## 1 Introduction
From the large-scale data of different kinds, low-rank representations can be found using tensor factorization models; e.g., the PARAFAC, Tucker decomposition and linked CANDECOM/PARAFAC models [18]. Further insights into different tensor factorization models and their applications can be traced in [19; 20]. However, few studies are found interlinking both tensor factorization and vibration signals. [PERSON] et al. estimated latent sources from single-channel vibration signal for fault diagnosis with the help of the unmixing feature of tensor decomposition [21]. The tensor was constructed from one-dimensional time series after segmentation. Furthermore, [PERSON] et al. utilized multiple TFDs for tensor construction and presented a non-negative tensor factorization (NTF)-based source separation framework for multi-channel vibration signals in [22]. [PERSON] and [PERSON] proposed the noise reduction model based on non-negative tensor factorization (NTF) for vibration signals of rotary machinery, which was verified efficacious in fault detection through the experiments on gears and bearings [12]. [PERSON] et al. gave consideration to the information loss of transforming a TFD into a vector and proposed the tensor manifold learning algorithm for feature extraction of bearing fault, in which they skillfully projected each TFD on a projection feature vector for manifold learning [23]. Tensors were introduced, yet limitations on the low dimensionality remained. To accurately extract the fault impulses, as well as to make the best use of the advantage of tensor factorization, we present a novel feature extraction finesse for vibration signals based on sparse non-negative tensor factorization (SNTF). Sparseness constraints are imposed based on L1 norms to improve the parts-based representation of the decomposition. The method can be summarized into four major steps: PSR, STFT, tensor decomposition via SNTF, and envelope analysis. Multiple TFD are organized to build a three-way time-frequency tensor. Following this, the hierarchical alternating least squares (HALS) algorithm based on SNTF is introduced to find the optimal factor matrices, based on which different feature subspaces are separated for TFD reconstruction. In the last step of the method, envelope analysis is carried out to extract the characteristic frequency of the impulse fault. Main contributions of this work are summarized as follows: (1) The proposed method made up for the weaknesses of existing fault extraction models, which are incapable or inflexible of dealing with multi-way data; (2) by the virtue of the three-way time-frequency tensor, the SNTF is capable of finding dominant components within multiple time-frequency distributions. The remaining content of the paper is organized as follows. Section 2 begins by laying out the basic unfolding model of NTF with the update rules based on alternating least squares (ALS), and also illustrates the principle of SNTF solved by the HALS algorithm. The procedure of fault feature extraction is thoroughly demonstrated in Section 3. The experiments on the vibration signal of fault bearings, as well as the comparisons to conventional methods, are presented in Section 4. Finally, conclusions are drawn in Section 5.