## 1 Introduction  
A rolling bearing is one of the core components of rotating machinery. Its normal operation is of great significance to the whole mechanical system [1, 2, 3]. Status monitoring and fault diagnosis have always been an important part of maintaining health [4, 5, 6]. The main fault types of rolling bearings include inner race faults, cage faults, rolling element faults, and outer race faults [7, 8, 9, 10, 11]. Because the rolling element is inside the rolling bearing, the transmission path of its fault signal is more complex, which makes the fault information of the rolling element fault signal is more difficult to be extracted [12, 13, 14, 15]. Therefore, the research on the enhancement and the effective separation of fault features is the key step in fault diagnosis. In the past few decades, many scholars have presented various signal processing methods to extract fault features from signals, such as wavelet transform (WT), empirical mode decomposition (EMD), local mean decomposition (LMD), and so on [16, 17, 18, 19, 20, 21, 22, 23, 24]. These methods are applied to better extract fault features from the signal, but they also have some shortcomings. WT needs to early select wavelet basis function before the signal is processed. The EMD and LMD have mode mixing. Although the ensemble EMD(EEMD) alleviates the mode mixing, the number of iterations will increase during signal processing, which results in an increase in the amount of calculation [25; 26; 27; 28; 29; 30; 31]. Therefore, [PERSON] et al. [32] presented a VMD method in 2014. The central frequency and bandwidth of intrinsic mode functions (IMFs) are iteratively updated by solving variational problems to decompose signals into a series of IMFs. This method not only overcomes the mode mixing and end effect but also has suitable robustness. Therefore, the VMD has been extensively applied in fault diagnosis. [PERSON] et al. [33] used VMD to extract bearing fault features. However, the VMD requires to early set the parameter K, and improper selection of \(k\) will affect the decomposition effect [34]. Therefore, the proper selection of K is of great significance for the improvement of VMD. [PERSON] et al. [35] presented a dual-threshold method to obtain the number of decomposition modes and used this improved VMD to extract features of partial discharge signal. [PERSON] et al. [36] presented a fault feature method based on VMD with a stationary wavelet. This method selects K by kurtosis criterion. However, the method using the kurtosis criterion to determine the decomposition mode number is not very accurate in practical application, and under strong noise interference, the VMD is not enough to extract weak fault features. [PERSON] et al. [37] presented MCKD in 2012. This method aims at maximizing the correlation kurtosis to highlight the periodic fault information submerged by noise in the signal. It has a suitable extraction effect on weak fault information. [PERSON] et al. [38] used MCKD to extract the weak gear fault features. However, the deconvolution effect of MCKD is greatly affected by the values of L and T. The adaptive selection of the two parameters will greatly improve the algorithm effect. [PERSON] et al. [39] used the grid search to select two parameters adaptively, and Shannon entropy is used as the evaluation index. This improved MCKD method has a suitable result in processing fault signals of rolling bearings. In addition, the other some feature extraction methods and fault diagnosis methods are also proposed in recent years. [PERSON] et al. [40] presented a new fault feature extraction method based on integrating ensemble empirical mode decomposition, mode selection, and multi-scale fuzzy entropy to accurately diagnose faults. [PERSON] et al. [41] presented a center frequency method of double thresholds to improve the variational mode decomposition (VMD) method for extracting the features of signals. [PERSON] et al.