## 1 Introduction  
The development of smart manufacturing calls for efficient and effective production, necessitating advanced approaches to ensure the reliable and efficient operation of machinery [1]. To better monitor the condition of machinery and address these challenges, Prognostics and Health Management (PHM) has emerged as a comprehensive approach to assess the health status of systems, predict potential failures, and enable proactive maintenance [2, 3]. PHM has facilitated the transition from time-based or scheduled maintenance to condition-based maintenance (CBM), enabling more efficient and predictive maintenance strategies [4]. PHM plays a central role in CBM by utilizing advanced fault detection techniques to support more reliable and adaptive maintenance practices [5]. Data-driven methods are a key focus in fault detection, forming a vital part of PHM [6]. These methods utilize advancements in machine learning, deep learning, and statistical analysis to analyze large volumes of operational data effectively. By enabling the identification of intricate fault patterns and anomalies through real-time processing, these methods have steadily gained traction [7]. Over time, data-driven process monitoring has gained increasing popularity, gradually evolving into a mainstream methodology that provides a robust framework for continuous fault detection and system health evaluation. Among the various data-driven process monitoring approaches, traditional multivariate statistical methods have played a pivotal role in extracting low-dimensional representations from high-dimensional process data [8]. Methods such as Principal Component Analysis (PCA) [9], Partial Least Squares (PLS) [10], and Canonical Correlation Analysis (CCA) [11] are widely employed to identify process faults by analyzing variable correlations. These techniques form the foundation for systematic fault detection, leveraging data-driven insights to monitor process states effectively. PCA reduces high-dimensional data into a lower-dimensional space while retaining essential variance. PLS focuses on maximizing input-output relationships to improve the prediction accuracy in processes with strong dependency structures. CCA, on the other hand, identifies the most correlated representations between two sets of variables for robust multivariate feature extraction. This method offers unique advantages over PCA and PLS by simultaneously incorporating both input and output data to uncover latent patterns across variable sets [12]. To further enhance its applicability, Kernel CCA (KCCA) [13] extends CCA to model nonlinear relationships by mapping data into higher-dimensional feature spaces through kernel functions. The sparse CCA [14] incorporates sparsity constraints to improve feature interpretability and efficiently manage high-dimensional datasets. Recent advancements have further integrated CCA with deep learning techniques, enabling the modeling of complex relationships in high-dimensional and nonlinear data. Deep Canonical Correlation Analysis (DCCA) [15] uses deep feedforward neural networks to learn nonlinear transformations of paired data views, improving the extraction of correlated features. Despite these advancements, several challenges remain in the application of these methods to real-world industrial processes. Traditional CCA-based methods, while effective in capturing linear relationships, struggle with the complexity of dynamic and nonlinear environments [16]. Kernel learning methods (e.g., the KCCA) can address nonlinearities, but are often unsuitable for industrial big data scenarios due to their computational complexity, as each sample must perform kernel operations with all reference samples [17]. Additionally, their representational power is constrained by predefined kernel functions, and the identification of optimal kernel parameters remains a challenge. Similarly, DCCA and its variants have demonstrated notable success in modeling nonlinear relationships by overcoming several limitations inherent in kernel learning methods.