## 1 Introduction  
Rolling bearings are the foundation and key components of mechanical equipment; their performance is directly related to the health status of the whole equipment, and their operation failure often leads to disasters and accidents [1, 2]. Because of the operating conditions' variability, the interference of multiple vibration sources and the weak impulse characteristics in the fault initiation stage, it is difficult to directly diagnose early faults in many situations, which requires in-depth research on modern diagnostic technology [3, 4, 5, 6]. Machine learning is a reasoning process that simulates human thinking. By effectively obtaining, transmitting and processing diagnostic information, it can simulate human experts and make intelligent judgments and decisions on the operation status and faults of monitored objects with flexible diagnostic strategies. Intelligent diagnosis plays an important role in the field of mechanical fault diagnosis because of its learning function and the ability to automatically obtain diagnostic information for real-time fault diagnosis [7, 8, 9]. In addition, some potentially valuable modern signal processing technologies have been used in rolling bearing monitoring and diagnosis, and a series of practical theoretical and engineering applications have been achieved, effectively improving the service status and predictive maintenance of equipment. Variational mode decomposition (VMD), proposed by [PERSON] and [PERSON] [10], is a unique, innovative mathematical tool that can characterize the input signal as an intrinsic mode component with specific equivalent filtering and sparsity characteristics according to the input settings. However, the most important limitation of the VMD is the required explicit selection of the mode number and balancing parameter in the decomposition, like in EWT but as opposed to EMD [11]. To address the aforementioned shortcomings, scholars have carried out some very valuable research. In the past few years, these studies mainly focused on the independence of mode components and the selection of optimization indicators, among which kurtosis and correlation coefficient are the most widely used in parameter optimization [12; 13; 14; 15]. A representative study is to construct the objective function by considering the kurtosis and correlation coefficient of the envelope power spectrum [16]. The energy criterion was also widely used in parameter selection. The transient state corresponds to the state of energy distribution deviating from the system, which means that the pulse-like frequency characteristics can be measured by energy changes. The varying features of the center frequency of extracted modes are thereby investigated [17; 18], with initial center frequencies, in which the converging U-shape phenomenon is found. In the same direction, an energy-driven VMD method based on spectrum division and the Gini index is proposed for the adaptive decomposition of rolling bearing fault signals [19]. Sparsity is a very useful concept that has become a research hotspot in the field of mechanical signal processing in recent years. Due to the introduction of compressed sensing, sparse diagnosis has aroused the great interest of scholars. Since it can be assumed that this corresponds to mechanical failure rather than normal behavior, the sparsity feature can also be used to find and optimize VMD parameters [20]. However, it is still difficult to characterize the fault attribute only by sparsity since the impulse behavior induced by local damage of bearing is periodic fluctuation and attenuation oscillation. In view of this, some research work has explored the strategy of multiple index fusion [21; 22; 23]. A permutation entropy and orthogonality-based mode number adjusting procedure was conducted by [24]. Ref. [25] presented a parameter extraction method based on the characteristics of the intrinsic mode function.