Multi-view Clustering Based on Deep Non-negative Tensor Factorization

Venue: acmmm_org_ACMMM_2024_Conference
Published: 2024-04-08 04:58:08

Abstract: Multi-view clustering (MVC) methods based on non-negative matrix factorization (NMF) have gained popularity owing to their ability to provide interpretable clustering results. However, these NMF-based MVC methods generally process each view independently and thus ignore the potential relationship between views. 
Besides, they are limited in the ability to capture the nonlinear data structures. 
To overcome these weaknesses and inspired by deep learning, we propose a multi-view clustering method based on deep non-negative tensor factorization (MVC-DNTF). With deep tesnor factorization, our method can well exploit the spatial structure of the original data and is capable of extracting more deep and nonlinear features embedded in different views. To further extract the complementary information of different views, we adopt the weighted tensor Schatten $p$-norm regularization term. An optimization algorithm is developed to effectively solves the MVC-DNTF objective. Extensive experiments are performed to demonstrate the effectiveness and superiority of our method.

Main paper: 
 introduction: 
 In recent years, the advancement of information technology has led to an exponential increase in data generated across various domains. This data usually comes from different sources [1] . Multiview data processing have demonstrated significant applications in scenarios such as image processing, social network analysis, and bioinformatics. As an unsupervised learning method, multi-view clustering gains increasing attention due to its ability to utilize the multiple data views to improve the clustering performance by exploiting the consistent and complementary information among them [15] . So far, numerous multi-view clustering methods are proposed, which can be roughly divided in four categories [41] : Cotraining based methods [20] , multi-kernel learning [34] , multi-view graph clustering [39] , and multi-view subspace clustering [45] . Among these methods, the co-training approach [22] leverages the co-training framework to ensure consistency in either prior knowledge or clustering outcomes. Furthermore, Tzortzis et al [34] proposed a multi-view clustering technique based on kernel functions, which weights all views using kernel functions. This approach assigns different weights to various views, achieving improved clustering results. Pan et al [30] proposed a generalized framework for clustering multi-view attribute graph data and introduced the Multi-view Comparison Graph Clustering (MCGC) method, which enhances graph clustering by filtering noise and learning consensus graphs with contrast loss.Additionally, some approaches focus on individually optimizing each view to identify the most informative subspaces, enabling clustering within these subspaces. This strategy enhances the handling of high-dimensional data and noise, thereby improving the robustness and effectiveness of clustering algorithms [6, 35] .Notably, non-negative matrix factorization (NMF) has received widespread attention owing to its outstanding clustering performance and interpretability [8, 23] , and many improved methods were developed based on NMF [3, 7, 9, 10, 37, 38] , etc. For example, Cai et al [4] proposed a graph-regularized NMF (GNMF) method, which mines the flow structure of NMF data for effective clustering. Its effectiveness motivates researchers to extend NMF to multi-view NMF [46] . For example, Ou et al [29] introduced NMF into multiview clustering by considering the geometric structure of each view to mine the potential information of each view; Semi-supervised multi-view clustering method based on NMF was proposed in [28] , which adopts semi-supervised approach for clustering, and the effectiveness of the method is verified by several sets of experiments. Effective as they, these methods are incapable of extracting nonlinear features.To overcome this weakness, Zhao et al. proposed a multi-view clustering via deep matrix factorization, which fully explores the potential attributes of the data in a layer-wise fashion and explore the geometrical structure with a graph-based regularizer [43] . Similarly, Zhang et al. developed a deep NMF based multi-view clustering method with partition alignment (MVC-DMF-PA), which employs deep NMF to learn the clustering partition matrix for each view and then fuse them into the final partition result with a partition algiment [42] . These deep NMF-based MVC methods exhibit impressive feature extraction ability. Besides, they realize better efficiency when compared with other MVC methods (e.g., graph-based methods or deep methods). Nonetheless, they essentially apply NMF on each single view and then integrate them into a consistent clustering assignment matrix, resulting in the the loss of the original inter-view spatial information [24] . For another, since they directly decompose the original data, their efficiency decreases dramatically when the original data dimensions are large.Kilmer et al. introduced the factorization of nonnegative tensors [21] , which has been proven to be an effective to learn the inter-view information from multi-view data and numerious tensorial MVC methods are developed [24, 27, 31, 44] . Inspired by their impressive performance, we in this paper extend deep NMF for multi-view clustering to deep non-negative tensor factorization (DNTF) and propose a novel multi-View clustering with deep tensor factorization (MVC-DTF). MVC-DTF not only takes into account the differences and complementarities between views but also enables the learning of deeper representations of the data through the deep tensor factorization method, so as to capture and utilize the complex structural information in the multi-view data in a more effective way. In addition to this, we use weighted tensor Schatten 𝑝-norm to describe the clustering structure of the view data, which fully utilizes the complementary information between views and avoids the problem that the use of Schatten 𝑝-norm in [24] leads to the same contribution of singular values across views. To further improve the efficiency, we perform tensor factorizaton on the anchor graph instead of the original data tensor. The main contributions of this paper are as follows:• We extend deep non-negative matrix factorization to deep tensor factorization , based on which we propose a novel multi-view clustering method, called MVC-DTF. By further introducing a Schatten 𝑝-norm based regularizer, MVC-DTF can better exploit the inter-view spatial relationship and the complementary information of differentviews. • We adopt anchor graph to construct the tensor to be factorized to reduce the computational complexity. Besides, we develop an optimization algorithm to solve the MVC-DTF optimization problem.• Extensive experiments are conducted on several multi-view datasets to evaluate the prformance of our method, and the comparison with existing MVC methods demonstrate its effectiveness and superiority. 
related work: 
 In this section, we introduce the concepts and formulas of multiview clustering using non-negative matrix factorization and nonnegative tensor factorization respectively, which is tightly related to our method.NMF has become an important feature dimensionality reduction tool in clustering. NMF provides an intuitive way to explore the low-dimensional structure of the data by decomposing the nonnegative data matrix X into a weight matrix W and an indicator matrix H. Its formula is expressed as follows:minH,W ∥X -HW∥ 2 𝐹 , s.t. W ≥ 0, H ≥ 0 (1)where X ∈ R 𝑛×𝑑 , H ∈ R 𝑛×𝑘 and W ∈ R 𝑘 ×𝑑 . Furthermore, Semi-NMF improvies NMF by relaxing the non-negative constraint of W, which has been proven to be a relaxation of the K-means clustering approach [10] . Specifically, Semi-NMF can be expressed as follows:minH,W ∥X -HW∥ 2 𝐹 , s.t. H ≥ 0 (2)Kilmer et al [21] extended NMF to third-order tensors. Specifically, they present a new third-order tensors representation and introduce a closed tensor multiplication operation, based on which they defined tensor transpose, inverse, and unit tensor.Definition 1 ( [21] ). For a 3-way tensor 𝒜 ∈ R 𝑛 1 ×𝑛 2 ×𝑛 3 , the Frobenius norm of 𝒜 is defined as ∥𝒜∥ 𝐹 = √︃ 𝑖 𝑗𝑘 𝑎 𝑖 𝑗𝑘 2 , and theconjugate transpose of 𝒜 ∈ R 𝑛 1 ×𝑛 2 ×𝑛 3 is 𝒜 𝑇 ∈ R 𝑛 2 ×𝑛 1 ×𝑛 3 .). For a 3-way tensor 𝒜 ∈ R 𝑛 1 ×𝑛 2 ×𝑛 3 , we denote A as a block diagonal matrix with each block on the diagonal as the frontal slice A (𝑖 ) of 𝒜. A has the following form:EQUATIONA. . . [21] ). : The 𝑡-product between two 3 -order tensors with matched dimensions, ℳ ∈ R 𝑛 1 ×𝑚×𝑣 and 𝒩 ∈ R 𝑚×𝑛 2 ×𝑣 , is defined asA (𝑛 3 )            Definition 3 (T-Productℳ * 𝒩 ∈ R 𝑛 1 ×𝑛 2 ×𝑣 , i.e., ℳ * 𝒩 = ifft(bdiag(𝑀𝑁 ))where ℳ is the discrete Fourier transform of ℳ along the third dimension, ℳ = fft(ℳ, [], 3) , 𝑀 = bdiag(ℳ) and bdiag(•) denotes the block diagonal matrix.  
the proposed method: 
 In this section, we first present the motivation for our proposed scheme. After that, we present the details of our proposed multiview clustering scheme based on deep tensor factorization and the optimization method of the scheme. Finally, we summarize our scheme and give the time complexity analysis of the scheme. We show some key notations used in the scheme in Table 1 for easier presentation.In recent years, tensor factorization has emerged as a powerful tool for multi-view data processing, since it can better explore The F-norm of tensor 𝑀 𝒮 ∈ R 𝑛×𝑚×𝑣 Data tensor after anchor processingℋ 𝑖 ∈ R 𝑛×𝑙 𝑖 ×𝑣 Indication tensor for layer 𝑖 𝒲 1 ∈ R 𝑙 1 ×𝑚×𝑣Weight tensor for layer 1𝒲 𝑖 ∈ R 𝑙 𝑖 ×𝑙 𝑖 -1 ×𝑣Weight tensor for layer 𝑖 > 1𝜓 𝑖 Short for 𝒲 𝑖 -1 * 𝒲 𝑖 -2 * . . . * 𝒲 1 ∥𝒳 ∥ 𝝎,𝑆 𝑝 Weighted tensor Schatten 𝑝-norm ℳ, 𝒩 Auxiliary Variables 𝒴 1 , 𝒴 2Lagrange multipliers the relationship between different views and the spatial structure within the view, which provides the possibility to deal with the correlation between multiple views. Similar to NMF, the objective function of tensor factorization can be described as follows:minEQUATIONwhere 𝒳 ∈ R 𝑛×𝑑 ×𝑣 is the original data tensor obtained by combining the data matrices of the different views. ℋ ∈ R 𝑛×𝑘 ×𝑣 is the cluster indicator tensor. 𝒲 ∈ R 𝑘 ×𝑑 ×𝑣 is the cluster weight tensor. This fusion of data from different views into a tensor for factorization can naturally leverage the relationships between views [24] . Fig. 1 depicts the construction of tensor by taking ℋ as an exmple. However, despite tensor factorization's theoretical superiority, its ability to handle nonlinear structures and learn deep representations is still limited. Inspired by deep learning, this paper presents an innovative deep tensor factorization framework to overcome the above challenges. By incorporating the concepts of deep learning into the tensor factorization process, our approach is able to not only capture the multidimensional structure of data but also learn deep and nonlinear representations of data.minEQUATIONThe deep tensor factorization method enables us to more effectively process multi-view data by extracting the deep and potential correlations between different views. The following experiments section proves that our method can significantly improve the performance of multi-view clustering.Considering that directly performing deep tensor factorization on the original data matrix incurs high computational complexity, we choose to construct a anchor graph for each view, and a anchor tensor 𝒮 ∈ R 𝑛×𝑚×𝑣 constructed from the anchor graphs of different views to replace the original data tensor 𝒳 ∈ R 𝑛×𝑑 ×𝑣 as the input to the tensor factorization, and 𝑚 is the number of anchors. Since the number of anchors is much smaller than the original data dimension (i.e., 𝑚 << 𝑑), it can effectively reduce the computational overhead.To further exploit the complementary information of different views, we further minimize the tensor schatten 𝑝 norm of the tensorial form of the cluster indicator matrix, and Eq. ( 4) becomes: minEQUATIONThe second term of objective function is regularization term based on weighted tensor Schatten 𝑝-norm of ℋ 𝑙 , and 𝜆 is a balancing hyperparameter. The weighted tensor Schatten 𝑝-norm is defined as follows Definition 4 (Weighted tensor Schatten 𝑝-norm [16] ). Given 𝒳 ∈ R 𝑛 1 ×𝑛 2 ×𝑛 3 , Weighted tensor Schatten 𝑝-norm of 𝒳 is defined asEQUATION)where 0 ≤ 𝑝 ≤ 1, 𝜎 𝑗 (𝒳 (𝑖 ) ) is the 𝑗-th singular value of 𝒳 (𝑖 ) , 𝜔 𝑗 is the weight of 𝜎 𝑗 (𝒳 (𝑖 ) ).It is worth noting that the aim of weighted tensor Schatten 𝑝norm for the regularization is to avoid the problem of all singular values contributing equally in ordinary tensor Schatten 𝑝-norm, which provides better robustness to data noise or outliers. Based on [27] , we update the weights by𝜔 𝑘 𝑗 𝑡 +1 = 1 𝛿 𝑝 𝑗 X 𝑘 𝑓 𝑡 +𝜀, where 𝑡 and 𝜀 denote the number of iteration and a small constant.Before optimizing the objective equation, following the work on deep non-negative matrix factorization for multi-view clustering [42, 43] , we first initialize layer after layer of views. We first get 𝒲 (𝑣)1 by 𝒮 (𝑣) = ℋ (𝑣) 1 * 𝒲 (𝑣)1 and then factorize ℋ(𝑣) 1 = ℋ (𝑣) 2 * 𝒲 (𝑣)2 layer by layer up to 𝒲 𝑙 . Besides, We initialize 𝒴 1 = 𝒴 2 = 0 and 𝑁 as the unit matrix.We use the Augmented Lagrange Multiplier (ALM) to optimize the objective Eq. ( 5), To perform an alternating direction minimization strategy, we introduce two auxiliary tensor variables ℋ 𝑙 = 𝒩 , ℋ 𝑙 = ℳ where 𝒩 ⩾ 0 are used, and then, the objective function can be reformularized to the following unconstrained optimization problem: EQUATIONEQUATIONwhere the tensors 𝒴 1 ∈ R 𝑛×𝑘 ×𝑣 and 𝒴 2 ∈ R 𝑛×𝑘 ×𝑣 are Lagrange multipliers and 𝜇, 𝜌 are the penalty parameters. The optimization process can therefore be separated into five steps: 𝒲 𝑖 -subproblem: By fixing 𝒲 1 , 𝒲 2 , . . . , 𝒲 𝑖 -1 , 𝒩 , ℋ 𝑖 , ℳ, the optimization Eq.( 7) becomes:EQUATION)where𝜓 𝑖 -1 = 𝒲 𝑖 -1 * 𝒲 𝑖 -2 * . . . * 𝒲 1 .First, implement the discrete Fourier transform (DFT) along the third dimension, the equivalent representation of Eq.( 8) in the frequency domain becomes:min 𝑉 ∑︁ 𝑣=1 ∥𝒮 (𝑣) -ℋ 𝑖 (𝑣) 𝒲 𝑖 (𝑣) 𝜓 𝑖 -1 (𝑣) ∥ 2 𝐹where 𝒲 = ftt(𝒲, [], 3), and the others in the same way. So Making the equation zero gets the solution of Eq.( 8)is:EQUATIONwhere ℋ 𝑙(𝑣) †denotes MP inverse of ℋ 𝑙 (𝑣) [10] . ℋ 𝑖 -subproblem: By fixing 𝜓 𝑖 , 𝒩 , ℳ,the optimization Eq.( 7)becomes:min ∥𝒮 -EQUATION)Same as 𝒲 𝑖 . First, perform the DFT of the equivalent representation of Eq. (10) in the frequency domain becomes:min 𝑉 ∑︁ 𝑣=1 ∥𝒮 (𝑣) -ℋ 𝑖 (𝑣)𝜓 𝑖 (𝑣) ∥ 2According to [10] the update rule for ℋ 𝑖 (𝑣) (𝑖 < 𝑙) isEQUATIONWhere[𝐴] + = (|𝐴| + 𝐴)/2, [𝐴] -= (|𝐴| -𝐴)/2.ℋ 𝑙 -subproblem: By fixing 𝒩 ,𝜓 𝑙 , ℳ Eq.( 7) becomes: minEQUATIONEQUATION) Eq.( 12) can be reduced to maxEQUATIONwhereEQUATION𝜇 and 𝒬 (𝑣)2 = ℳ (𝑣) - 𝒴 (𝑣) 2According to [24] , the solution of Eq.( 13)isEQUATION)where𝚲 (𝑣)and 𝑽(𝑣) can be obtained by SVD ℬ(𝑣) = 𝚲 (𝑣)X 𝑽(𝑣) T 𝒩 -subproblem: By fixing ℋ, 𝒢, ℳ. Eq.( 7) becomes:EQUATIONAccording to [40] , the solution of Eq.( 15) is:EQUATIONℳ-subproblem: By fixing 𝒩 , ℋ, 𝒢. Eq.( 7) becomes:min 𝜆∥ℳ∥ 𝜔,𝑆𝑝 + 𝜌 2 ℋ -ℳ + 𝒴 2 𝜌 2 𝐹A simple transformation gives usEQUATIONwhere 𝒵 = 𝒴 2 𝜌 -ℳ. According to [17] , the following lemma can give the solution of Eq.( 17)Lemma 1. Let 𝒵 ∈ R 𝑛 1 ×𝑛 2 ×𝑛 3 satisfy an increasing sequence of 0 ≤ 𝜔 1 ≤ 𝜔 2 ≤ • • • ≤ 𝜔 min and have a t-SVD 𝒵 = 𝒰 * 𝒮 * 𝒱 𝑇 .For the following optimization model:EQUATIONa global optimal solution to the 18 is 𝜎 𝑖 (𝒵), 𝜏 * 𝜔 𝑖 ), the introduction of GST can be found in [17] .𝒳 * = Υ 𝜏 * 𝜔 (𝒵) = 𝒰 *Through Lemma1, we get the solution of Eq.( 17) asEQUATIONThe optimization of our multi-view clustering method based on deep tensor factorization can be summarized as Algorithm 1. It is worth noting that we need to optimize ℋ 𝑖 and ℳ 𝑖 layer by layer before obtaining ℋ 𝑙 .To simplify the analysis, we set the dimension of each layer to be uniformly 𝑑 and assume that the number of iterations of the Initialization process and training process are 𝑡 𝑝𝑟𝑒 and 𝑡 𝑡𝑟𝑎𝑖𝑛 . The computational complexity of constructing the anchor data tensor from the original data tensor is 𝒪(𝑣𝑛𝑚𝑑 + 𝑣𝑛𝑚𝑙𝑜𝑔(𝑚)). The computational complexity of is 𝒪(𝑣𝑡 𝑝𝑟𝑒 𝑛𝑚𝑑 2 𝑙) after approximate simplicity. The computational complexity of the update process for 𝒲 𝑖 , ℋ 𝑖 , ℋ 𝑙 , ℳ, 𝒩 are 𝒪(𝑣𝑙𝑡 𝑡𝑟𝑎𝑖𝑛 (𝑑𝑛𝑚+𝑑 2 𝑚+𝑑 2 𝑛)), 𝒪(𝑣𝑙𝑡 𝑡𝑟𝑎𝑖𝑛 𝑑 2 𝑛+ 𝑣𝑙𝑡 𝑡𝑟𝑎𝑖𝑛 𝑑𝑛𝑚),𝒪(𝑡 𝑡𝑟𝑎𝑖𝑛 𝑣 (𝑚 2 𝑘+𝑚𝑘 2 )), 𝒪(𝑡 𝑡𝑟𝑎𝑖𝑛 𝑣𝑛𝑘), 𝒪(𝑡 𝑡𝑟𝑎𝑖𝑛 (2𝑣𝑛𝑘𝑙𝑜𝑔(𝑣𝑘)+ 𝑣 2 𝑘𝑛)). As these parameters are small constants and 𝑚 < 𝑛, 𝑘 < 𝑑,input: multi-view data {X (𝑣) } 𝑉 𝑣=1 ; anchor number 𝑚; cluster number 𝑘; number of layers and dimension for each layer; output: Cluster labels Y of each sample. Update 𝒩 (𝑣) by solving Eq.( 16). Update ℳ by using Eq.( 19).12:Update 𝒴 1 , 𝒴 2 , 𝜇 and 𝜌: 10 13 ), 𝜌 = min(𝜂𝜌, 10 13 ). 13: end while 14: Calculate the 𝐾 clusters by using H = 𝑉 𝑣=1 ℋ (𝑣) /𝑉 . 15: return Clustering result (In the resulting indicator matrix, the index of the largest element in each row corresponds to the cluster label of the respective sample. ).𝒴 1 = 𝒴 1 + 𝜇 (ℋ -𝒩 ), 𝒴 2 = 𝒴 2 + 𝜇 (ℋ -ℳ), 𝜇 = min(𝜂𝜇,and in general 𝑛 is the largest number among them, the final computational complexity of our proposed scheme is𝒪(𝑣𝑡 𝑝𝑟𝑒 𝑛𝑚𝑑 2 𝑙 + 𝑣𝑙𝑡 𝑡𝑟𝑎𝑖𝑛 𝑑𝑛𝑚 + 𝑡 𝑡𝑟𝑎𝑖𝑛 𝑣𝑚 2 𝑘) 
experiments 4.1 dataset: 
 We evaluate the performance of the proposed method on eight widely adapted multi-view learning benchmark datasets, which are 3-sources, BBCSport [18] , HW [12] , Sonar [32] , Yale, Vehicle Sensor [11] , Caltech-5V [14] ,SentencesNYU v2(RGB-D) [33] . Details of these datasets are shown in Table 2 .All experiments were conducted on a desktop computer equipped with a 13th Gen Intel(R) Core(TM) i5-13400 processor at 2.50 GHz, and 32 GB RAM. The experiments were executed using MATLAB 2023a (64-bit) as the primary software environment. For experimental setup, we have selected the following nine representative multiview clustering algorithms to compare with our proposed method: Two multi-view clustering methods based on graph learning GMC [36] and UDBGL [13] , Three sub-space multi-view learning methods DiMSC [5] , MvLRSSC [2] and RMSL [26] , a fast calculation method FastMICE [19] and two non-negative matrix factorization methods MvDGNMF [25] and MVC-DMF-PA [42] , and a method for orthogonal non-negative tensor factorization Orth-NTF [24] .Before performing clustering, for all the methods including the proposed method and all the comparison methods, we first preprocess the data of all the datasets, i.e., we normalize the data of different views. For the method we proposed, 𝜆 and 𝑝 are hyper-parameters.According to [42] , when performing a two-layer tensor factorization the layer size is [𝑙 1 , 𝑘],where 𝑙 1 is chosen from [4𝑘, 5𝑘, 6𝑘], when performing a three-layer tensor factorization the layer size is [𝑙 1 , 𝑙 2 , 𝑘], where 𝑙 1 is chosen from [8𝑘, 10𝑘, 12𝑘] and 𝑙 2 is chosen from [4𝑘, 5𝑘, 6𝑘] and so on. We used three widely used metrics to evaluate clustering performance i.e., 1) Accuracy (ACC); 2)Normalized Mutual Information(NMI); 3) Purity(PUR). For all the metrics mentioned previously, a higher value indicates better clustering outcomes. For each experiment to avoid the effect of random initialization, we repeated each experiment 10 times.The experimental results in Tables 3 and 4 demonstrate the performance of our method MVC-DNTF on eight datasets and are compared with several benchmark algorithms. The best results are bolded and the second best results are underlined. The main results of our analysis of these results are as follows:(1) Our method performs better than the benchmark methods on most of the datasets. We believe that this superior performance is largely attributable to the unique design of our proposed method, which performs deep factorization based on a data tensor consisting of multi-view data anchor graphs. This design allows our method to effectively utilize spatial structure information and depth and hidden features from different views. Meanwhile the imposition of orthogonal and non-negative constraints enhances the interpretability of our clustering, and each row of the metrics matrix for each view is explicitly mapped to a cluster without the need for post-processing for label determination. (2) Compared with DMVC and AwDMVC methods that rely on the deep semi-NMF framework, our method achieves superior results. This demonstrates that the tensor factorization method we adopt better preserves the structural information of the data in multiple dimensions and effectively captures the complementary information and intrinsic connections among different views. Compared to Orth-NTF, which also employs tensor factorization, our method also achieves superior performance, which suggests that our proposed deep tensor factorization can automatically learn deeper feature representations from multi-view data, and thus better capture the essential properties and intrinsic structure of the data. In conclusion, the experimental results highlight the effectiveness of our multi-view clustering method based on deep tensor factorization.We performed convergence experiments on four different datasets. These experiments were aimed at observing the evolution of the clustering accuracy of the algorithm and the convergence behavior of the algorithm's objective function during successive iterations. It is noteworthy that the algorithm converges rapidly after 60 iterations, with a significant decrease in the difference between the objective matrices ℋ 𝑙 , ℳ, and 𝒩 . At the same time, the clustering accuracy improves dramatically, which is consistent with the convergence of the objective values. This proves the effectiveness of our proposed method since the convergence of the objective function is equivalent to the improvement of the clustering performance.Our proposed algorithm depends on three key variable parameters: the number of anchors 𝑚, 𝜆 and 𝑝. 𝜆 and 𝑝 regulate the sparsity and orthology of the potential representation ℋ 𝑙 . The hyperparameter 𝜆 plays an important role in the trade-off between the accuracy show that 𝜆 has a significant effect on the efficacy of the algorithm, probably due to its key role in preventing overfitting through regularization, while 𝑝 has a relatively stable effect on performance, but tends to perform better when 𝑝 takes smaller values than when 𝑝 takes larger values. Empirical results from various datasets indicate that the best results are obtained when 𝜆 is calibrated in the middle range 10 to 100 and 𝑝 is taken to be 0.1. This range allows the algorithm to achieve sufficient sparsity for generalization without compromising the ability to capture important data structures.We set the anchor point rate from 0.1 to 1 with a step size of 0.1, and conduct experiments on the four datasets 3-sources , BBCSport, Yale , RGB-D to test the effect of anchor point rate on the clustering results. As shown in Fig. 4 , we can find that the clustering performance is improved with the increase of the number of anchors, but the anchor rate has little effect on the experimental results after the anchor rate is greater than 0.3. The four datasets achieve the best clustering results on 1.0, 0.9, 1.0, 0.8 respectively.We primarily focused on exploring how different depths of deep tensor factorization affect clustering performance for multi-view data. We executed numerous experiments using eight benchmark datasets as described in Section 4.1. These experiments varied the number of layers in the models to examine their influence on clustering results. The depths ranged from one to four layers, with configurations from [𝑘] to [𝑙 1 , 𝑙 2 , 𝑙 3 , 𝑘]. This setup allowed us to evaluate the effectiveness of deep tensor factorization at various layer depths. Results detailing the experimental accuracy rates are presented in Table 5 . However, overfitting may occur as the number of factorization layers increases . On the Caltech-5V dataset, the accuracy using two layers was essentially the same as when using only one layer , increasing to three layers the accuracy went up but then declined when going to four layers and the same happens on the BBCSport and HW datasets.On the Vehicle Sensor and Sonar datasets, the performance improvement with increasing layers is not significant, which we attribute to the simpler structure of these datasets, where the accuracy of the basic tensor factorization is already close to 100, and thus it is difficult to achieve a large improvement in the deeper factorization.In general, the deep tensor factorization method significantly improves the experimental results, which proves that our proposed multi-view clustering method with deep tensor factorization can effectively extract the deep and hidden features of the data.Besides, We conducted ablation experiments on deep tensor factorization and weighted tensor Schatten 𝑝-norm in Table 6 . case1 indicates whether deep tensor factorization or single-layer tensor factorization was used. due to the lack of consistency constraints of the tensor factorization methods alone, it is difficult to learn consistent cluster representation matrices, and the effect is poor and difficult to compare, so we set case2 as the use of the weighted tensor Schatten 𝑝-norm or the use of the ordinary Schatten 𝑝norm.It can be found that the weighted Schatten 𝑝-norm is overall superior to the Schatten 𝑝-norm because the weighted Schatten 𝑝norm capitalizes on certain important information corresponding to larger singular values while portraying complementary information from different views [27] .  
 
 Appendix:  


Decision: In the paper, the authors proposed a Deep Non-negative Tensor Factorization technique for multi-view data clustering task. The paper is well-structured and the presented multi-view deep tensor factorization approach is interesting and innovative. The experiments are also sufficient and convincing. The paper also receives four positive recommendations (A, A, WA, WA). Therefore, based on the recommendations of reviewers and the novelty of the paper, the paper can be accepted and presented on ACM MM.

Reviews: review: paper_summary: paper_summary: The paper presents a tensor-based multi-view clustering method termed MVC-DTF, which is derived from the non-negative matrix factorization based multi-view clustering (MVC-NMF). Nevertheless, it outperforms MVC-NMF since it can process multi-view data together within a tensor, rather than process each view independently. Thus it could better reveal the relationship between different views. Weighted tensor schatten-p norm and anchor graph are also employed to improve clustering performance and computational efficiency, respectively. An optimization to solve MVC-DTF is developed. The paper also includes a series of experimental and comparison results.
strength_weakness: strength_weakness: (1) Novelty: The idea of multi-view clustering based on deep non-negative tensor factorization is novel which employs deep tensor factorization to learn the deep and nonlinear features and simultaneously process all the views together in the form of tensor. 
(2) Sufficient Contribution: Several contributions are made in this work to multi-view clustering. The utilization of deep non-negative tensor factorization is useful in promoting the clustering performance. Besides, the authors further introduce weighted tensor schatten-p norm based term to explore the complementary inter-view spatial information for better clustering performance, and take advantage of anchor for better efficiency. 
(3) Technically soundness: The objective and optimization of the proposed MVC-DTF is obtained via a series of theoretical derivation. 
(4) Experiments: Various experiments are illustrated in this paper, including clustering performance evaluation and comparison, parameter sensitivity test, and the ablation study, which evaluates the MVC-DTF from different views.
limitations: limitations: (1) Typos: There exist some typos in the paper. For example, some symbols in Table 1 seem not proper. The matrix M in Table 1 should be in bold, which should be corrected as \bf{M}. Besides, it is recommended to use a consistent term of MVC-DNTF or MVC-DTF throughout the paper.
(2) Method: the description of the method might benefit from providing more details. Some derivation is somehow simple, and some details should be added. For example, Eq. 12 (in line 397 of page 4) is reduced to Eq. 13, but the authors did not give the reason. Therefore, this part should be enriched to help readers to learn the paper better.
paper_summary: paper_summary: This paper proposes a multi-view clustering method based on deep non-negative tensor factorization (MVC-DNTF). The approach aims to overcome limitations in existing non-negative matrix factorization methods by capturing nonlinear data structures and inter-view relationships. It introduces a weighted tensor Schatten p-norm regularization, which enhances the clustering results to effectively leverage the complementary information from different views. A deep non-negative tensor factorization method is developed to learn the deep information. The integration of the weighted tensor Schatten p-norm regularization and deep non-negative tensor factorization contributes to the improvement of the multi-view clustering performance. A comprehensive experiment is designed that evaluates the clustering performance of the work as well as the contribution of each component.
strength_weakness: strength_weakness: The paper is well-organized and presents an innovative idea of multi-view deep tensor factorization. Besides, a comprehensive experimental analysis is designed. In summary, its advantages include:

(1) The paper exhibits good organization and readability, with a well-structured presentation of the research. The literature review is relatively comprehensive, providing a sufficient overview of the relevant works in the field of multi-view clustering.

(2) The paper contributes by integrating deep non-negative Matrix Factorization concepts into non-negative tensor factorization, effectively capturing complex and nonlinear relationships within multi-view data. In this way, it effectively tackles limitations of conventional NMF-based methods, achieving improved clustering performance through deep feature extraction. Besides, the weighted tensor schatten-p norm works as a low-rank constraint to further enhance the clustering performance.

(3) The experimental results demonstrate the superiority of the proposed method over several state-of-the-art clustering algorithms across multiple datasets, thus validating its effectiveness for practical applications.
limitations: limitations: The paper confronts several limitations in writing and paper formatting, as summarized below:

(1) The writing would benefit from further improvement, particularly by simplifying lengthy sentences that can be difficult to understand. Additionally, there are a few typos that need to be addressed. In line 883 of Page 8, “case 1” should be corrected as “Case 1”.

(2) It is advisable for the authors to pay more attention to the formatting issue. Specifically, in Line 462, the formula extends beyond the boundary and requires adjustment to ensure proper display and readability.
paper_summary: paper_summary: The paper introduces a multi-view clustering method based on deep non-negative tensor factorization (MVC-DNTF). The work aims to address the weakness of classical deep non-negative matrix factorization based multi-view clustering (MVC) method that they generally process each view independently and thus ignore the potential relationship between views. Therefore, the authors propose to extend the deep non-negative matrix factorization into deep non-negative tensor factorization (DNTF). To further explore complementary information between different views, a weighted tensor schatten-p norm based constraint is introduced. An optimization algorithm is developed to effectively solves the MVC-DNTF objective.  Experimental results on multiple benchmark datasets validate the proposed method's superior performance compared to existing multi-view clustering techniques.
strength_weakness: strength_weakness: (1) Tensorized methods have emerged as a popular way to process multi-view data in recent years. This work considers extending deep non-negative matrix factorization into deep non-negative tensor factorization, which is an innovative method for multi-view clustering tasks. The proposed deep non-negative tensor factorization helps to extract deep and nonlinear features from multi-view data and thus benefits the clustering performance.
(2) Apart from deep non-negative tensor factorization, a weighted tensor schatten-p norm based constraint is introduced to further improve the clustering results. This constraint can better exploit the complementary information between different views as well as the spatial structure. 
(3) Deep non-negative tensor factorization seems to be more difficult to be solved. The work develops an effective optimization algorithm to solve the MVC-DNTF objective, which greatly strengthens the method's practical applicability.
(4) Extensive experiments conducted on eight benchmark datasets provide comprehensive validation of the proposed method's effectiveness and superiority over state-of-the-art algorithms.
limitations: limitations: 1.	There are some typos and grammar mistakes in this paper to be corrected. For example, (1) there is a “tesnor” in page 1, which should be revised as “tensor”. (2) In table 1 (notations), \mathbfcal{\psi}_{i} should be \mathbfcal{\psi}_{i-1}. 
2.	There are also some layout issues, especially in Section 3.4 when analyzing the computational complexity.
3.	Tensor construction could be further specified. The description of the tensor construction and the Schatten $p$-norm can be added in conjunction with Fig.1 for better presentation.
paper_summary: paper_summary: The main idea of the work is multi-view clustering based on deep non-negative tensor factorization. The method stacks the matrix into a tensor and then decomposes it into the multiplication of several tensors, so as to extract the deep and nonlinear features of multi-view data. Weighted tensor schatten $p$-norm based regularization helps to exploit the spatial structure across different views. Additionally, anchor graph is introduced to reduce the computational complexity, and an optimization algorithm for deep non-negative tensor factorization is developed. The performance of the method is evaluated on several multi-view datasets, and the comparison with existing multi-view clustering algorithms demonstrate the superiority of the work.
strength_weakness: strength_weakness: The pros of the paper are:
1.The contributions seem to be sufficient, and the core idea (i.e., DNTF) is novel. The utilization of deep tensor factorization enables the extraction of deep and nonlinear features, which play a crucial role in attaining high-quality clustering results. 
2.The extensive experiments on eight benchmark datasets are performed to evaluate the performance of the method. The results obtained from these experiments indicate the superior performance of the proposed method when compared to existing multi-view clustering approaches.
3.The ablation studies and parameter analyses demonstrate the robustness of the proposed method and the influence of different model components. These analyses provide a comprehensive evaluation of the method's performance under various conditions, which better supports the contributions of the paper.
limitations: limitations: The cons of the paper are:
1.The Introduction reviews most works related to multi-view clustering, but there are still some references to be further analyzed. The authors should add more related references. 
2.Figure 2 and Figure 4 are somehow blur, and their quality could be further improved. Besides, the two figures are with different styles, and if possible, it is recommended to adjust them to follow a same style.
3.The paper is well-written with well-structured organization. However, there are still some typos. For example, the organization of the subsection “4.7 Ablation Study” might require adjustment; there are both MVC-DNTF and MVC-DTF in this paper, and the authors should use a same name.