###### Abstract
The data distribution generated by this method is not the same as the test data distribution and cannot fit the real data distribution. [PERSON] et al. [19] proposed a minority clustering SMOTE algorithm for the problem of unbalanced data synthesis in order to achieve oversampling of the minority data, and the algorithm has been validated in wind turbine blade failures. In addition, [PERSON] et al. [20] proposed the adaptive synthetic technique (ADASYS), which improves the learning of the data distribution by enforcing larger synthetic weights on the minority class samples. Although the above methods can synthesize valid data, these methods are based on self-replication or the interpolation of samples, and do not take into account the distribution and label information of the test data. The Generative Adversarial Network [21] is also a data synthesis algorithm in terms of the data level. Its core idea is an adversarial game: Gaussian noise is input into the generator to generate fake data, and the discriminator is used to identify real data and fake data. When the discriminator reaches a Nash equilibrium, the generated data distribution is considered to be consistent with the true distribution. Different from the methods described above, GAN has no requirement regarding the minimum number of samples and can be used flexibly to generate images and enhance resolution [22]. However, it is difficult for the basic GAN structure to reach the Nash equilibrium. To this end, [PERSON] et.al [23] proposed a deep convolutional generative adversarial network (DCGAN), which greatly improves the model training stability. Additionally, the Wasserstein and Gradient Penalized-based GAN model (WGAN-GP) developed by [PERSON] et al. [24] can address the vanishing gradient problem while improving the training performance. [PERSON] et al. [25] proposed ACGAN using label information, which can synthesize multilabel data at the same time, solving the defect that the unlabeled GAN model can only synthesize single data. Thesemodels greatly improve the possibility of GAN application in engineering and provide new ideas for rolling bearing fault data synthesis and fault diagnosis tasks under S & I data. In recent years, there have been many valuable works on GAN-based fault data synthesis. [PERSON] et al. [26] studied the automatic enhancement technology of mechanical data based on vibration signals and GANs and realized high-precision fault diagnosis. [PERSON] et al. [13] utilized the basic structure of ACGAN and convolutional layers to train the generator, augmented with few-shot fault data. [PERSON] et al. [27] combined the Wasserstein distance and gradient penalty to improve ACGAN and proposed ACWGAN-GP, which was verified in the bearing dataset. The data synthesis and fault diagnosis of an induction motor under unbalanced data are realized by using the DCGAN structure [28]. Although the current GAN-based bearing fault data synthesis research has made good progress, under S & I data, there are still problems of unstable model training and gradient disappearance or explosion. More importantly, the data synthesis methods mentioned above have not been studied under extremely limited sample data, and the amount of data for each fault type in most studies are no less than five. This paper explores the problem of fault diagnosis in the case of unbalanced data calculation and insufficient data volume, assuming that there are enough normal data, and there is only one labeled datum under each fault type. The spectral normalization technique [29] can impose constraints on the parameters of the model to achieve the Lipschitz continuum condition, thereby enhancing the training stability of the model. Therefore, this paper proposes a novel bearing fault method based on an auxiliary classifier generative adversarial network with spectral normalization (ACGAN-SN) under S & I data. This method can realize the synthesis of data under extremely unbalanced samples to use the synthesized data to train the model and realize fault diagnosis.