## 1 Introduction  
With the arrival of modern manufacturing systems, machines have become more automatic and efficient, which has led to increased demands on their reliability, quality, and availability [1, 2]. As a result, machinery fault diagnosis systems, which focus on the detection of health conditions after the occurrence of certain faults, have attracted considerable attention [3, 4]. Additionally, with recent developments in both industry and the Internet, data acquisition has exponentially increased. Thus, fault diagnosis has entered the era of Big Data [5, 6, 7]. Because mechanical big data is typically characterized as large-volume, diverse, and of high-velocity [8], methods of extracting features rapidly and accurately from mechanical big data has become an urgent subject of research [9, 10]. Existing fault diagnosis methods can be divided into two major categories [11]: physics-based models and data-driven ones [12, 13]. Physics-based models overly rely on high-quality domain knowledge and necessitate massive computation costs, which reduces the overall efficiency of fault diagnosis. Thus, it is unsuitable for big data [14, 15, 16]. The data-driven models [15], such as Artificial Neural Networks (ANN) [17, 18], Autoencoders [5], Restricted Boltzmann Machine (RBM) [19], Convolutional Neural Networks (CNN) [20, 21], and \(k\)-Nearest Neighbor [22], depend less on human knowledge and can effectively diagnose faults in mechanical big data. However, these intelligent fault diagnosis methods pose specific challenges, e.g., a difficulty in adjusting various hyperparameters, and good diagnosis accuracy can be obtained only when the hyperparameters are set properly. For example, Autoencoder has four hyperparameters to tune while RBM has six. [PERSON] [23] proposed a sparse filtering method: an unsupervised two-layer network which optimizes the sparsity distribution of the features calculated by the collected data instead of modeling the distribution of the data. It also scales well with the dimension of the input [24]. Only the number of features needs to set; as a result, it is extremely simple to tune and easy to implement. Thanks to this strong performance, sparse filtering has been successfully adopted in several image recognition cases [25; 26; 27]. Recently, sparse filtering also has been introduced into the field of rotary machinery fault diagnosis, delivering excellent performance in feature extraction from complex fault signals. As is well known, [PERSON] et al. [28] first made a constructive attempt to apply sparse filtering to fault diagnosis by a two-stage learning method. As a result, sparse filtering was proved to be an excellent tool for feature extraction. Subsequently, a physical interpretation to sparse filtering was achieved, in which trained weight vectors could be compared directly to the Gabor filters. [PERSON] et al. [29] used sparse filtering to extract multi-domain sparse features and adopted it for fault identification in a planetary gearbox. [PERSON] et al. [30] proposed a multiscale representation learning (MSRL) framework, which was based on sparse filtering, to learn useful features from a direct interpretation of raw vibration signals; the aim was to capture rich and complementary fault pattern information at different scales. [PERSON] et al. [31] combined sparse filtering with a Support Vector Machine optimized by a Improved Particle Swarm to simplify the hyperparameters. However, as shown in [PERSON] et al. [28], when the important parameter of input dimension increased, an initial increase in testing accuracies was followed by a marked decrease (in this paper, the rule is called non-monotonicity). This indicates considerable time is required to optimize the input dimension to increase diagnosis accuracy. To avoid this unnecessary work, the motivation behind non-monotonicity should be clearly explained. Therefore, the interpretation of input dimensions should be studied first.