)

respectively, where $\omega_{1(2)}=\sqrt{m_{\bar{D}_{1}(K)}^{2}+p_{t}^{2}}$
(we have defined $p_{t}^{2}=-p_{t}\cdot p_{t}$).

In the BS equation approach, the interaction between $\bar{D}_{1}$ and $K$
mesons arises from the light vector-meson ($\rho$ and $\omega$) exchange.
Based on the heavy quark symmetry and the chiral symmetry, the relevant
effective Lagrangian used in this work is shown in the following Ding:2008gr :

$\begin{split}\mathcal{L}_{D_{1}D_{1}V}=&ig_{D_{1}D_{1}V}(D^{\nu}_{1b}\overleftrightarrow{\partial}_{\mu}D_{1a\nu}^{\dagger})V_{ba}^{\mu}+ig^{\prime}_{D_{1}D_{1}V}(D_{1b}^{\mu}D_{1a}^{\nu{\dagger}}-D_{1a}^{\mu{\dagger}}D_{1a}^{\nu})(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu})_{ba}\\\
&+ig_{\bar{D}_{1}\bar{D}_{1}V}(\bar{D}_{1b\nu}\overleftrightarrow{\partial}_{\mu}\bar{D}_{1a}^{\nu{\dagger}})V_{ab}^{\mu}+ig^{\prime}_{\bar{D}_{1}\bar{D}_{1}V}(\bar{D}_{1b}^{\mu}\bar{D}_{1a}^{\nu{\dagger}}-\bar{D}_{1a}^{\mu{\dagger}}\bar{D}_{1b}^{\nu})(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu})_{ab}\\\
\mathcal{L}_{KKV}=&ig_{KKV}(K_{b}\overleftrightarrow{\partial}_{\mu}K_{a}^{\dagger})V_{ba}^{\mu}+ig_{\bar{K}\bar{K}V}(\bar{K}_{b}\overleftrightarrow{\partial}_{\mu}\bar{K}_{a}^{\dagger})V_{ba}^{\mu},\\\
\end{split}$ (6)

where $a$ and $b$ represent the light flavor quark ($u$ and $d$), $V_{\mu}$ is
a $3\times 3$ Hermitian matrix containing $\rho$, $\omega$, $K^{\ast}$, and
$\phi$:

$\displaystyle V$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&\rho^{+}&K^{*+}\\\
\rho^{-}&-\frac{\rho^{0}}{\sqrt{2}}+\frac{\omega}{\sqrt{2}}&K^{*0}\\\
K^{*-}&\bar{K}^{*0}&\phi\end{array}\right).$ (10)

The coupling constants involved in Eq. (6) are related to each other as
follows Ding:2008gr :

$\begin{split}&g_{D_{1}D_{1}V}=-g_{\bar{D}_{1}\bar{D}_{1}V}=\frac{1}{\sqrt{2}}\beta_{2}g_{V},\\\
&g^{\prime}_{D_{1}D_{1}V}=-g^{\prime}_{\bar{D}_{1}\bar{D}_{1}V}=\frac{5\lambda_{2}g_{V}}{3\sqrt{2}}m_{D_{1}},\\\
&g_{KKV}=g_{V}/2,\\\ \end{split}$ (11)

where the parameters $\beta_{2}g_{V}$ and $\lambda_{2}g_{V}$ are given by
$2g_{\rho NN}$ and $\frac{3}{10m_{N}}(g_{\rho NN}+f_{\rho NN})$, respectively,
with $g_{\rho NN}^{2}/4\pi=0.84$ and $f_{\rho NN}/g_{\rho NN}=6.10$
Wang:2019aoc . The parameter $g_{V}=5.8$ is determined by the Kawarabayashi-
Suzuki-Riazuddin-Fayyazuddin relations Ding:2008gr .

Figure 1: One-particle exchange diagrams induced by vector mesons $\rho$ and
$\omega$.

Then, at the tree level, in the $t$-channel the kernel for the BS equation of
the $\bar{D}_{1}K$ system in the lader approximation includes the following
term (see Fig. 1):

$\begin{split}K^{\tau\sigma}_{direct}(P,p,q;m_{V})=&-(2\pi)^{4}\delta^{4}(p^{\prime}_{1}+p^{\prime}_{2}-p_{1}-p_{2})c_{I}\Big{\\{}g_{D_{1}D_{1}V}g_{DDV}(p_{1}+q_{1})_{\gamma}(p_{2}+q_{2})_{\rho}g^{\tau\sigma}\\\
&\times\Delta^{\rho\gamma}(k,m_{V})+g^{\prime}_{D_{1}D_{1}V}g_{DDV}(p_{2}+q_{2})_{\rho}\left[k^{\tau}\Delta^{\rho\sigma}(k,m_{V})-k^{\sigma}\Delta^{\rho\tau}(k,m_{V})\right]\Big{\\}},\\\
\end{split}$ (12)

where $m_{V}$ ($V=\rho,\omega$) represents the mass of the exchanged light
vector meson $\rho$ or $\omega$ , $c_{I}$ is the isospin coefficient:
$c_{0}=3,1$ and $c_{1}=-1,1$ for $\rho$ and $\omega$, respectively,
$\Delta^{\mu\nu}$ represents the propagator for the light vector meson.

In order to describe the phenomena in the real world, we should include a form
factor at each interacting vertex of hadrons to include the finite-size
effects of these hadrons. For the meson-exchange case, the form factor is
assumed to take the following forms:

$\begin{split}F_{M}(k)&=\frac{\Lambda_{M}^{2}-m^{2}}{\Lambda_{M}^{2}-k^{2}},\\\
F_{D}(k)&=\frac{(\Lambda_{D}^{2}-m^{2})^{2}}{(\Lambda_{D}^{2}-k^{2})^{2}},\\\
F_{E}(k)&=e^{(k^{2}-m^{2})/\Lambda_{E}^{2}},\\\ \end{split}$ (13)

in the monopole ($M$), dipole ($D$), and exponential ($E$) models,
respectively, where $\Lambda$, $m$ and $k$ represent the cutoff parameter, the
mass of the exchanged meson and the momentum of the exchanged meson,
respectively. The value of $\Lambda$ is near 1 GeV which is the typical chiral
symmetry breaking scale.

In general, for an axial-vector meson ($D_{1}$) and a pseudoscalar meson ($K$)
bound state, the BS wave function $\chi_{P}^{\mu}(p)$ has the following form:

$\chi_{P}^{\mu}(p)=f_{0}(p)p^{\mu}+f_{1}(p)P^{\mu}+f_{2}(p)\epsilon^{\mu}+f_{3}(p)\varepsilon^{\mu\nu\alpha\beta}p_{\alpha}P_{\beta}\epsilon_{\nu},$
(14)

where $f_{i}(p)$ $(i=0,1,2,3)$ are Lorentz-scalar functions and
$\epsilon^{\mu}$ represents the polarization vector of the bound state. After
considering the constraints imposed by parity and Lorentz transformations, it
is easy to prove that $\chi_{P}^{\mu}(p)$ can be simplified as

$\chi_{P}^{\mu}(p)=f(p)\varepsilon^{\mu\nu\alpha\beta}p_{\alpha}P_{\beta}\epsilon_{\nu},$
(15)

where the scalar function $f(p)$ contains all the dynamics.

In the following derivation of the BS equation, we will apply the
instantaneous approximation, in which the energy exchanged between the
constituent particles of the binding system is neglected. In our calculation
we choose the absolute value of the binding energy $E_{b}$ of the
$\bar{D}_{1}K$ system (which is defined as $E_{b}=M-m_{D_{1}}-m_{K}$) less
than 30 MeV. In this case the exchange of energy between the constituent
particles can be neglected.

Substituting Eqs. (4), (5), (12) and (13) into Eq. (3) and using the covariant
instantaneous approximation in the kernel, $p_{l}=q_{l}$, one obtains the
folowing expression:

$\begin{split}f(p)=&\int\frac{d^{4}q}{(2\pi)^{4}}\frac{i}{6\omega_{1}(\lambda_{1}M+p_{l}-\omega_{1}+i\epsilon)[(\lambda_{2}M-p_{l})^{2}-\omega_{2}^{2}+i\epsilon][-(p_{t}-q_{t})^{2}-m_{V}^{2}]}\\\
&\Big{\\{}g_{\bar{D}_{1}\bar{D}_{1}V}g_{KKV}\left[4(\lambda_{1}M+p_{l})(\lambda_{2}M-p_{l})+(p_{t}+q_{t})^{2}+(p_{t}^{2}-q_{t}^{2})^{2}/m_{V}^{2}\right]\\\
&+g^{\prime}_{\bar{D}_{1}\bar{D}_{1}V}g_{KKV}\omega_{2}(p_{t}\cdot
q_{t}-q_{t}^{2})/(\lambda_{2}M-\omega_{2}))\Big{\\}}F^{2}(k_{t})f(q),\end{split}$
(16)

where $k_{t}=p_{t}-q_{t}$ is the momentum of the exchanged meson in the
covariant instantaneous approximation.

In Eq. (16) there are poles in the plane of $p_{l}$ at
$-\lambda_{1}M+\omega_{1}-i\epsilon$, $\lambda_{2}M+\omega_{2}-i\epsilon$ and
$\lambda_{2}M-\omega_{2}+i\epsilon$. By choosing the appropriate contour, we
integrate over $p_{l}$ on both sides of Eq. (16) in the rest frame of the
bound state, then we obtain the following equation:

$\begin{split}\tilde{f}(p_{t})=&\int\frac{dq_{t}^{3}}{(2\pi)^{3}}\frac{1}{12\omega_{1}\omega_{2}(M-\omega_{1}-\omega_{2})\left[-(p_{t}-q_{t})^{2}-m_{V}^{2}\right]}\\\
&\times\Big{\\{}3g_{\bar{D}_{1}\bar{D}_{1}V}g_{KKV}\left[4\omega_{2}(M-\omega_{2})+(p_{t}+q_{t})^{2}+(p_{t}^{2}-q_{t}^{2})^{2}/m_{V}^{2}\right]\\\
&+2g^{\prime}_{\bar{D}_{1}\bar{D}_{1}V}g_{KKV}\omega_{2}(p_{t}\cdot
q_{t}-q_{t}^{2})/(\lambda_{2}M-\omega_{2})\Big{\\}}F^{2}(k_{t})\tilde{f}(q_{t}),\end{split}$
(17)

where $\tilde{f}(p_{t})\equiv\int dp_{l}f(p)$.

Now, we can solve the BS equation numerically and study whether the $S$-wave
$\bar{D}_{1}K$ bound state exists or not. It can be seen from Eq. (17) that
there is only one free parameter in our model, the cutoff $\Lambda$, which
enters through various phenomenological form factors in Eq. (13). It contains
the information about the extended interaction due to the structures of
hadrons. The value of $\Lambda$ is of order 1 GeV which is the typical scale
of nonperturbative QCD interaction. In this work, we shall treat $\Lambda$ as
a parameter and vary it in a wide range 0.8-4.8 GeV when the binding energy
$E_{b}$ is in the region from -5 to -30 MeV to see if the BS equation has
solutions.

To find out the possible molecular bound states, one only needs to solve the
homogeneous BS equation. One numerical solution of the homogeneous BS equation
corresponds to a possible bound state. The integration region in each integral
is discretized into $n$ pieces, with $n$ being sufficiently large. In this
way, the integral equation is converted into an $n\times n$ nmatrix equation,
and the scalar wave function will now be regarded as an $n$-dimensional
vector. Then, the integral equation can be illustrated as
$\tilde{f}^{(n)}(p_{t})=A^{(n\times n)}(p_{t},q_{t})\tilde{f}^{(n)}(q_{t})$,
where $\tilde{f}^{(n)}(p_{t})(\tilde{f}^{(n)}(q_{t}))$ is an $n$-dimensional
vector, and $A^{(n\times n)}(p_{t},q_{t})$ is an $n\times n$ matrix, which
corresponds to the matrix labeled by $p_{t}$ and $q_{t}$ in each integral
equation. Generally, $p_{t}$ ($q_{t}$) varies from 0 to $+\infty$. Here,
$p_{t}$ ($q_{t}$) is transformed into a new variable $t$ that varies from $-1$
to 1 based on the Gaussian integration method,

$p_{t}=\mu+w\log\left[1+y\frac{1+t}{1-t}\right],$ (18)

where $\mu$ is a parameter introduced to avoid divergence in numerical
calculations, $w$ and $y$ are parameters used in controlling the slope of wave
functions and finding the proper solutions for these functions. Then one can
obtain the numerical results of the BS wave functions by requiring the
eigenvalue of the eigenvalue equation to be 1.0.

In our calculation, we choose to work in the rest frame of the bound state in
which $P=(M,0)$. We take the averaged masses of the mesons from the PDG
Zyla:2020zbs , $m_{D_{1}}=2420.8$ MeV, $m_{K}=494.98$ MeV, $m_{\rho}=775.26$
MeV, and $m_{\omega}=782.65$ MeV. After searching for possible solutions in
the isoscalar channel of the $\bar{D}_{1}K$ system, we find the bound state
exists. We list some values of $E_{b}$ and the corresponding $\Lambda$ for the
three different form factor models in Table 1.

Table 1:  Values of $E_{b}$ and corresponding cutoff $\Lambda_{M}$, $\Lambda_{D}$, and $\Lambda_{E}$ for $I=0$ and $I=1$ $\bar{D}_{1}K$ bound states for the monopole, dipole, and exponential form factors, respectively. |  $E_{b}$(MeV) | -5 | -10 | -15 | -20 | -25 | -30
---|---|---|---|---|---|---|---
I=0 |  $\Lambda_{M}$(MeV) | 1208 | 1261 | 1297 | 1327 | 1352 | 1375
$\Lambda_{D}$(MeV) | 1668 | 1756 | 1817 | 1867 | 1910 | 1948
$\Lambda_{E}$(MeV) | 1159 | 1231 | 1280 | 1321 | 1356 | 1386
I=1 |  $\Lambda_{M}$(MeV) | 1541 | 1649 | 1723 | 1783 | 1835 | 1880
$\Lambda_{D}$(MeV) | 1897 | 2371 | 2492 | 2589 | 2671 | 2744
$\Lambda_{E}$(MeV) | 1571 | 1708 | 1804 | 1881 | 1947 | 2006

##  III The Normalization Condition of the BS wave function

To find out whether the bound state of the $\bar{D}_{1}K$ system exists or
not, one only needs to solve the homogeneous BS equation. However, when we
want to calculate physical quantities such as the decay width we have to face
the problem of the normalization of the BS wave function. In the following we
will discuss the normalization of the BS wave function $\chi^{\mu}_{P}(p)$.

In the heavy quark limit, the normalization of the BS wave function of the
$\bar{D}_{1}K$ system can be written as Guo:2007qu

$i\int\frac{d^{4}pd^{4}q}{(2\pi)^{8}}\bar{\chi}^{\mu}_{P}(p)\frac{\partial}{\partial
P_{0}}[I_{P\mu\nu}(p,q)]\chi^{\nu}(q)=2E_{P},$ (19)

where
$I_{P\mu\nu}(p,q)=(2\pi)^{4}\delta^{4}(p-q)S^{-1}_{\mu\nu}(p_{1})S^{-1}(p_{2})$.

In the rest frame, the normalization condition can be written in the following
form:

$\begin{split}-i\int\frac{d^{4}p}{(2\pi)^{4}}\Big{\\{}&4M^{2}p_{t}^{2}\big{[}\lambda_{1}^{2}(6\lambda_{2}^{2}M^{2}-6\lambda_{2}Mp_{l}+p_{l}^{2}-\omega_{2}^{2})\\\
&+2\lambda_{1}\lambda_{2}p_{l}(3\lambda_{2}M-2p_{l})+\lambda_{2}^{2}(p_{l}^{2}-\omega_{1}^{2})\big{]}f^{2}(q)=1.\end{split}$
(20)

From Eqs. (16) and (17), we obtain

$\begin{split}f(p)=&\frac{i\omega_{2}(M-\omega_{1}-\omega_{2})}{\pi(\lambda_{1}M+p_{l}-\omega_{1}+i\epsilon)(\lambda_{2}M-p_{l}+\omega_{2}-i\epsilon)(\lambda_{2}M-p_{l}-\omega_{2}+i\epsilon)}\tilde{f}(p_{t}).\\\
\end{split}$ (21)

Then, one can recast the normalization condition for the BS wave function into
the form

$\begin{split}&-\int\frac{d^{3}p_{t}}{8\pi^{5}}\frac{M^{2}p_{t}^{2}\omega_{1}}{\omega_{2}^{2}(M-\omega_{1}-\omega_{2})^{2}}\Big{\\{}\lambda_{2}^{2}(p_{t}^{2}-\omega_{1}^{2})(\lambda_{2}M-\omega_{1}-3\omega_{2})+\lambda_{1}^{3}(\lambda_{2}^{2}M^{3}-2M\omega_{2}^{2})\\\
&+\lambda_{1}\lambda_{2}[2\lambda_{2}^{3}M^{3}+\lambda_{2}M(p_{t}^{2}-\omega_{1}^{2})-4\omega_{2}^{2}(\omega_{1}-\omega_{2})-2\lambda_{2}^{2}M^{2}(\omega_{1}+3\omega_{2})]\\\
&+\lambda_{1}^{2}[3\lambda_{2}^{3}M^{3}-6\lambda_{2}M\omega_{2}^{2}+2\omega_{2}^{2}(\omega_{1}+\omega_{2})-\lambda_{2}^{2}M^{2}(\omega_{1}+3\omega_{2})]\Big{\\}}\tilde{f}^{2}(p_{t})=1.\end{split}$
(22)

The wave function obtained in the previous section (which is calculated
numerically from Eq.(17)) can be normalized by Eq. (22).

In our case, the binding energy
$E_{b}=M_{X_{1}(2900)}-(M_{{\bar{D}_{1}}}+M_{K})\simeq-12.4$ MeV, where we
have used the mass of the $X_{1}(2900)$ as 2904 MeV. From our calculations, we
find the $I=0$ $\bar{D}_{1}K$ system can be assigned as the $X_{1}(2900)$
state when the cutoff $\Lambda$ = 1280 MeV, 1788 MeV, and 1257 MeV for the
monopole, dipole, and exponential form factors, respectively, the $I=1$
$\bar{D}_{1}K$ system can be the $X_{1}(2900)$ state when the cutoff $\Lambda$
= 1688 MeV, 2434 MeV, and 1758 MeV for the monopole, dipole, and exponential
form factors, respectively. The corresponding numerical results of the
normalized Lorentz scalar function, $\tilde{f}(p_{t})$, are given in Figs. 2
and 3 for the $X_{1}(2900)$ states with $I=0$ and $I=1$ in the $\bar{D}_{1}K$
molecule picture for the monopole, dipole, and exponential form factors,
respectively.

Figure 2: Numerical results of the normalized Lorentz scalar function
$\tilde{f}(p_{t})$ for the $X_{1}(2900)$ in the $I=0$ $\bar{D}_{1}K$ molecular
picture with (a) the monopole form factor, (b) the dipole form factor, and (c)
the exponential form factor.

Figure 3: Numerical results of the normalized Lorentz scalar function
$\tilde{f}(p_{t})$ for the $X_{1}(2900)$ in the $I=1$ $\bar{D}_{1}K$ molecular
picture with (a) the monopole form factor, (b) the dipole form factor, and (c)
the exponential form factor.

##  IV The decay of $X_{1}(2900)\rightarrow D^{-}K^{+}$

Figure 4: The diagrams contributing to the $X_{1}(2900)\rightarrow D^{-}K^{+}$
decay process induced by $\rho$ and $\omega$.

Besides investigating whether the bound state of the $\bar{D}_{1}K$ system can
be $X_{1}(2900)$ or not, we can also study the decay of the $X_{1}(2900)$ as
the $S$-wave $\bar{D}_{1}K$ bound state. The $X_{1}(2900)$ can decay to
$D^{-}K^{+}$ via the Feynman diagrams in Fig. 4. which are induced by the
effective Lagrangians for the $D_{1}DV$ and $KKV$ vertices (which have been
given in Eq. (6)) as the following Ding:2008gr :

$\begin{split}\mathcal{L}_{DD_{1}V}=&g_{DD_{1}V}D_{1b}^{\mu}V_{\mu
ba}D_{a}^{\dagger}+g^{\prime}_{DD_{1}V}(D_{1b}^{\mu}\overleftrightarrow{\partial}^{\nu}D_{a}^{\dagger})(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu})_{ba}\\\
&+g_{\bar{D}\bar{D}_{1}V}\bar{D}_{a}^{\dagger}V_{\mu
ba}\bar{D}_{1b}^{\mu}+g^{\prime}_{\bar{D}\bar{D}_{1}V}(\bar{D}_{1b}^{\mu}\overleftrightarrow{\partial}^{\nu}\bar{D}_{a}^{\dagger})(\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu})_{ba}+H.c.,\\\
\end{split}$ (23)

where the coupling constants are given as
$g_{DD_{1}V}=-g_{\bar{D}\bar{D}_{1}V}=-\frac{2}{\sqrt{3}}\zeta_{1}g_{v}\sqrt{m_{D}m_{D_{1}}}$,
$g^{\prime}_{DD_{1}V}=-g^{\prime}_{\bar{D}\bar{D_{1}}V}=\frac{1}{\sqrt{3}}\mu_{1}g_{V}$
Ding:2008gr , with the two parameters $\zeta_{1}$ and $\mu_{1}$ being involved
in the coupling constants, about which the information is very scarce leading
them undetermined. However, in the heavy quark limit, we can roughly assume
that the coupling constants $g_{D{D_{1}}V}$ and $g^{\prime}_{DD_{1}V}$ are
equal to $g_{D^{\ast}D_{0}V}$ (=$\zeta g_{V}\sqrt{2m_{D^{\ast}}m_{D^{0}}}$)
and $g^{\prime}_{D^{\ast}D_{0}V}$ (=$1/\sqrt{2}\mu g_{V}$), respectively. The
parameters $\mu=0.1$ $\mathrm{GeV}^{-1}$ and $\zeta=0.1$ are taken in Ref.
Casalbuoni:1996pg .

According to the above interactions, we can write down the amplitude for the
decay $X_{1}(2900)\rightarrow D^{-}K^{+}$ induced by light vector meson
($\rho$ and $\omega$) exchanges as shown in Fig. 4, as the following:

$\begin{split}\mathcal{M}=&g_{KKV}\sqrt{2E}\int\frac{d^{4}p}{(2\pi)^{4}}F^{2}(k)\big{[}g_{\bar{D}\bar{D}_{1}V}(p_{2}+p^{\prime}_{2})^{\alpha}\Delta_{\alpha\mu}(k,m_{V})\\\
&+g^{\prime}_{\bar{D}\bar{D}_{1}V}(p_{1}+p^{\prime}_{1})^{\nu}(p_{2}+p^{\prime}_{2})^{\alpha}\left(k_{\mu}\Delta_{\alpha\nu}(k,m_{V})-k_{\nu}\Delta_{\alpha\mu}(k,m_{V})\right)\big{]}\chi_{P}^{\mu}(p)\end{split}$
(24)

In the rest frame, we define
$p_{1}^{\prime}=(E_{1}^{\prime},-\mathbf{p}^{\prime}_{1})$ and
$p_{2}^{\prime}=(E_{2}^{\prime},\mathbf{p}^{\prime}_{2})$ to be the momenta of
$D$ and $K$, respectively. According to the kinematics of the two-body decay
of the initial state in the rest frame, one has

$\begin{split}E_{1}^{\prime}&=\frac{M^{2}-m_{2}^{{}^{\prime}2}+m_{1}^{{}^{\prime}2}}{2M},\quad\quad
E_{1}^{\prime}=\frac{M^{2}-m_{1}^{{}^{\prime}2}+m_{2}^{{}^{\prime}2}}{2M},\\\
|\mathbf{p}^{\prime}_{1}|&=|\mathbf{p}^{\prime}_{2}|=\frac{\sqrt{[M^{2}-(m^{\prime}_{1}+m^{\prime}_{2})^{2}][M^{2}-(m^{\prime}_{1}-m^{\prime}_{2})^{2}]}}{2M},\end{split}$
(25)

and

$d\Gamma=\frac{1}{32\pi^{2}}|\mathcal{M}|^{2}\frac{|\mathbf{p}^{\prime}|}{M^{2}}d\Omega,$
(26)

where $|\mathbf{p}^{\prime}_{1}|$ and $|\mathbf{p}^{\prime}_{2}|$ are the norm
of the 3-momentum of the particles in the final states in the rest frame of
the initial bound state and $\mathcal{M}$ is the Lorentz-invariant decay
amplitude of the process.

Substituting the normalized numerical solutions of the BS equation, and the
cutoff $\Lambda$ are 1280 MeV, 1788 MeV, and 1257 MeV with $I=0$ and 1688 MeV,
2434 MeV, and 1758 MeV with $I=1$ for the monopole, dipole, and exponential
form factors, respectively. The decay widths of the $X_{1}(2900)$ to
$D^{-}K^{+}$ can be obtained as following:

$\Gamma_{X_{1}(2900)(I=0)\rightarrow D^{-}K^{+}}=\left\\{\begin{aligned}
&70.73\ \mathrm{MeV}\ \mathrm{with\ monopole\ form\ factor},\\\ &98.75\
\mathrm{MeV}\ \mathrm{with\ dipole\ form\ factor},\\\ &60.38\ \mathrm{MeV}\
\mathrm{with\ exponential\ form\ factor},\end{aligned}\right.$ (27)

and

$\Gamma_{X_{1}(2900)(I=1)\rightarrow D^{-}K^{+}}=\left\\{\begin{aligned}
&28.14\ \mathrm{MeV}\ \mathrm{with\ monopole\ form\ factor},\\\ &18.13\
\mathrm{MeV}\ \mathrm{with\ dipole\ form\ factor},\\\ &12.78\ \mathrm{MeV}\
\mathrm{with\ exponential\ form\ factor}.\end{aligned}\right.$ (28)

From our calculation results, we can see that different form factors have a
great influence on the decay width, and different cutoff $\Lambda$ for the
same form factor also have a great influence on the decay width.

##  V summary

In this paper, we studied the $X_{1}(2900)$ with the hadronic molecule
interpretation by regarding it as a bound state of $\bar{D}_{1}$ and $K$
mesons in the BS equation approach. In our model, we applied the ladder and
instantaneous approximations to obtain the kernel containing one-particle-
exchange diagrams and introduced three different form factors (the monopole
form factor, the dipole form factor, and the exponential form factor) at the
interaction vertices. From the calculating results we find that there exist
bound states of the $\bar{D}_{1}K$ system. The binding energy depends on the
value of the cutoff $\Lambda$. For the $I=0$ $\bar{D}_{1}K$ system, we find
the cutoff regions in which the solutions (with the binding energy $E_{b}\in$
(-5, -30) MeV) for the ground state of the BS equation can be found (in units
of MeV): $\Lambda_{M}\sim$ (1208, 1375), $\Lambda_{D}\sim$ (1668, 1948), and
$\Lambda_{E}\sim$ (1159, 1386) for the monopole form factor, the dipole form
factor, and the exponential form factor, respectively. For the $I=1$
$\bar{D}_{1}K$ system, we find two the regions (in units of MeV):
$\Lambda_{M}\sim$ (1541, 1880), $\Lambda_{D}\sim$ (1897, 2744), and
$\Lambda_{E}\sim$ (1571, 2006) in which the solutions of the BS equation can
be found. Thus, we can confirm that $X_{1}(2900)$ can be regarded as the
$S$-wave $\bar{D}_{1}K$ molecules in our model.

We applied the numerical solutions for the BS wave functions with the
corresponding cutoff ($\Lambda$ = 1280 MeV, 1788 MeV, and 1257 MeV for $I=0$
and $\Lambda$ = 1688 MeV, 2434 MeV, and 1758 MeV for $I=1$ with the monopole,
dipole, and exponential form factors, respectively.) to calculate the decay
widths of $X_{1}(2900)\rightarrow D^{-}K^{+}$ induced by $\rho$ and $\omega$
exchanges. We predict the decay widths are 70.73, 98.75, and 60.38 MeV and
28.14, 18.13, and 12.78 MeV for $X_{1}(2900)$ as $I=0$ and $I=1$
$\bar{D}_{1}K$ molecules with the corresponding cutoff in the decay process,
respectively. From our study, the $X_{1}(2900)$ is suitable as $I=0$
$\bar{D}_{1}K$ molecular state. There are two uncertain factors in the
calculation of the decay width, one is that the parameters $\xi_{1}$ and
$\mu_{1}$ have not been determined since the information about them is very
scarce, the other is that we can not give the definite value of the cutoff
$\Lambda$.

###### Acknowledgements.

This work was supported by National Natural Science Foundation of China
(Projects No. 11775024, No.11605150 and No.11947001), the Natural Science
Foundation of Zhejiang pvovince (No. LQ21A050005), the Ningbo Natural Science
Foundation (No.2019A610067), the Fundamental Research Funds for the Provincial
Universities of Zhejiang Province and K.C.Wong Magna Fund in Ningbo
University.

## References

  * (1) R. Aaij et al. [LHCb], Phys. Rev. D 102, 112003 (2020).
  * (2) R. Aaij et al. [LHCb], Phys. Rev. Lett. 117, 152003 (2016).
  * (3) A. M. Sirunyan et al. [CMS], Phys. Rev. Lett. 120, 202005 (2018).
  * (4) T. Aaltonen et al. [CDF], Phys. Rev. Lett. 120, 202006 (2018).
  * (5) M. Aaboud et al. [ATLAS], Phys. Rev. Lett. 120, 202007 (2018).
  * (6) F. K. Guo, C. Hanhart, U. G. Meißner, Q. Wang, Q. Zhao and B. S. Zou, Rev. Mod. Phys. 90, 015004 (2018).
  * (7) S. L. Olsen, T. Skwarnicki and D. Zieminska, Rev. Mod. Phys. 90, 015003 (2018).
  * (8) M. Karliner and J. L. Rosner, Phys. Rev. D 102, 094016 (2020).
  * (9) G. J. Wang, L. Meng, L. Y. Xiao, M. Oka and S. L. Zhu, [arXiv:2010.09395 [hep-ph]].
  * (10) Q. F. Lü, D. Y. Chen and Y. B. Dong, Phys. Rev. D 102, 074021 (2020).
  * (11) X. G. He, W. Wang and R. Zhu, Eur. Phys. J. C 80, 1026 (2020).
  * (12) H. X. Chen, W. Chen, R. R. Dong and N. Su, Chin. Phys. Lett. 37, 101201 (2020).
  * (13) Y. Tan and J. Ping, [arXiv:2010.04045 [hep-ph]].
  * (14) H. Mutuk, [arXiv:2009.02492 [hep-ph]].
  * (15) M. Z. Liu, J. J. Xie and L. S. Geng, Phys. Rev. D 102, 091502 (2020).
  * (16) Y. Huang, J. X. Lu, J. J. Xie and L. S. Geng, Eur. Phys. J. C 80, 973 (2020).
  * (17) C. J. Xiao, D. Y. Chen, Y. B. Dong and G. W. Meng, [arXiv:2009.14538 [hep-ph]].
  * (18) X. K. Dong and B. S. Zou, [arXiv:2009.11619 [hep-ph]].
  * (19) J. He and D. Y. Chen, [arXiv:2008.07782 [hep-ph]].
  * (20) G. J. Ding, Phys. Rev. D 79, 014001 (2009).
  * (21) F. L. Wang, R. Chen, Z. W. Liu and X. Liu, Phys. Rev. D 99, 054021 (2019).
  * (22) P. A. Zyla et al. [Particle Data Group], PTEP 2020, no. 8, 083C01 (2020).
  * (23) X. H. Guo, K. W. Wei and X. H. Wu, Phys. Rev. D 77, 036003 (2008).
  * (24) R. Casalbuoni, A. Deandrea, N. Di Bartolomeo, R. Gatto, F. Feruglio and G. Nardulli, Phys. Rept. 281, 145-238 (1997).