## 1 Introduction
However, SE still has some restrictions, such as compromised estimation accuracy when dealing with time series data of high complexity, resulting in inadequate distinguishability [22]. Since sample entropy assesses the similarity between vectors by phase space reconstruction, while solely employing the plane Chebyshev distance to measure the distance of the time series, its outcome is vulnerable to noise interference and mutations. High-dimensional space mapping theory is extensively employed in the field of machine learning pattern recognition [23]. The vectors linearly indistinguishable in the original low-dimensional vector space are discerned through appropriate high-dimensional mapping, suppressing noise information, highlighting effective features, and achieving effective differentiation between vectors. The original planar data space is transformed into a high-dimensional vector space using a high-dimensional mapping function \(\varphi\)(x), which captures the high-dimensional feature representation of the data [24]. This mapping function improves the separability of the sample data in the kernel space and increases the variability among the feature vectors. To accurately assess the similarities and differences between different sequences, the Euclidean distance calculation is employed to reorganize the vector samples in the high-dimensional kernel space. This approach effectively attenuates interfering factors, suppresses noise, and highlights the relevant information by reevaluating the vector distances. Therefore, combined with the sample entropy algorithm, a new high-dimensional mapping entropy (HDME) method is proposed. The HDME method proposed in this study focuses on the problems of noise interference and parameter sensitivity in sample entropy analysis methods. Compared with traditional entropy research, HDME can effectively suppress the noise and highlight the features after mapping in the proper high-dimensional vector kernel function space, thus improving the accuracy and stability of the fault features. The innovation of this paper is to introduce a new high-dimensional mapping method to improve the fault diagnosis effect. Meanwhile, combined with signal pre-processing and intelligent optimization technology, a set of fault feature extraction strategies is formed. Furthermore, this research achieves an accurate diagnosis of faults in reciprocating compressor bearings. The research motivation for this method comes from the in-depth analysis of the traditional sample entropy method. When the vector distance is measured in the planar phase space, the sample entropy is easily affected by the noise disturbance. In this paper, the theory of high-dimensional mapping is applied to the process of entropy feature extraction, and a new method, HDME, is formed. Another major innovation of this approach is that it combines the simplicity of linear processing with the power of non-linear mapping. In addition, this thesis develops multi-scale high-dimensional mapping entropy (MHDME) and refined composite multi-scale high-dimensional mapping entropy (RCMHDME). We examine the application of the proposed HDME under the presumption that it possesses strong noise resistance and differentiation capabilities. The primary objectives of this research are outlined as follows: First, we propose a method aimed at enhancing the stability of signal complexity descriptions and increasing the accuracy of data separability. Second, we address the limitations inherent in single-scale complex datasets. Third, we formulate a novel fault feature extraction strategy by amalgamating signal preprocessing with an intelligent preference technique. Lastly, we apply the proposed fault diagnosis methodology to the bearings of reciprocating compressors. In this study, we focus on the entropy calculation of high-dimensional vector spaces. In Section 2, we review the sample entropy approach to present the idea of high-dimensional mapping entropy.