## 1 Introduction  
Rolling bearings are integral components of high-speed and high-power systems, and are prone to failure due to fatigue, wear, excessive loads, and other factors. Among the variety of bearings available, ceramic bearings are distinguished by their exceptional performance in high-speed environments, which is attributed to their low density, high hardness, resistance to wear, and thermal expansion. These characteristics not only reduce energy loss due to friction, but also enhance the longevity of machinery components, making ceramic bearings an ideal choice for a plethora of advanced engineering applications. Therefore, accurate fault diagnosis is essential to prevent undue operational interruptions and minimize economic losses [1]. [PERSON] et al. [2] introduced a discrete-time model for ceramic bearings considering incipient faults, and proposed a modified observer with enhanced design freedom compared to traditional Luenberger observers. [PERSON] et al. [3] developed an ultra-high-speed hybrid ceramic rolling element triboelectric bearing to enable the real-time monitoring of dynamic behavior and stability. Industrial failures typically manifest as weak periodic shocks in bearing signals, exhibiting non-stationary and nonlinear characteristics that can be obscured by background noise [4]. Therefore, signal processing techniques are crucial for noise reduction in vibration signals. Techniques like empirical mode decomposition (EMD) [5], ensemble empirical mode decomposition (EEMD) [6; 7], wavelet threshold denoising (WTD) [8], and local mean decomposition (LMD) [9] have been applied for noise elimination in vibration signals, proving their efficacy in signal processing. However, EMD and LMD struggle with issues like mode mixing and endpoint effects, while EEMD, although it addresses mode mixing by introducing white noise to each decomposition, does not entirely resolve mode mixing and endpoint challenges. WTD's noise reduction success is highly dependent on the threshold value selection. VMD [10] emerges as a non-recursive, adaptive technique that decomposes signals into specific frequency-centered modes with finite bandwidths, overcoming the limitations of EMD and related methods. [PERSON] et al. [11] enhanced VMD's efficacy in diagnosing bearing faults by adaptively determining the signal modes and optimizing the initial center frequencies. [PERSON] et al. [12] explored VMD's equivalent filtering characteristics and applied their insights to diagnose faults in rotors and stators. [PERSON] et al. [13] advanced the application of VMD by integrating it with a kernel extreme learning machine for enhanced fault diagnosis in rolling bearings. However, VMD's effectiveness is heavily dependent on certain parameters, such as the number of decomposition layers K and the penalty factor \(\alpha\)[14; 15]. As demonstrated above, there is a considerable amount of research on parameter-optimized VMD [16; 17; 18]. Incorrect settings can result in over or under decomposition, highlighting the importance of selecting suitable objective functions and optimizing the parameters for reliable results. The key objective functions in this domain include envelope entropy [19], ensemble kurtosis [20], and envelope spectrum kurtosis [21], which, despite their decomposition efficacy, are sensitive to noise. Correlated kurtosis (CK) [22; 23], which considers the periodicity of bearing fault signals and effectively isolates non-periodic components, represents a potential solution to this problem. Additionally, the correlation coefficient offers insights into signal similarities [24]. Leveraging the strengths of correlated kurtosis and the correlation coefficient, weighted correlated kurtosis (WCK) has been devised as a comprehensive objective function. However, VMD's computational demands and sensitivity to mode count underscore the importance of careful parameter selection and optimization to ensure accurate and reliable fault diagnosis outcomes. To address the challenges posed by VMD, [PERSON] et al.