###### Abstract  
The data-driven intelligent fault diagnosis method of rolling bearings has strict requirements regarding the number and balance of fault samples. However, in practical engineering application scenarios, mechanical equipment is usually in a normal state, and small and imbalanced (S & I) fault samples are common, which seriously reduces the accuracy and stability of the fault diagnosis model. To solve this problem, an auxiliary classifier generative adversarial network with spectral normalization (ACGAN-SN) is proposed in this paper. First, a generation module based on a deconvolution layer is built to generate false data from Gaussian noise. Second, to enhance the training stability of the model, the data label information is used to make label constraints on the generated fake data under the basic GAN framework. Spectral normalization constraints are imposed on the output of each layer of the neural network of the discriminator to realize the Lipschitz continuity condition so as to avoid vanishing or exploding gradients. Finally, based on the generated data and the original S & I dataset, seven kinds of bearing fault datasets are made, and the prediction results of the Bi-directional Long Short-Term Memory (BiLSTM) model is verified. The results show that the data generated by ACGAN-SN can significantly promote the performance of the fault diagnosis model under the S & I fault samples. fault diagnosis; small and imbalanced data; ACGAN; spectral normalization 2022 20 July 20221 22 20221 22 20221 22 202  
of rolling bearings. The authors of [11] proposed a performance evaluation method for bearing performance degradation based on a long short-term memory network (LSTM), which accurately predicted the remaining bearing life. [PERSON] et al. [12] extracted non-Euclidean spatial features of vibration signals based on a graph convolutional neural network, which expanded the research ideas of fault diagnosis. Although the abovementioned intelligent diagnosis methods have achieved great success, these methods have a strong dependence on fault data. That is, a large amount of data and labels are necessary to train the model, which is beneficial for fitting nonlinear fault classification functions. Unfortunately, this contradicts the actual situation in industrial scenarios: mechanical equipment is generally in normal working conditions, there are sufficient normal data for research, and rolling bearing failure data are very sparse. In the field of fault diagnosis, S & I data [7] refer to the number ratio between normal data and fault data not equal to 1, and the data volume of the fault samples is very small. S & I data reduce the performance of deep learning models and have attracted the attention of many scholars [13; 14]. At present, the analysis methods of S & I data problems can be divided into two categories: algorithm-based and data-based methods. From an algorithmic point of view, researchers are committed to improving the model structure and classification loss function to reduce the interference of S & I data on classification performance. For example, [PERSON] et al. [15] proposed a nonlinear support tensor machine containing a dynamic penalty factor (DC-NSTM), which optimizes the support vector machine classifier by using the dynamic penalty factor and is applied to the fault diagnosis of unbalanced data of rotating machinery. [PERSON] et al. analyzed the fault diagnosis of weighted loss [16] and focal loss [17] under the imbalanced data of a high-speed train, and the experimental results show that the weighted loss is helpful for the improvement of the diagnosis performance. From the data point of view, the imbalance of the sample data is reduced by augmenting or enhancing the original data to enhance the classification performance of the model, which is represented by the synthetic minority oversampling technique (SMOTE) [18]. SMOTE performs random interpolation sampling in the minority sample data to generate data within the minority sample data distribution.