Adversarial Bayesian Augmentation for Single-Source Domain Generalization
Sheng Cheng 1
Tejas Gokhale 1,2
Yezhou Yang 1
1
Arizona State University 2 University of Maryland, Baltimore County
scheng53@asu.edu, gokhale@umbc.edu, yz.yang@asu.edu

Abstract
Generalizing to unseen image domains is a challenging problem primarily due to the lack of diverse training
data, inaccessible target data, and the large domain shiftthat may exist in many real-world settings. As such data
augmentation is a critical component of domain generalization methods that seek to address this problem. We present
Adversarial Bayesian Augmentation (ABA), a novel algorithm that learns to generate image augmentations in thechallenging single-source domain generalization setting.
ABA draws on the strengths of adversarial learning andBayesian neural networks to guide the generation of diverse
data augmentations – these synthesized image domains aid
the classifier in generalizing to unseen domains. We demonstrate the strength of ABA on style shift. ABA outperforms all previous state-of-the-art methods, including prespecified augmentations, pixel-based and convolutionalbased augmentations. Full paper: https://arxiv.
org/abs/2307.09520. Code: https://github.
com/shengcheng/ABA.

1. Introduction
Improving the generalization of deep neural networks
to out-of-distribution samples is a fundamental yet challenging problem in machine learning and computer vision [21, 10, 15]. Typically, neural networks are trained andtested on data samples from the same distribution (under thei.i.d. assumption); under this setting, image classifiers haveachieved impressive performances. However, in real-world
applications, the distribution of test samples can drasticallydiffer from the training samples [20, 17]. This is especiallyproblematic when the process of acquiring labeled samples
from the target test domain is expensive or infeasible, making it difficult to apply semi-supervised learning for domain
adaptation [26, 25]. Therefore, there is a need to develop
techniques that enable deep neural networks to capture thedomain-invariant patterns in the data [15, 23], facilitating
improved generalization to out-of-distribution samples.
In the multi-source domain generalization (MSDG) set-

SourceTargetFigure 1: An illustration of the diversity introduced by Adversial Bayesian Augmentations. The blue and orange surfaces represent the source (seen) and target (unseen) domains respectively. The red dots represent the samples augmented by ABA; these augmentations expose the classifier
to regions closer to the target domain, thereby improving
image classifiers’ generalization to unseen domains.

ting, where there are multiple source domains for training,domain label information can be leveraged to learn the domain shift [15, 4, 23]. Prior information about the target
domain is also useful to design specific data augmentation
methods to tackle domain shift. However, in the singlesource domain generalization (SSDG) setting, where only
one domain is available for training, it is more challenging
to address the domain shift issue. In this paper, we focus on
the strict SSDG setting, where only one source domain isavailable for training and no prior knowledge is available
about the target domain. Recent work in SSDG focuses
on augmenting the data to simulate the presence of outof-distribution domains. One way involves learning-freedata augmentation methods, such as RandConv [24], Augmix [9] and JiGen [1] – here the data augmentation is prespecified and does not evolve or adapt during training. Another approach is based on adversarial perturbations, which
involves generating adversarial samples to improve generalization, such as Augmax [22], ADA [20], M-ADA [18],and ALT [7]. Although the Bayesian neural networks as the
backbone of the classifier show good generalization abilityto out-of-distribution samples intrinsically [14, 23, 2], andsome papers [19] use Bayesian neural networks for generating images, none of the work directly augments the data
by Bayesian neural network for domain generalization.
In this paper, we present a novel approach called Adversarial Bayesian Augmentation, dubbed ABA, which draws
on the strengths of adversarial learning and Bayesian neuralnetworks to generate more diverse data and improve generalization on different domains. Specifically, the adversarial learning-based methods, which explore a wider augmentation space, already outperforms learning-free methods [7, 22] on SSDG. The introduction of weight uncertainty by the Bayesian neural network further enhances thestrength of data augmentation, as shown in Figure 1. Our
experimental results demonstrate ABA’s superior performance compared to existing methods.
The key contributions and findings of the paper thus are:
• We introduce a novel data augmentation method, dubbed
ABA, which combines adversarial learning and Bayesianneural network, to improve the diversity of training data
for single-source domain generalization setting.
• We empirically validate the effectiveness of our proposed
method on style generalization. Our method outperforms
all existing state-of-art methods.

Algorithm 1: Learning with Adversarial Bayesian
Augmentation (ABA)
Input : {xi , yi }N
i=1
Output : Classifier f parameters θ∗
1 for t ← 1 to T do
2
if t < Twarmup then3
θ ← θ − γ ▽ Lcls
4
else
5

/* Training ABA6Φ ← Φ0for m ← 1 to Tadv do
yg = f (g(x, Φ), θ)Φ ← Φ − η ▽ LELBO // See (1)end for

7
8
9
10
11
12
13

*/

/* Train classifier*/Φ ← µ + σ ⊙ ϵ // Sample parametersθ ← θ − γ ▽ (Lcls + αLKL ) // See (2),(3)end ifend for
16 Return θ14

15

p(ϕl |x, y). To enable efficient sampling of the variational
distribution, we re-parameterize as ϕl = µl + σl ϵl , where
ϵl is a sample from the standard normal distribution, whichallows us to compute the gradients of µl and σl . We denoteLµ = {µl }L
l=1 and σ = {σl }l=1 . So Φ = {µ, σ}.
The optimization of ABA is formulated as a min-max
problem. Initially, we optimize the g network using adversarial optimization to augment images that can fool theclassifier f . To achieve this, we use the evidence lower
bound (ELBO) of the variational Bayesian network as theloss function. ELBO is a lower bound on the log marginal
likelihood of the observed data and is defined as follows:

2. Proposed Method
Let S and T represent the source and target domains respectively, which share the same label space. The training
set is a subset in the source domain and contains N training pairs, denoted as {(xi , yi )}N
i=1 ⊂ S. The objective of
SSDG is to use S to learn parameters θ of a classifier fwhich also can generalize well to target domain T .

2.1. Adversarial Bayesian AugmentationN

To accomplish SSDG, since no information is available
from the target domain T , previous works focus on data
augmentation, denoted as g. In this paper, we design g as a
L-layer Bayesian convolutional neural network, parameterkl ×kl ×Cin(l) ×Cout(l)ized by Φ = {ϕl }Lare
l=1 , where ϕi ∈ R
the parameters of each Bayesian convolutional layer. Following the setting in [24], we randomly sample kl fromK = {1, 3, ...n}. Cin(l) and Cout(l) represent the number of
input and output channels for each layer convolutional kernel. Since g is an image augmentation function, the numberof input channels for the first and last layer are equal to thenumber of image channels (3 for color images and 1 forgrayscale images).
To perform Bayesian inference, we need to estimate
the posterior distribution p(ϕl |x, y), which is intractable inclosed form. To approximate it, we adopt the variational
Bayesian inference approach and use a variational distribution q(ϕl ). This distribution is obtained by minimizing the
KL divergence between q(ϕl ) and true posterior distribution

LELBO =1 X
Eg∼q(Φ) [log(yi |g(xi ), θ)]
N i=1
−β

L
X

KL(q(ϕl )||p(ϕl )),

(1)

l=1

where the prior distribution p(ϕl ) of each layer follows
1), which is used in network initializaN (0, kl ×kl ×Cin(l)tion [8]. Theoretically, the coefficient β for the KL term
should be 1. However, in practice, for small datasets orlarge models, smaller β (0 < β < 1) is preferred [13].
Starting from a random initialization, the parameters of
g are iteratively updated by maximizing the negative of
ELBO. In contrast to adv-BNN [13], which constrains theadversarial samples bounded by ℓp norm, we control thestrength of adversarial samples by adjusting the learning
rate η and the number of adversarial steps Tadv . The final
augmented images xg are obtained through Bayesian inference using the optimized parameters Φ∗ and clamped to the2Method

MNIST-10K

MNIST-M

SVHN

USPS

SYNTH

Target Avg.

ERM
ADA
M-ADA
ESDA
AdvBNN
Augmix98.40 (0.84)
N/A
99.30
99.30 (0.10)
98.23 (0.08)
98.53 (0.18)

58.87 (3.73)
60.41
67.94
81.60 (1.60)
71.79 (0.69)
53.36 (1.59)

33.41 (5.28)
35.51
42.55
48.90 (5.20)
44.85 (0.55)
25.96 (0.80)

79.27 (2.70)
77.26
78.53
84.00 (1.20)
46.05 (0.53)
96.12 (0.72)

42.43 (5.46)
45.32
48.95
62.20 (1.30)
44.99 (0.54)
42.90 (0.60)

53.50 (4.23)
54.62
59.49
69.12 (2.33)
51.92 (0.51)
54.59 (0.50)

98.85 (0.04)
98.41 (0.15)
98.54 (0.10)
98.82 (0.09)
98.78 (0.09)

87.76 (0.83)
72.80 (2.06)
75.77 (1.51)
78.81 (1.64)
78.62 (0.92)

57.62 (2.09)
47.07 (1.88)
49.90 (1.62)
51.88 (1.93)
52.04 (1.13)

83.36 (0.96)
94.79 (0.88)
95.64 (0.62)
96.22 (0.26)
96.16 (0.16)

62.88 (0.78)
66.27 (1.56)
68.61 (1.75)
71.25 (1.27)
71.23 (0.93)

72.88 (0.58)
70.23 (1.22)
72.47 (1.18)
74.57 (0.52)
74.51 (0.70)

98.73 (0.10)
98.67 (0.11)

80.94 (0.39)
80.05 (0.81)

55.88 (0.70)
56.87 (1.05)

96.34 (0.54)
96.55 (0.34)

73.09 (0.34)
73.40 (0.19)

76.56 (0.06)
76.72 (0.41)

98.46 (0.27)
98.46 (0.25)
98.55 (0.11)
98.78 (0.06)
98.76 (0.12)
98.66 (0.16)

74.28 (1.36)
76.90 (1.42)
75.98 (0.89)
80.54 (0.53)
79.69 (0.35)
80.24 (0.51)

52.25 (1.54)
53.78 (1.97)
55.01 (1.34)
52.45 (1.21)
54.09 (1.27)
56.43 (0.59)

94.99 (0.68)
95.40 (0.72)
96.17 (0.45)
95.81 (0.47)
96.42 (0.35)
96.14 (0.64)

68.44 (0.98)
69.40 (1.07)
68.93 (2.17)
70.25 (1.21)
71.55 (0.96)
70.91 (0.83)

72.49 (0.87)
73.87 (1.03)
74.38 (0.86)
74.76 (0.52)
75.44 (0.61)
75.93 (0.60)

1-layer convolutional-based augmentations

RandConv
ALT1-layer
ALT1-layer+RandConv
ABA1-layer
ABA1-layer+RandConv
3-layer convolutional-based augmentations

ABA3-layers
ABA3-layers+RandConv
5-layer convolutional-based augmentations

ALT5-layer
ALT5-layer+RandConv
ALT5-layer+Augmix
ABA5-layer
ABA5-layer+RandConv
ABA5-layer+Augmix

Table 1: SSDG accuracy on Digits dataset. The source domain is MNIST-10K. The target domains are MNIST-M, SVHN,
USPS, SYNTH. We report the mean (and standard deviation) of 5 runs.
MethodPhoto

CartoonArt

Sketch

Avg.

ERM
JiGen
SagNetADA
AdvBNN
Augmix

38.93
41.70
48.53
44.63
45.93 (0.41)
45.24 (1.12)

70.00
72.23
75.66
71.96
60.24 (0.95)
74.66 (1.09)

68.83
67.70
73.20
72.43
75.33 (0.95)
71.47 (0.64)

39.36
36.83
50.06
45.73
26.19 (1.23)
47.72 (1.72)

54.28
54.61
61.86
58.68
51.92 (1.15)
60.51 (1.14)

49.80 (4.23)
50.83 (2.13)
52.24 (0.82)
54.49 (1.35)
52.32 (1.82)

67.90 (1.55)
75.00 (0.62)
75.16 (0.67)
75.61 (0.89)
76.01 (0.56)

69.63 (2.15)
73.87 (1.31)
73.46 (1.29)
75.59 (1.56)
75.77 (1.64)

54.06 (1.96)
47.83 (1.95)
49.21 (2.14)
52.84 (2.80)
50.20 (1.93)

60.34 (2.47)
61.88 (1.50)
62.51 (1.23)
64.63 (1.65)
63.58 (1.49)

58.86 (0.83)
56.95 (0.80)

77.49 (0.57)
77.21 (0.85)

75.34 (0.89)
75.34 (0.52)

53.76 (2.46)
53.52 (0.90)

66.36 (1.19)
65.76 (0.15)

54.33 (1.08)
55.66 (0.50)
55.09 (1.87)
59.04 (1.43)
57.59 (1.26)
57.87 (0.22)

75.96 (1.12)
76.23 (0.80)
77.36 (0.73)
77.16 (0.35)
76.66 (0.24)
77.29 (0.78)

74.06 (1.09)
73.96 (0.54)
75.69 (1.21)
74.71 (0.76)
75.61 (1.02)
74.70 (0.96)

50.03 (2.41)
50.86 (0.79)
50.72 (1.41)
53.18 (2.07)
54.12 (1.33)
52.35 (0.03)

63.60 (1.43)
64.18 (0.66)
64.72 (1.30)
66.02 (1.15)
66.00 (0.96)
65.55 (0.49)

1-layer convolutional-based augmentations

RandConv
ALT1-layer
ALT1-layer+RandConv
ABA1-layer
ABA1-layer+RandConv
3-layer convolutional-based augmentations

ABA3-layers
ABA3-layers+RandConv
5-layer convolutional-based augmentations

ALT5-layer
ALT5-layer+RandConv
ALT5-layer+Augmix
ABA5-layer
ABA5-layer+RandConv
ABA5-layer+Augmix

Table 2: SSDG accuracy on PACS. Each column is the average accuracy on the target domains trained on the given source
domain. We report the mean (and standard deviation) of 5 runs. More details about the accuracy of the source domain toeach target domain are in the Appendix.
image range. Note that we can sample multiple augmented
images from Bayesian inference, and we sample twice denoted as xg1 and xg2 . These augmented images can be used
for classifier learning in the presence of domain shift.
Next, we optimize the classifier f with a loss function
consisting of two terms: a cross-entropy loss, which is
Lcls = CrossEntropy(f (xg1 , θ), y),and a consistency regularization loss, which helps to keep
the prediction consistent on augmented data, defined as:
LKL = KL(pc ||p̄) + KL(pg1 ||p̄) + KL(pg2 ||p̄),(3)

where pc , pg1 , pg2 denotes the softmax prediction of f on
clean image x and augmented images xg1 , xg2 respectively.
p̄ is the average of pc , pg1 , and pg2 .

(2)3

Input Image
RandConv

with RandConv [24], we use ABA1-layer , i.e. ABA with a
1-layer Bayesian convolutional neural network. To matchthe number of convolutional layers in ALT [7], we use
ABA5-layer , i.e. ABA with a 5-layer Bayesian convolutional
neural network. In the variants ABA5-layer+RandConv andABA5-layer+Augmix , the second augmented image is generated
by RandConv or Augmix instead of Bayesian inference.

ABA

ALT

Results.
For Digits dataset, Table 1 shows that pixel-level adversarial perturbation methods such as ADA and M-ADA,
and the composition of image augmentation method like
Augmix, only marginally improve SSDG performance,
while AdvBNN even downgrades the performance. However, convolutional-based augmentations, even with just one
layer, can significantly enhance performance. Among the
1-layer convolutional augmentations, ALT do not performbetter than RandConv. However, our 1-layer ABA outperforms both. A 5-layer ABA performs better than 1-layer
ABA and adding a RandConv or Augmix module can further improve performance. We achieve state-of-art resultsby 3-layer ABA with RandConv to 76.72%.
Our experiments on the PACS dataset are summarizedin Table 2. As PACS contains images with different
styles, methods such as SagNet and RandConv that preserve
shape and texture information can improve performance.
In comparison, JiGen and ADA only marginally improveaccuracy, while AdvBNN downgrades the performance.
Similarly to the Digits dataset, leveraging convolutionalbased augmentations provides significant performance improvements, with four variants of ALT performing better
than other baseline models. However, our proposed ABA
method outperforms ALT on both 1-layer and 5-layer cases.
The addition of RandConv or Augmix modules does notyield further performance improvement, but it still outperforms the corresponding ALT models, respectively. We
achieve the state-of-art results by 3-layer ABA, with an accuracy of 66.36%. We show the qualitative results of augmented images by RandConv, ALT and ABA in Figure 2.

Figure 2: Qualitative comparison of PACS images augmented by RandConv, ALT and our ABA.
Implementation. Algorithm 1 depicts the implementation
details. For network design, the activation of multiple layers
ABA is LeakyRelu. The second augmented image xg2 canbe obtained not only through Bayesian inference, but also
obtained from other data augmentation techniques, such as
RandConv [24], Augmix [9]. We train the classifier for a
total of T iterations. At first Twarmup iterations, we train
the classifier without any data augmentation methods. After Twarmup , for each iteration, we randomly initialize theg and update its parameters by adversarial Bayesian training. The learning rate of adversarial learning is η. After
Tadv steps learning, we sample the augmented images via
Bayesian inference and clamp them to the image range. We
then use the augmented images, along with the clean image,to train the classifier f using the classification loss and consistency regularization. The learning rate of the classifieris γ and the weight of consistency regularization is α. The
implementation details of each dataset are in Appendix.

3. Experiments
In this section, we validate our method on two popular style-shift benchmark datasets: (1) Digits is composedof digit images from MNIST-10K [11], MNIST-M [6],SVHN [16], USPS [3], SYNTH [5]. Following the setting in [20], MNIST-10K is the source domain containing 10,000 images from MNIST, and the other four datasets
are target domains. (2) PACS [12] consists of images fromfour domains: photo, art painting, sketch, and cartoon, and7 classes. We select one domain as the source domain andthe other three as the target domains.
We compare our approach against several state-of-theart methods 1 using seven variants. For fair comparison

4. Conclusion
In this paper, we demonstrate how adversarial learning
combined with Bayesian convolutional neural network cangenerate more diverse samples, leading to an improvementin the performance of image classifiers on the single-source
domain generalization task. Our method, ABA, outperforms all existing methods on style shift. The promising
results from this work spark potential future research, suchas exploring whether the Bayesian neural network as a feature extractor can improve SSDG.

1 In Tabs. 1 and 2 we highlight the previous best model in gray, variantsof ABA better than the previous best in blue, and the best accuracy in bold.

4

References[13] Xuanqing Liu, Yao Li, Chongruo Wu, and Cho-Jui Hsieh.
Adv-BNN: Improved adversarial defense through robust
bayesian neural network. In International Conference on
Learning Representations, 2019. 2[14] Christos Louizos, Kevin Swersky, Yujia Li, Max Welling,
and Richard S. Zemel. The variational fair autoencoder. In
Yoshua Bengio and Yann LeCun, editors, International Conference on Learning Representations, 2016. 2[15] Krikamol Muandet, David Balduzzi, and Bernhard
Schölkopf.
Domain generalization via invariant feature representation. In International Conference on Machine
Learning, pages 10–18. PMLR, 2013. 1[16] Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Ng. Reading digits in natural
images with unsupervised feature learning. In Advances in
Neural Information Processing Systems, 01 2011. 4[17] Anh Nguyen, Jason Yosinski, and Jeff Clune. Deep neuralnetworks are easily fooled: High confidence predictions forunrecognizable images. In IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 427–436, 2015.
1[18] Fengchun Qiao, Long Zhao, and Xi Peng. Learning to learn
single domain generalization. In IEEE/CVF Conference on
Computer Vision and Pattern Recognition, pages 12556–
12565, 2020. 1[19] Yunus Saatci and Andrew G Wilson. Bayesian gan. In
Advances in Neural Information Processing Systems, volume 30, 2017. 2[20] Riccardo Volpi, Hongseok Namkoong, Ozan Sener, John C
Duchi, Vittorio Murino, and Silvio Savarese. Generalizing
to unseen domains via adversarial data augmentation. In
Advances in Neural Information Processing Systems, volume 31, 2018. 1, 4[21] Haohan Wang, Zexue He, and Eric P. Xing. Learning robust
representations by projecting superficial statistics out. In International Conference on Learning Representations, 2019.
1[22] Haotao Wang, Chaowei Xiao, Jean Kossaifi, Zhiding Yu, Anima Anandkumar, and Zhangyang Wang. Augmax: Adversarial composition of random augmentations for robust training. In Advances in Neural Information Processing Systems,
volume 34, pages 237–250, 2021. 1, 2
[23] Zehao Xiao, Jiayi Shen, Xiantong Zhen, Ling Shao, andCees Snoek. A bit more bayesian: Domain-invariant learningwith uncertainty. In International Conference on Machine
Learning, pages 11351–11361. PMLR, 2021. 1, 2
[24] Zhenlin Xu, Deyi Liu, Junlin Yang, Colin Raffel, and Marc
Niethammer. Robust and generalizable visual representation
learning via random convolutions. In International Conference on Learning Representations, 2021. 1, 2, 4[25] Ting Yao, Yingwei Pan, Chong-Wah Ngo, Houqiang Li, andTao Mei. Semi-supervised domain adaptation with subspacelearning for visual recognition. In IEEE/CVF Conference
on Computer Vision and Pattern Recognition, pages 2142–
2150, 2015. 1[26] Yabin Zhang, Haojian Zhang, Bin Deng, Shuai Li, Kui
Jia, and Lei Zhang. Semi-supervised models are strong[1] Fabio Maria Carlucci, Antonio D’Innocente, Silvia Bucci,
Barbara Caputo, and Tatiana Tommasi. Domain generalization by solving jigsaw puzzles. In IEEE/CVF Conference
on Computer Vision and Pattern Recognition, pages 2229–
2238, 2019. 1[2] Erik Daxberger and José Miguel Hernández-Lobato.
Bayesian variational autoencoders for unsupervised out-ofdistribution detection. arXiv preprint arXiv:1912.05651,
2019. 2[3] John Denker, W Gardner, Hans Graf, Donnie Henderson, R
Howard, W Hubbard, Lawrence D Jackel, Henry Baird, andIsabelle Guyon. Neural network recognizer for hand-written
zip code digits. In Advances in Neural Information Processing Systems, volume 1, 1988. 4[4] Antonio D’Innocente and Barbara Caputo. Domain generalization with domain-specific aggregation modules. In Pattern Recognition: 40th German Conference, GCPR 2018,
Stuttgart, Germany, October 9-12, 2018, Proceedings 40,
pages 187–198. Springer, 2019. 1[5] Yaroslav Ganin and Victor Lempitsky. Unsupervised domain
adaptation by backpropagation. In International Conference
on Machine Learning, pages 1180–1189. PMLR, 2015. 4[6] Yaroslav Ganin, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario
Marchand, and Victor Lempitsky. Domain-adversarial training of neural networks. The Journal of machine learning
research, 17(1):2096–2030, 2016. 4[7] Tejas Gokhale, Rushil Anirudh, Jayaraman J Thiagarajan,
Bhavya Kailkhura, Chitta Baral, and Yezhou Yang. Improving diversity with adversarially learned transformations fordomain generalization. In IEEE Winter Conference on Applications of Computer Vision, pages 434–443, 2023. 1, 2,
4[8] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun.
Deep residual learning for image recognition. In IEEE/CVF
Conference on Computer Vision and Pattern Recognition,
pages 770–778, 2016. 2[9] Dan Hendrycks*, Norman Mu*, Ekin Dogus Cubuk, Barret
Zoph, Justin Gilmer, and Balaji Lakshminarayanan. Augmix: A simple method to improve robustness and uncertainty
under data shift. In International Conference on Learning
Representations, 2020. 1, 4
[10] David Krueger, Ethan Caballero, Joern-Henrik Jacobsen,
Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi
Le Priol, and Aaron Courville. Out-of-distribution generalization via risk extrapolation (rex). In International Conference on Machine Learning, pages 5815–5826. PMLR, 2021.
1[11] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick
Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
4[12] Da Li, Yongxin Yang, Yi-Zhe Song, and Timothy M
Hospedales. Deeper, broader and artier domain generalization. In IEEE International Conference on Computer Vision,
pages 5542–5550, 2017. 4

5

unsupervised domain adaptation learners.
arXiv:2106.00417, 2021. 1

arXiv preprint

6