Enhanced Tensorial Self-representation Subspace Learning for Incomplete Multi-view Clustering

Venue: acmmm_org_ACMMM_2024_Conference
Published: 2024-04-08 16:22:25

Abstract: Incomplete Multi-View Clustering (IMVC) is a promising topic in multimedia as it breaks the data completeness assumption. Most existing methods solve IMVC from the perspective of graph learning. In contrast, self-representation learning enjoys a superior ability to explore relationships among samples. However, only a few works have explored the potentiality of self-representation learning in IMVC. These self-representation methods infer missing entries from the perspective of whole samples, resulting in redundant information. In addition, designing an effective strategy to retain salient features while eliminating noise is rarely considered in IMVC. To tackle these issues, we propose a novel self-representation learning method with missing sample recovery and enhanced low-rank tensor regularization. Specifically, the missing samples are inferred by leveraging the local structure of each view, which is constructed from available samples at the feature level. Then an enhanced tensor norm, referred to as Logarithm-p norm is devised, which can obtain an accurate cross-view description. Our proposed method achieves exact subspace representation in IMVC by leveraging high-order correlations and inferring missing information at the feature level. Extensive experiments on several widely used multi-view datasets demonstrate the effectiveness of the proposed method.

Main paper: 
 related work 2.1 self-representation clustering: 
 Multi-view self-representation clustering relies on all instances to construct coefficient matrices from multiple views. Multi-view self-representation clustering can reflect the relationships among instances, leading to excellent clustering performance. According to [19, 57] , the self-representation clustering for complete data can be formulated into minEQUATIONwhere 𝑍 𝑣 is the subspace corresponding to 𝑣th view, the first term denotes the loss of subspace coefficient reconstruction, and the second term is the regularity of 𝑍 𝑣 . In literature, L and R can be replaced with various norms, e.g. 𝑙 2 -norm, 𝑙 1 -norm, 𝑙 2,1 -norm, graph regularity, and nuclear norm, etc. Luo et al. [33] exploit consistency and specificity jointly for subspace. Zhang et al. [56] utilize naive low-rank regularity to learn the subspace structure. Moreover, Cai et al. [3] introduce a high-order manifold regularized term to capture the manifold information of data in subspace learning. However, this naive strategy neglects the impact of the missing instances without any completion term. To infer the missing information, our ELRSSL leverages a local structure at the feature level which is restricted to similar features and ensures the inferred features are meaningful.Low-rank tensor learning has been empirically proven to be effective in capturing high-order correlation [8, 13, 35, 51, 53] . The formula for low-rank tensor learning is given asEQUATIONwhere Z and Y denote the learned low-rank tensor and approximation tensor, respectively. The 𝑟𝑎𝑛𝑘 () is the estimation of tensor rank which is usually replaced with various nuclear norms, e.g. TNN [53] , weighted TNN [13] etc. Since the assumption that the large singular values are generally associated with some salient parts while the small singular values are the contrary, a natural idea is to shrink large singular values less to preserve the main information. Some remarkable approaches are developed, one popular strategy is the WTNN [13] [14] [15] . Besides, the nonconvex low-rank tensor approximation is widely used in various fields [8, 9] . Distinct from the above works, our proposed ELRSSL adopts an enhanced low-rank tensor regularity, enables the important information can be preserved adaptively.Given an incomplete dataset {𝑋 𝑣 } 𝑉 𝑣=1 and indicator vector {𝑒 𝑣 } 𝑉 𝑣=1 , where 𝑉 is the number of views, each element of 𝑒 𝑣 is defined as follows.EQUATIONThe goal of IMVC is to partition the instances under the incomplete condition. Based on the NMF, some simple but effective methods are devised [17, 21, 36, 59 ]. Beyond, graph-based methods [20, 22, 37] also achieve impressive performance, e.g. work in [38] learns consensus graph and clustering indicator simultaneously, Liu et al. [27] propose a graph propagation strategy to infer missing entries by graph propagation, Guo et al. [16] captures non-linear relations by kernel on computed anchor graph. Based on anchor technology, the work in [30] proposes an algorithm with linear time complexity and space complexity. Wen et al. [49] simultaneously consider the local information, the complementary information, and the discriminatory power of all views. Liu et al. [29] first introduce selfrepresentation subspace clustering to IMVC. Unlike prior works, we step further by mining the high-order information while preserving the salient feature as well as ensuring that the inferred information is useful.EQUATION)where 𝑇 𝑣 = 𝑋 𝑣 + 𝐸 𝑣 𝑊 𝑣 and Ž 𝑣 = 𝐼 -𝑍 𝑣 . The optimal X (𝑣) is obtained byEQUATION) 
the proposed method: 
 Incomplete multi-view data presents a great challenge to clustering tasks. To implement self-representation subspace learning to IMVC, the missing information completion and consistent correlation learning are crucial. Therefore, we devise a novel self-representation method for IMVC as follows.To complete the missing instances, the local structure is leveraged. The completion matrix 𝐸 𝑣 ∈ R 𝑑 𝑣 ×𝑛 𝑚 𝑣 and index matrix 𝑊 𝑣 ∈ R 𝑛 𝑚 𝑣 ×𝑛 corresponding to 𝑣th view are introduced, where 𝑑 𝑣 and 𝑛 𝑚 𝑣 denote the dimensions of feature and number of missing instances.The index matrix 𝑊 𝑣 is constructed byEQUATIONThe completion matrix 𝐸 𝑣 is learned from the existing information by minEQUATIONwhere 𝐺 𝑣 ∈ R 𝑑 𝑣 ×𝑑 𝑣 is a guidance matrix used to ensure the recovered missing samples come from similar features. 𝐺 𝑣 is a similarity matrix constructed in the feature perspective. EQUATIONThe above term completes samples from the feature level while ensuring the recovered features are meaningful by restricting similar features [4] . According to the normalization of the Laplacian matrix, Formula (5) can be reformulated as minEQUATIONwhere𝐿 𝑣 𝑔 ∈ R 𝑑 𝑣 ×𝑑 𝑣 is a Laplacian matrix computed by 𝐿 𝑣 𝑔 = 𝐷 𝑣 𝑔 -𝐺 𝑣 . The 𝐷 𝑣 𝑔 which is calculated as 𝐷 𝑣 𝑔 = 𝑑𝑖𝑎𝑔({ 𝑑 𝑣 𝑗 𝐺 𝑣 𝑖,𝑗 } 𝑑 𝑣 𝑖=1), is a degree matrix of 𝐺 𝑣 . Then, we learn the complete feature view with the following model. minEQUATIONwhere X 𝑣 ∈ R 𝑑 𝑣 ×𝑛 is the obtained 𝑣th complete view. For selfrepresentation subspace learning with multi-view data, the learned multiple subspaces present a high-order structure, which can be represented by a low-rank tensor. As shown in Figure 2 , the multiview subspace representation exhibits a high-order correlation. Thus, many works establish the following model to capture the high-order correlation [8, 13, 53] .EQUATIONwhere ∥ • ∥ * is t-SVD based TNN, 𝜇 and 𝜆 are balance parameters, 𝐸 𝑣 denotes the 𝑣th noise matrix, and R is the regularizations on 𝐸. Ordinary TNN treats different singular values equally. This, however, ignores the fact that different singular values hold different contributions. For instance, given an image that is represented as a matrix, the large singular values may contain the salient feature, while small values consist of noise. Therefore, Inspired by the success of weighted TNN and nonconvex low-rank tensor approximation, we designed an enhanced TNN, called Logarithm-𝑝 norm. The Logarithm-𝑝 norm is formulated as follows.EQUATIONwhere 𝜎 𝑖 (𝑍 ) is 𝑖th singular values of matrix 𝑍 with non-descending ordering. For tensor Z ∈ R 𝑛 1 ×𝑛 2 ×𝑛 3 , its Logarithm-𝑝 norm is given asEQUATION)where Z𝑘 denotes the 𝑘th frontal slice of Z and Z is obtained by conducting Fast Fourier Transformation (FFT) on Z among third dimension. In the next section, a detailed proof is given to show how the Logarithm-𝑝 norm shrinks singular values to preserve the salient feature while eliminating noise.Finally, the overall objective can be formulated as follows.minEQUATIONwhere 𝛼 and 𝜇 are balance parameters, Z ∈ R 𝑛×𝑉 ×𝑛 is a tensor generated by stacking each 𝑍 𝑣 and then rotating its dimensionality to 𝑛 × 𝑉 × 𝑛. Nevertheless, the complete feature views { X 𝑣 } 𝑉 𝑣=1 are imputed from the observed information, and the learned subspaces are constructed as a rotated tensor to capture the latent high-order correlation. Once the optimal results are computed, offthe-shelf spectral clustering is applied on 𝑆 =  
optimization: 
 The Formula ( 12) is difficult to optimize due to the coupling of variable Z. Thus, an auxiliary variable J is introduced as follows.minEQUATIONTo solve Formula (13), we devise an iterative optimization algorithm in which each variable is alternately optimized while others are fixed. The overall procedure is summarised in Algorithm 1.When 𝐸 𝑣 , 𝑍 𝑣 , and J 𝑣 are fixed, the optimization respects to X 𝑣 can be written as follows.minEQUATIONTo obtain the local optimal solution of Formula ( 14), we take the partial derivative concerning X 𝑣 and set it to zero.When X (𝑣) , 𝑍 𝑣 , and J 𝑣 are fixed, the objective function of 𝐸 𝑣 is minEQUATIONSince 𝑊 𝑣 denotes the mapping of absent instances to all instances, we can obtain 𝑊 𝑣 (𝑊 𝑣 ) 𝑇 = 𝐼 . Due to the unitarily invariant of Frobenius norm, Formula ( 17) can be transformed intoEQUATIONThe partial derivative of 𝐸 𝑣 is set to zero while the following equation is obtained.EQUATION)The optimal solution to 𝐸 𝑣 isEQUATION)For the optimization of 𝑍 𝑣 , the other variables are fixed. The minimization problem can be formulated as follows.minEQUATIONwhere 𝐽 𝑣 denotes the 𝑣th frontal slice of tensor J . In order to transfer Formula ( 21) to an unconstrained optimization function, an augmented Lagrangian function is introduced.EQUATIONwhere Λ 𝑣 is Lagrange multiplier, 𝜌 is a penalty parameter. Then, the partial derivative concerning 𝑍 𝑣 is taken to zeros.EQUATION)The optimal 𝑍 𝑣 is obtained by reformulated Equation (23) as follows.EQUATIONTo update J , the following theorem is introduced. Theorem 1 Given a minimization problem as minEQUATIONwhere J and P are both tensor with size 𝑚 × 𝑉 × 𝑛. The optimal J can be obtained by the following steps.(1) Transform P into P by performing FFT on P among third dimension.(2) Updated Ĵ𝑖 can be obtained by Ĵ𝑖 =𝑈 𝑖 𝑆 ∇𝜙/𝜌 (Σ 𝑖 )(𝑌 𝑖 ) 𝑇 , where P𝑖 = 𝑈 𝑖 Σ 𝑖 𝑌 𝑖 , ∇𝜙 denotes the derivative of log(1+𝑥) 𝑝 , and the formulation of 𝑆 ∇𝜙/𝜌 (Σ 𝑖 ) is𝑆 ∇𝜙/𝜌 (Σ 𝑖 ) = 𝑑𝑖𝑎𝑔(max{Σ 𝑖 𝑣,𝑣 - ∇𝜙 (Σ 𝑖 𝑣,𝑣 ) 𝜌 , 0} 𝑉 𝑣=1 )(3) Then, stacking each Ĵ𝑖 to Ĵ . The updated J can be computed by performing inverse FFT on Ĵ among third dimension.Proof. Inherited from Equation ( 11), the Formula (25) can be divided into 𝑛 subproblems, while the 𝑖th subproblem can be written as minEQUATIONAccording to the non-ascending order of singular values and the monotonicity of Equation (11) , Formula (26) can be relaxed as the following form. minEQUATIONwhere 𝜎 𝑣 denotes the 𝑣th singular value of current Ĵ𝑖 , 𝜎 𝑣 ( Ĵ𝑖 ) represents the 𝑣th singular value of updated Ĵ𝑖 . All singular values are non-decreasing. Since 𝜙 (𝜎 𝑣 ) and 𝜎 𝑣 are constants, we rewrite the above problem as minEQUATIONThe optimal solution of Ĵ 𝑣 can be obtained by generalized weighted singular value thresholding [31] , which can be formulated asEQUATIONwhere ∇𝜙 𝑖 = 𝑝/(1 + Σ 𝑖 ), meaning the shrinkage weights are governed by Σ 𝑖 . Thus, we can intuitively obtain that ∇𝜙 𝑖 is non-decreasing due to Σ 𝑖 being non-increasing, which leads to large singular values being preserved. □ By fixing X (𝑣) , 𝐸 𝑣 , and 𝑍 𝑣 , the optimization of J can be formulated as minEQUATIONwhere T ∈ R 𝑛×𝑉 ×𝑛 is a 3-order tensor constructed by stacking each Λ 𝑣 and rotating its dimension into 𝑛 ×𝑉 × 𝑛. The updated rule of Formula ( 30) is shown in Theorem (1).The Lagrange multiplier and penalty parameter can be updated byEQUATIONEQUATIONwhere 𝛽 denotes the growth rate and is set to 1.5. 𝜌 𝑚𝑎𝑥 is maximum value of penalty parameter [2] .Similar to [4, 29] , the convergence condition is prescribed asEQUATIONwhere 𝑜𝑏 𝑗 𝑡 represents the value of objective function at the 𝑡th iteration, and 𝜖 is a pre-defined parameter. 13), the designed iterative alternation algorithm ensures the proposed method converges to a local optimum solution.Following, we analyze the computational complexity in detail. The main complexity of Algorithm 1 depends on the iteration procedure. Firstly, the computation of 𝐸 𝑣 concerning the matrix inversion. However, the inverse of (𝐼 + 𝐿 𝑣 𝑔 ) can be computed in advance. Therefore, the computational complexity of updating 𝐸 𝑣 is O (𝑑 2 𝑣 𝑛 +𝑑 𝑣 𝑛𝑛 𝑚 𝑣 ) resulting from matrix multiplication. The optimization of X 𝑣 , involves computing the inverse matrix, with a complexity of O (𝑛 3 ). For updating 𝑍 𝑣 , the main complexity is O (𝑛 3 ), which comes from computing inverse matrix. Here J ∈ R 𝑛×𝑉 ×𝑛 , the updating J needs FFT, singular value decomposition, and inverse FFT. The computational complexity of FFT and IFFT is O (𝑉 𝑛 2 𝑙𝑜𝑔(𝑛)), and the step of singular value decomposition costs O (𝑛 2 𝑉 2 ). The total complexity of Formula ( 30) is O (𝑉 𝑛 2 𝑙𝑜𝑔(𝑛) + 𝑛 2 𝑉 2 ). Thus, the overall computational complexity of the proposed method is O (𝑡 (𝑑 2 𝑣 𝑛 + 𝑛 3 )), where 𝑡 is the number of iterations. The computational complexity of Algorithm 1 is dependent on the cubic of instances and the dimension of the feature, while the number of iterations also affects the computation consumption. 
experiment: 
 In this section, to verify the performance of the proposed method, several widely-used datasets are chosen. Meanwhile, eleven stateof-the-art IMVC algorithms are adopted as comparison methods. In addition, model analyses, including ablation study, parameters analysis, and convergence analysis, are performed to explore the proposed model. At the very beginning, the experiment setup is introduced.5.1.1 Datasets Description. Nine multi-view datasets are adopted, including 3source, MSRC v1, Yale, ORL, BBC, BBCSport, Caltech7, Leaves, and UCI digits. Specifically, the detailed information of all datasets is summarised in Table 1 .Following the principle that each sample appears in at least one view, incomplete multi-view datasets are generated with missing ratios at 0.2 intervals from 0.1 to 0.9. Particularly, for an incomplete sample, it is randomly absent in some views, however, present in at least one view.Eleven state-of-the-art models are chosen, including APMC [16] , BGIMVSC [38] , CBG [44] , DAIMC [17] , FIMVC [30] , FLSD [49] , HCP [25] , IMSR [29] , PGP [27] , PIC [42] , and SIMVC [50] .For compared methods, parameters are tuned in the range advised in the literature to report the best results. For the proposed method, three parameters, including 𝛼, 𝛾, and 𝜇 are adjusted to [10 -4 , 10 -3 , 10 -2 , 10 -1 , 10 0 , 10 1 , 10 2 , 10 3 , 10 4 ], [5, 10, 15, 20, 50, 100, 150, 200, 300] , and [10 -4 , 10 -3 , 10 -2 , 10 -1 , 10 0 , 10 1 , 10 2 , 10 3 , 10 4 ], respectively. We run each method 10 cycles to calculate average performance and standard deviation for reporting.To assess the performance of clustering, three metrics, Accuracy (ACC), Normalized Mutual Information (NMI), and Purity are adopted.Experiments are conducted on MATLAB 2022b (64-bit) with Intel Xeon Gold 6248R CPU, 256G RAM.Table 2 reports the experiment results on all datasets with 10% samples are incomplete. The bold and underlined indicate the best and suboptimal results, respectively, while '-' denotes the unavailable results due to program error. Figure 3 shows the results on all datasets with different missing rates. To visualize the running time of different methods, we convert real runtime to relative Logarithm runtime, which is reported in Figure 4 . These provide the following observations.(1) The proposed method achieves superior performance on all datasets with 10% incomplete samples. Meanwhile, PGP surpasses other methods except ELRSSL. For instance, compared to the suboptimal approach, ELRSSL has 14.14%, 1.1%, 6.87%, 6.19%, 8.45%, 19.82, 3.48%, 9.64%, and 13.92% improvement on terms of ACC. The excellent performance demonstrates the critical role of missing instances completion and lowrank tensor regularity. (2) Compared to IMSR, which is a pioneering work in selfrepresentation learning for IMVC, the proposed method is superior. The difference between IMSR and our ELRSSL is that ELRSSL infers samples from the perspective of feature and efficiently explores high-order information among multiple views, resulting in a more accurate representation of subspaces. Compared to HCP, which controls the tensor tubal rank by tensor factorization, the proposed method offers better clustering results. It demonstrates the effectiveness of the proposed Logarithm-𝑝 norm. (3) As shown in Figure 3 , the proposed method is superior on several datasets, such as ORL, Yale, Leaves, and UCI digits. However, ELRSSL shows great fluctuations in performance on BBCSport and Caltech7 datasets. We conjecture that this is because the ORL, Yale, Leaves, and UCI digits datasets enjoy well-structured subspace, e.g. the face datasets (ORL, Yale) hold low-rank properties and are more suitable for self-representation subspace learning [1, 58] . Overall, the proposed ELRSSL is superior on most datasets with various missing ratios, demonstrating its superiority in IMVC task. (4) Compared to anchor-based methods, ELRSSL requires more time consumption but has better performance. Furthermore, the proposed method is more time-consuming than the suboptimal method PGP, due to the computation of matrix inversion and tensor singular value decomposition. In general, the extra time consumption is worthwhile as the proposed method achieves remarkable clustering performance. There are three main hyperparameters for the proposed method, including 𝛼, 𝛾, and 𝑝. Particularly, the number of neighbors for 𝑘-nearest neighbor in Equation ( 6) is fixed to 7 due to its hardness in selecting the optimal value. The tuning range of these parameters is given in the section 5.1. As shown in Figure 6 and Figure 7 , the proposed method enjoys stable performance.As shown in Figure 8 , the objective value decreases in each iteration, which proves the proposed method can converge rapidly. 
conclusion: 
 In this paper, an effective self-representation subspace learning method is devised for IMVC. In particular, a local structure in each view is introduced to recover the missing samples. Meanwhile, we designed an enhanced low-rank tensor regularity to efficiently capture the high-order correlation by assigning different weights to different singular values. Extensive experiments demonstrate the superiority of the proposed method and the improvement compared with previous self-representation subspace learning approaches. We plan to explore an efficient mechanism without precision reduction for self-representation on the IMVC task in future.  
: 
 Roughly, current multi-view clustering approaches can be categorized into the following four types: matrix factorization clustering [11, 43] , graph-based clustering [23, 24, 34, 41] , deep multiview clustering [28, 52, 54] , and multiview subspace clustering [5, 10, 32, 45, 57] . In particular, self-representation learning methods based on low-rank tensor regularization achieve commendable performance, where they mine complementary information from multiple views by tensor Singular Value Decomposition (t-SVD) based nuclear norm [8, 13, 51, 53] .Most of the existing methods are based on the completeness assumption, which denotes that samples in all views are observed. However, in real-world scenarios, collected instances are usually missing in certain views due to malfunctions in sensors or data corruption during transmission [12] . The absent views create a significant challenge for multi-view clustering where the consistent information across multiple views is difficult to learn. To tackle this limitation, numerous researchers have devoted themselves to IMVC, leading to the flourishing development of IMVC. As a foundational work in the field of IMVC, Partial Multi-view Clustering (PVC) [21] utilizes the Non-negative Matrix Factorization (NMF) to establish a latent subspace under incomplete conditions. Expanding from PVC, several impressive algorithms have emerged [17, 36, 59] . The work in [59] introduces a novel graph Laplacian term to bridge the connection of missing samples across multiple views. In [17] and [36] , enhanced NMF, including weighted NMF and weighted semi-NMF are adopted to improve the ability to handle incomplete multi-view data. The works in [48, 55] reduce the impact of incomplete instances and capture consistent information by inferring the missing entries.Apart from the above matrix factorization based methods, the graph learning framework is introduced into IMVC. Perturbationoriented IMVC [42] transfers the missing problem from the perspective of instances to the graph and reduces the spectral perturbation risk among different views. Wen et al. [47] propose an adaptive graph completion strategy and unify the graph completion and consensus graph learning into one framework. Liang et al. [26] devise a graph fusion-based IMVC model that alleviates the impact of missing instances by sample-level auto weights. Liu et al. [27] infer the missing entries in partial graphs by graph propagation strategy. The work in [46] directly learns the clustering results from a consensus probability representation. Zhao et al. [60] introduce a heterogeneous-graph learning and embedding strategy to adopt high-order structures and recover the incomplete graph for each view.Although graph-based approaches have achieved impressive performance, they still endure high computational complexity. In order to enhance efficiency, anchor technology is introduced. Nie et al. [6] and Wang et al. [40] leverage anchor graph for complete multiview clustering. To expand anchor graph learning into IMVC, the following works have emerged. Anchor-based partial multi-view clustering [16] reconstructs similarity among instances by anchors. The work in [44] proposes a highly efficient multi-view clustering method based on a bipartite graph framework for incomplete data. Recently, a scalable IMVC method [50] has been proposed, which constructs the view-specific anchor graph for capturing complementary information and devises a novel structure alignment module to eliminate the misalignment of anchors. The methods mentioned above have improved the performance of IMVC from various perspectives, most of them are based on matrix factorization or graph learning. Numerous multi-view clustering methods have empirically demonstrated that self-representation subspace learning and low-rank tensor constraint are beneficial for clustering [7, 13, 15, 18, 29, 39, 53] . For complete multi-view data, the work in [53] is the pioneer in adopting low-rank tensor in self-representation subspace learning. Motivated by [53] , t-SVD based methods are well-developed [3, 35, 51] . To overcome the gap of self-representation approaches in IMVC task, missing data imputation and self-representation are unified into a framework [29] . Further, a tensor factorization term is adopted to preserve highorder correlation in subspaces [25] . However, the used strategy is to infer absent samples by summarising information from whole observed samples, resulting in redundant information. In addition, the reservations about salient features and the elimination of noise are neglected during exploring high-order correlation.To tackle this issue, we propose an Enhanced Low-Rank tensor Self-representation Subspace Learning (ELRSSL) for IMVC. In detail, our ELRSSL recovers the missing instance by restricting similar features which ensures the inferred entries are meaningful, and index matrices are utilized to ensure the inferred entries belong to missing samples. Meanwhile, a low-rank tensor learning strategy is adopted to explore the high-order correlation across multiple views. In addition, the large singular values are considered to be important because they contain main feature information, while small singular values represent the noise embedded in data. Weighted Tensor Nuclear Norm (WTNN), which shrinks singular values by assigned weights, is a popular technology for low-rank tensor learning [13, 15] . However, WTNN needs the a priori information about data to assign reasonable weights. For the proposed method, an enhanced nuclear norm is devised, which assigns weight to various singular values with a self-adaptive weight strategy. The overall framework is depicted in Figure 1 . The main contribution of this paper is summarised as follows.• We present ELRSSL, a self-representation approach for IMVC which enhances the expressive capability of subspace on incomplete data by exploring high-order correlation and missing information inferences at the feature level. • An enhanced nuclear norm is devised, which assigns weight to different singular values. The scale of weights is determined by corresponding singular values. Thus, the important features across various views are retained while the noises embedded in multiple views are eliminated. • To solve the proposed method, an alternating optimization algorithm is designed with a computational complexity analysis. The extensive experiments demonstrate the superiority of the proposed method. 
 
 Appendix:  


Decision: In this paper, for the challenging issue of incomplete multi-view clustering, the authors proposed a novel Tensorial Self-representation Subspace Learning method. All reviewers unanimously recognized the innovation and quality of this paper and gave an acceptance opinion. The final scores of the paper are Borderline Accept, Weak Accept, and two Accepts after rebuttal. Based on the recommendations of reviewers and the quality of the paper, the final recommendation to the paper is acceptance.

Reviews: review: paper_summary: paper_summary: This paper introduces an Enhanced Tensorial Self-representation Subspace Learning (ELRSSL) method for Incomplete Multi-view Clustering (IMVC). The proposed method addresses the challenge of incomplete data in multi-view clustering by recovering missing samples through local structure inference and leveraging enhanced low-rank tensor regularization.
strength_weakness: strength_weakness: - **Clarity and Readability**: The paper is well-written and easy to understand, making complex concepts accessible to a broader audience.
- **Innovative Approach**: The method shows certain novelty, particularly in integrating local structure inference with enhanced low-rank tensor regularization for missing sample recovery.
- **Extensive Experiments**: The superiority of the proposed method is demonstrated with extensive experiments across multiple datasets, highlighting its effectiveness and robustness.
- **Enhanced Low-rank Tensor Norm**: The introduction of the Logarithm-𝑝 norm for low-rank tensor regularization is a significant contribution, as it allows for better preservation of important features and elimination of noise.
limitations: limitations: * Insufficient Theoretical Justification: The paper would benefit from a more detailed theoretical justification. Especially, it would be better if it could be proved theoretically that the proposed enhanced TNN outperforms TNN .
* Typographical and Writing Errors: There are several minor typographical and grammatical errors throughout the paper. Examples include:"Incompete" should be corrected to "Incomplete" (Table 1 title); The symbol in Eq. 5, $Ej,:$ should be $E_{j,:}$.

* Despite that the work shows several contributions with the enhanced TNN, the overall novelty seems still limited, since several design such as Laplacian Matrix based regularization is has been used by several works.
paper_summary: paper_summary: This paper proposes a novel self-representation learning method for incomplete multi-View clustering. The core motivation is to learn an exact self-representation for IMVC. To compensate for missing information, the authors propose constraining similar features to ensure the recovered data is meaningful, alongside devising an enhanced tensor norm to accurately delineate cross-view relationships. Experiments on various real-world datasets demonstrate the effectiveness of the proposed method.
strength_weakness: strength_weakness: (1) The paper is well-written. The working mechanism and motivation of each module are clearly explained. 
(2) The proposed method achieves superior results on numerous benchmarks. 
(3) Ablation studies are conducted to demonstrate the effectiveness of the proposed components, while comparisons of running times provide insight into the method's efficiency relative to others.
limitations: limitations: (1) The authors exclusively focus on self-representation subspace learning, overlooking previous works addressing noise in representations. It would be beneficial for the authors to expound upon their rationale for this omission. 
(2) The number of clusters is directly specified in experiments. However, it is unknown in the real scenario. Further discussion of the impact of cluster counts is necessary. 
(3) Grammatical errors, such as "Guo et al. [16] captures" instead of "Guo et al. [16] capture," should be rectified. 
(4) Some papers are incorrectly cited, such as: "Xiaojun Chen, Weijun Hong, Feiping Nie, Dan He, Min Yang, and Joshua Zhexue Huang. 2018. Spectral Clustering of Large-scale Data by Directly Solving Normalized Cut. In Proceedings of the 24th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, KDD 2018, London, UK, August 19-23, 2018 1206–1215"
paper_summary: paper_summary: This paper proposes an incomplete multi-view clustering method called ELRSSL. Specifically, the authors employ self-representation learning to explore relationships among samples. They introduce a strategy for missing sample recovery and enhanced low-rank tensor regularization. Through leveraging local structure and high-order correlations across multiple views, ELRSSL achieves precise subspace representation in IMVC. Extensive experiments on various multi-view datasets demonstrate the effectiveness of the proposed method.
strength_weakness: strength_weakness: 1. This paper is well-organized and easy to follow. 
2. By incorporating local structure and high-order correlations across multiple views, ELRSSL achieves precise subspace representation, demonstrating its effectiveness in handling complex multiview datasets. 
3. This paper proposes a novel self-adaptive weight strategy to retain salient features while eliminating noise.
limitations: limitations: 1. Clarity: The flowchart shown in the paper is not clear enough and the meaning of each matrix in Figure 1 is ambiguous. Authors are suggested to add more description. 
2. Motivation:
(a) While the concept that the large singular values may contain the salient feature and small values consist of noise is reasonable, the motivation of the proposed Logarithm-p norm is not convincing enough. Key points to explore include: Why can Logarithm-p norm better retain important features than normal TNN? What is the essential difference between the Logarithm-p norm and nonconvex low-rank tensor approximation? What are the physical significance and distribution trends of the proposed Logarithm-p norm? 
(b) In this paper, authors declare that the strategy of inferring missing samples used in previous works results in redundant information. To highlight the advantages of the strategy used in this paper, authors are encouraged to provide a detailed comparison of the two strategies. 
(c) The work in [Zhang et al., 2023] provides a generalized nonconvex tensor low-rank regularization to alleviate the biased approximation in the self-representation learning framework. A theoretical analysis of the difference between [Zhang et al., 2023] and the proposed method can enhance the comprehensiveness of the article. 
[Zhang et al., 2023] Enhanced Tensor Low-Rank and Sparse Representation Recovery for Incomplete Multi-View Clustering. In AAAI.

3. Experiments:
(a) It would be beneficial to categorize comparative methods to clarify the basic framework of the models in reporting experiment results. 
(b) Additional Investigation: Explore the impact of different cluster numbers on various metrics.
paper_summary: paper_summary: The authors introduce ELRSSL, a new method to group incomplete multi-view samples into corresponding clusters. ELRSSL works by (1) recovering the missing instance by restricting similar features which ensures the inferred entries are meaningful, and (2) devising an enhanced low-rank tensor learning strategy to explore the high-order correlation across multiple views. Comparative analysis against state-of-the-art methods underscores the effectiveness of ELRSSL, showcasing the potential of self-representation learning in incomplete multi-view clustering.
strength_weakness: strength_weakness: 1. The authors recognize and address drawbacks in prior self-representation learning based IMVC approaches, which are (1) redundant information resulting from incomplete data inference and (2) overlooking salient features in the self-representation subspace. 

2. The evaluation performed is comprehensive and the performances of the proposed method are impressive. 

3. The comparison results and the ablation study show that differential treatment of different singular values in the learned tensor is beneficial for clustering, reinforcing the author's assertions in Section 1. 

4. The paper's organization facilitates ease of comprehension.
limitations: limitations: These are not major weaknesses, but suggestions to strengthen the results:

1. There are inconsistencies in the notations. In Eq. (5), I think "$E{j :}^v$" is not what the authors want to represent, perhaps "$E_{j :}^v$". Authors are required some careful revisions. 

2. I would like the authors to further discuss the performances of IMVC methods if all samples are visible in only one view. 

3. The authors claim that the "-" in Table 2 is due to a program error, and I would like the authors to explain the cause of the error. 

4. In Figure 4, it is not clear what is relative running time in this paper. 

5. It is not clear how the number of neighbors in constructing guidance matrix G will individually impact the model's performance. 

6. The performances of CBG on the UCI digits and Leaves datasets are not promising. Please provide the explanation and analyses. 

7. Have you considered explaining the intrinsic structure of high-order correlation? In other words, why this structure is formed?