# SCS-JPGD: Single-Channel-Signal Joint Projected Gradient Descent
## 1 Introduction  
The reliable operation of industrial equipment is crucial for ensuring production efficiency and safety, particularly in the fault diagnosis of critical components such as rotating bodies. Bearing faults are often detected through vibration signals, which typically account for a significant portion of the fault diagnosis in industrial equipment. Bearing vibration signals can reflect their operational status, but in the complex and dynamic industrial environment, traditional fault diagnosis methods that rely on expert knowledge and manual feature extraction struggle to effectively address the noise and nonlinear characteristics inherent in vibration signals. In recent years, deep learning techniques, with their ability to automatically extract features and perform efficient classification, have been widely applied in fault diagnosis, especially in the analysis of bearing vibration signals [1, 2, 3, 4]. However, the security issues of deep learning models have increasingly attracted widespread attention. Specifically, deep learning models are vulnerable to adversarial attacks, leading to two main problems: on the one hand, malicious attacks may interferewith the model's judgment through carefully designed adversarial samples, misguiding the fault diagnosis system and potentially creating safety hazards [5; 6]; on the other hand, the model's inability to accurately recognize even slight perturbations can also impact the reliability of diagnostic results [7; 8]. Therefore, improving the robustness and generalization capability of deep learning models in fault diagnosis has become an urgent research topic. In recent years, deep learning-based fault diagnosis methods have gradually become an important tool for industrial equipment monitoring. These methods, through automatic feature extraction and efficient classification capabilities, have significantly improved the accuracy and efficiency of fault detection. Common deep learning models include Convolutional Neural Networks (CNNs) [1; 2; 3; 4], Long Short-Term Memory networks (LSTM) [3; 9; 10], and Recurrent Neural Networks (RNN) [11; 12], which can effectively learn key features from raw data and, in turn, enhance the accuracy of fault diagnosis. However, the robustness of deep learning models has gradually become a concern when facing the complex and dynamic industrial environment. In practical applications, factors such as noise, sensor faults, data loss, and temperature changes often exist in industrial settings, which can affect the quality and stability of data, making it difficult for traditional deep learning models to guarantee stability and accuracy under these conditions. To address these issues, researchers have proposed several methods to enhance robustness. Data augmentation techniques, by rotating, scaling, and adding noise to the training data, simulate the variable conditions in actual industrial environments, thereby improving the model's adaptability to uncertainty [13; 14]; adversarial training methods, by introducing adversarial samples with intentionally crafted perturbations during the training process, enable the model to learn more robust features and improve its resistance to disturbances [15; 16]; simultaneously, regularization techniques are widely applied in deep learning models to reduce overfitting and improve the generalization ability of the model across different environments [17; 18]. Additionally, techniques combining self-supervised learning and transfer learning have also been increasingly applied to fault diagnosis tasks, further enhancing the model's robustness in small-sample data scenarios and dynamically changing environments. Despite the improvements these methods have made in model robustness, the sensitivity of deep learning models to input data remains a critical issue. Especially when facing extreme disturbances or adversarial attacks, traditional methods still struggle to ensure model stability, potentially leading to severe deviations in fault diagnosis results and even posing safety risks.