## 1 Introduction
Methods based on dimensionality reduction do not need prior knowledge of the signal (such as characteristic frequency of the signal), and, at the same time, the parameter setting is less. Useful signal detection based on principal character extraction theory has currently become the research focus of dimensionality reduction and is used in methods such as principal component analysis (PCA) [12] and singular value decomposition (SVD) [13], which are used in image de-noising. The goal of noise reduction based on principal character extraction is to map the noisy signal into the feature subspace and select the features that represent the faulty signal, thus the noise and pure signal are separated successfully. SVD is a significant matrix decomposition method that can extract the main components that represent the useful signal from unknown small stationary or non-stationary signal components, which makes it more successful than other methods for image decomposition [13], dictionary learning [14], and de-noising of electronic noise data [15]. SVD has good stability in noise reduction. The characteristics of the de-noised signal based on SVD are zero phase shift and minimal waveform distortion. SVD gets better de-noising results than normal PCA methods through the bilateral decomposition method. Since 2004, SVD has also been developed for bearing fault signal processing [16; 17; 18]. In the application of de-noising based on SVD, the selection of effective singular values relying on the Hankel matrix is related to the performance of noise reduction and reconstruction of the pure signal [19]. This selection is made according to the difference between the singular value of noise and the fault characteristic signal. In the past, the number of effective singular values was determined through experience or trial-and-error methods. For unknown signals, these methods not only require a large amount of calculation, but also cause greater errors. Some researchers have found that this selection can be better realized by constructing the appropriate spectrum of singular values according to the different states of singular values between the noise and faulty signal. It has been verified that a major turning point appears in the curves of singular value of the pure signal, but none of noise, therefore, this turning point is a boundary to distinguish the useful signal from the noise. Recently, different methods have been proposed to capture this turning point. [PERSON] _et al._[20] put forward a concept of the difference spectrum that can describe the sudden change in status of singular values of a complicated signal. Using the difference spectrum by tracing the maximum peak, the hidden modulation feature caused by gear vibration in the head stock is isolated from a turning force signal. [PERSON] _et al._[21] provided an algorithm to search for the effective singular values based on the maximum peak of the curvature spectrum, which improves the accuracy of the location regarding bearing damage. The same method was used by [PERSON] _et al._ in [16] to distill the position of demarcation; [PERSON] _et al._ in [22] proposed a supervised feature selection algorithm based on SVD-entropy. However, SVD-entropy based methods have a limitation. These methods may not be able to discard even indifferent features having a constant value. However, the ability of providing an appropriate number of singular values based on the methods listed above is reduced under the exceedingly strong noise background (for example, the noise of the mechanical system), because these methods merely emphasize the status of the maximum peak of the curvature spectrum or difference spectrum on the number of effective singular values, which may lead to the loss of important information contained in other peaks [23; 24; 25].