Improved Weighted Tensor Schatten 𝑝-Norm for Fast Multi-view Graph Clustering

Authors: yinghui  sun
xingfeng  li
quansen  sun
min-ling  zhang
Venue: acmmm_org_ACMMM_2024_Conference
Published: 2024-04-07 15:37:30

Abstract: Recently, tensor Schatten $p$-norm has achieved impressive performance for fast multi-view clustering \cite{xia2023tensorized}. This primarily ascribes the superiority of tensor Schatten $p$-norm in exploring high-order structure information among views. Whereas, 1) tensor Schatten $p$-norm treats different singular values equally, such that the larger singular values corresponding to certain significant feature information (i.e., prior information) have not been utilized fully; 2) tensor Schatten $p$-norm also ignore ranking the core entries of core tensor, which may contain noise information; 3) existing methods select fixed anchors or averagely update anchors to construct the neighbor bipartite graphs, greatly limiting the flexibility and expression of anchors. To break these limitations, we propose a novel \textbf{Improved Weighted Tensor Schatten $p$-Norm for Fast Multi-view Graph Clustering (IWTSN-FMGC)}. Specifically, to eliminate the interference of the first two limitations, we propose an improved weighted tensor Schatten $p$-norm to dynamically rank core tensor and automatically shrink singular values. To this end, improved weighted tensor Schatten $p$-norm
has the potential to more effectively leverage low-rank structures and prior information, thereby enhancing robustness compared to current tensor Schatten $p$-norm methods. Further, the designed adaptive neighbor bipartite graph learning can more flexibly and expressively encode the local manifold structure information than existing anchor selection and averaged anchor updating.    
Extensive experiments validate our effectiveness and superiority across multiple benchmark datasets.

Main paper: 
 introduction: 
 Recently, technological advancements have facilitated the collection and storage of extensive data from diverse sources or feature extractors, resulting in the emergence of multi-view data [4, 26, 27, 32, 36] . For instance, a single news story could exist in various formats like video, audio, or text. Meanwhile, each news could be also reported in different languages across nations such as Chinese, English, Russian, or French. Multi-view clustering (MVC) leverages the similarities between samples and views to effectively integrate the consistent and complementary attributes of unlabeled multi-view data, thereby categorizing them into relevant clusters [2, 3, 38, 45, 52, 55] . MVC has witnessed significant growth in recent times [1, 32, 35, 44, 48, 49] .Despite the myriad of MVC methods, scalability remains a paramount challenge in real-world large-scale applications [24, 25, 46, 61] . Current MVC methods often exhibit quadratic or cubic complexity with respect to the number of instances n, posing significant obstacles in handling extensive datasets [6, 7, 31, 33, 37] . To address this challenge, the anchor graph strategy has emerged as an effective solution [5, 21, 47, 62] . Initially, it selects m independent anchor points to comprehensively represent instances within each view. The quality of these anchors significantly impacts performance and is typically determined through random sampling, k-means, or heuristic sampling strategies (such as VDA anchor selection [57] ). For instance, some studies leverage k-means operations and heuristic sampling on individual views, then integrate the outcomes into a consensus graph for final clustering results. However, these static anchor selection strategies limit the expressiveness and flexibility of anchors. In contrast, dynamic anchor learning methods aim to learn and optimize anchors alongside model variations, enhancing both performance and adaptability.Existing anchor learning strategies could be broadly divided into two categories [8, 18, 23] : The first one resembles the self-expressive property, which reconstructs each sample by combining learned anchors linearly from the original data [50, 51] . Then, the connections between samples and anchors produce a bipartite graph. Recent advancements in anchor-based multi-view clustering have stemmed from this strategy. Different from the first one, the second a small amount of neighbor bipartite graph learning methods to build the bipartite graph by assigning each sample some nearest anchors as neighbors, which could better capture the local manifold structure than the first one [19] . However, note herein that existing adaptive neighbor bipartite graph learning methods optimize anchors using average values across iterations, potentially leading to the loss of significant and flexible information. Adaptive neighbor bipartite graph learning is sensitive to noise or outliers since noise and outliers can destroy the local manifold structure in anchor-sample space. Moreover, these methods often overlook higher-order information that could enhance anchor quality. Due to the effectiveness of the tensor Schatten 𝑝-Norm in better approximating the target rank than the tensor nuclear norm, one may use tensor Schatten 𝑝-Norm to explore the high-order information of bipartite graphs [57] . However, existing tensor Schatten 𝑝-norm still suffer from the following limitations: (1) existing tensor Schatten 𝑝-norm treats singular values of tensor equally in each slice; (2) existing tensor Schatten 𝑝-norm cannot exploit the low-rank property fully in the core tensor space.To address these issues, we proposed a novel fast local multiview clustering method, termed Improved Weighted Tensor Schatten 𝑝-Norm for Fast Local Multi-view Clustering (IWTSN-FMGC). IWTSN-FMGC learns the bipartite graph by an adaptive neighbor bipartite graph learning, which usually better protects the local manifold structure than the widely-used subspace self-representation learning. And local manifold structure plays a significant role in clustering. Our adaptive neighbor bipartite graph learning adaptively updates the anchors with orthogonal constraints rather than averaging the value of anchors. Furthermore, our IWTSN-FMGC develops a new IWTSN to consider the prior information of different singular values. Then, IWTSN can also better exploit the low-rank property of core tensor by further ranking the diagonal core entries of core tensor. In this way, IWTSN of our IWTSN-FMGC could enhance the robustness of the adaptive neighbor bipartite graph learning to noise and outliers. Then, IWTSN-FMGC could learn more robust neighbor bipartite graphs to average as a final bipartite graph for obtaining the clustering results. In conclusion, this study makes the following contributions:• We propose a novel Improved Weighted Tensor Schatten 𝑝-Norm (IWTSN) to better rank the core tensor and take advantage of the prior information from singular values, enhancing robustness and validity compared to the widelyused tensor Schatten 𝑝-norm [57] . Meanwhile, an elegant solver is developed to optimize the proposed IWTSN. • By seamlessly coupling adaptive neighbor bipartite graph learning and IWTSN into a unified framework. Both procedures negotiate with each other to enhance the quality and robustness of neighbor bipartite graphs. Thus, the more refined and robust neighbor bipartite graphs could be learned to better capture the local manifold structure information. • Through extensive experiments, the proposed method showcases its superiority over state-of-the-art methods. 
related work 2.1 neighbor graph based mvc: 
 Generally, adaptive neighbor graph learning learns a neighbor graph by assigning a probability value to a sample as the neighbor similarity of another one [39, 53] . In this way, the local manifold structures between samples could be captured, which facilitates the clustering partition [11] . Although effective, the time and space complexities are cubic and quadratic to sample number, respectively. Thus, these methods limit the application for large-scale tasks. Furthermore, many advances employ the tensor nuclear norm or tensor Schatten 𝑝-norm to further explore the high-order information among neighbor graphs [19, 56, 57] , i.e., minEQUATIONwhere the larger distance criterion D has the smaller similarity value â𝑖 𝑗 of similarity graph Â ∈ R 𝑛×𝑛 [15] in the 𝑟 -th view. Control parameter 𝛽 tunes the contribution between adaptive neighbor graph learning and tensor constraints, such as tensor nuclear norm ∥ L∥ ⊛ or tensor Schatten 𝑝-norm ∥ L∥ 𝑝 𝑠 𝑝 [19, 54, 56, 57] . Regrettably, these methods also suffer from time-and space-prohibitive. More importantly, these methods cannot explore the high-order lowrank property fully since they fail to take advantage of the prior information of singular values in the core tensor space.To alleviate these limitations, neighbor bipartite graph construction is proposed. Neighbor bipartite graph A 𝑟 ∈ R 𝑚×𝑛 of 𝑟 -th view is constructed by linking their vertices by defining sample sets X 𝑟 ∈ R 𝑑 ×𝑛 and anchor sets B𝑟 ∈ R 𝑑 ×𝑚 . Vertices represent the relationship between 𝑚 anchors and 𝑛 samples [28, 29, 62, 63] in 𝑑-dimension space. Anchors are typically strategically selected from the original space to approximate all samples. Different strategies are then employed to construct the bipartite graph for clustering. Due to its efficiency, various methodologies have made promising progress in recent years, with anchor learning strategies broadly categorized into subspace graph construction and neighbor graph construction. Unlike subspace graph construction, neighbor graph construction emphasizes preserving locality, assuming that the primary high-dimensional feature space exists within a lowerdimensional manifold. Consequently, neighbor graph construction often excels in preserving the low-dimensional manifold structure. Specifically, for the 𝑖-th sample, the 𝑗-th anchor is connected as a neighbor with a probability represented by 𝑎 𝑖 𝑗 . Intuitively, shorter distances between anchor-node pairs correspond to higher probabilities 𝑎 𝑖 𝑗 , and vice versa, as explained by minA 𝑟 𝑚 ∑︁ 𝑖=1 𝑛 ∑︁ 𝑗=1 D( b𝑟 𝑖 , x 𝑟 𝑗 )𝑎 𝑟 𝑖 𝑗 s.t.(A 𝑟 ) ⊺ 1 = 1, A 𝑟 ≥ 0 (2)With the different criteria D, Eq. ( 2) captures the local manifold structures in different criterion spaces [19, 35, 55] . b𝑖 denotes the 𝑖-th anchor, and In Eq. ( 2), anchors are selected using k-means [13] or heuristic sampling methods [10, 22, 57] . These anchors remain fixed throughout the optimization process, known as static anchor selection, which limits their expressiveness and flexibility. Some researchers explore dynamic anchor learning, optimizing anchors alongside model variations. However, anchors learned from them are optimized using average values across iterations [1] , still limiting the diversity and flexibility of anchors. 
methodology 3.1 notations and preliminaries: 
 Notations and definitions of tensor and matrix Schatten 𝑝-norm and rank approximation [6, 57] are as follows.Vectors and their components are denoted by bold lowercase letters (k and 𝑘 𝑖 𝑗 ). Matrices are represented by bold uppercase letters (K), while tensors are indicated by bold calligraphy letters (K). For simplicity, we use the notation K (:, :, 𝑘) as K 𝑘 or K 𝑘 . The fast Fourier transformation (FFT) and its inverse along the third dimension of tensor K are expressed asK 𝑓 = fft(K, [ ], 3) and K = ifft( K 𝑓 , [ ],3), respectively. The block vectorization operation bvec(K) concatenates all K 𝑙 matrices into a single matrix in R 𝐼 1 𝐼 3 ×𝐼 2 , while its inverse, fold(bvec( K)), restores the original tensor K. Additionally, block circulant matrix is bcirc( K), and block diagonal matrix is bdiag(K 𝑓 ) [6] . Definition 1 (Tensor Product): Tensor product between two tensorsEQUATIONDefinition 2 (Tensor Singular Value Decomposition, 𝑡-SVD): 𝑡-SVD of a tensor K with dimensions 𝐼 1 × 𝐼 2 × 𝐼 3 is denoted asEQUATIONWhere two tensors [41] : Tensor nuclear norm ∥ K ∥ ⊛ represents the summation of singular values across all frontal slices of K 𝑓 , articulated asU K ∈ R 𝐼 1 ×𝐼 1 ×𝐼 3 and V K ∈ R 𝐼 2 ×𝐼 2 ×𝐼 3 are orthogonal, and tensor S K ∈ R 𝐼 1 ×𝐼 2 ×𝐼 3 is 𝑓 -diagonal. Definition 3 (∥ K ∥ ⊛ Tensor Nuclear NormEQUATIONwhere A 𝑙 𝐾 is obtained from Eq. ( 4). Definition 4x (Matrix Schatten 𝑝-Norm (𝑝 ∈ [0, 1])) [58] The matrix Schatten 𝑝-norm is defined as∥K∥ 𝑆 𝑝 =          ∥K∥ 𝑆 1 = ∥K∥ * = min{𝐼 1 ,𝐼 2 } 𝑖=1 𝛿 𝑖 (K), 𝑝 = 1 ( min{𝐼 1 ,𝐼 2 } 𝑖=1 𝛿 𝑝 𝑖 (K)) 1 𝑝 , 0 < 𝑝 < 1 ∥K∥ 𝑆 0 = rank(K), 𝑝 = 0 (6)where rank(•) denotes the operation that calculates the rank of a matrix. When addressing the minimization problem outlined in Eq. ( 6), it becomes evident that the Schatten 𝑝-norm ∥K∥ 𝑆 𝑝 with 0 < 𝑝 < 1 provides a more effective approach to approximating the rank compared to the nuclear norm ∥K∥ * .By substituting the matrix nuclear norm defined in Definition 3 with the matrix Schatten 𝑝-Norm as defined in Definition 4, we arrive at the subsequent formulation: Definition 5 (𝑡-SVD-based Tensor Schatten 𝑝-Norm, 𝑡-TSN [57] ) denotes asEQUATIONInspired by Definition 4, employing tensor Schatten 𝑝-norm with 0 < 𝑝 < 1 enables a more effective exploration of the low-rank characteristics compared to 𝑡-TNN [14] . As outlined in [57] , the subsequent derivation is as follows:EQUATION)which is solved via the generalized soft-thresholding algorithm [65] . Generally, smaller eigenvalues typically encapsulate less discriminative information than their larger counterparts [9, 57] . This motivates the development of weighted tensor Schatten 𝑝-norm asEQUATIONHere, 𝜃 𝑝 𝑗 (K 𝑘 𝑓 ) and 𝜔 𝑘 𝑗 represent the 𝑗-th singular value and its corresponding weight of K 𝑘 𝑓 , respectively. We determine the weights by inversely scaling them with respect to the singular values, reasoning that larger singular values should undergo less shrinkage. To achieve this, we update(𝜔 𝑘 𝑗 ) 𝑡 +1 = 1 (𝜃 𝑝 𝑗 (K 𝑘 𝑓 ) ) 𝑡 +𝜖using the previous iteration of 𝜃 𝑝 𝑗 (K 𝑘 𝑓 ) during optimization, where 𝑡 denotes the iteration number and 𝜖 represents a small constant. Consequently, Eq. ( 9) is transformed into minEQUATION)In contrast to the tensor Schatten 𝑝-norm, our weight tensor Schatten 𝑝-norm excels in capturing prior information by applying less shrinkage to larger singular values and dynamically updating their associated weights in each iteration. Although effective, Eq. ( 10) ignores ranking the diagonal core entries of core tensor, where the low-rank structure of core tensor is not fully extracted. To better exploit the low-rank structures hidden in the core tensor and alleviate the noise influence, Eq. ( 10) further becomes to minEQUATION)We devise a new solver for Eq. ( 11) by using Theorem 3, Theorem 4, and Lemma 1. Note herein that our IWTSN encompasses both the tensor nuclear norm and the tensor Schatten 𝑝-norm as special cases.Given multi-view data {X 𝑟 } 𝑣 𝑟 =1 ∈ R 𝑑 𝑟 ×𝑛 , we extend the formulation presented in Eq. ( 2) to incorporate adaptive neighbor bipartite graph learning. This extension enables the learning of the anchor matrix B 𝑟 ∈ R 𝑙 ×𝑚 and neighbor bipartite graph A 𝑟 ∈ R 𝑚×𝑛 within the 𝑙-dimension space of 𝑟 -th latent view. minEQUATIONwhere Tr(•) is the trace operation. Applying orthogonal constraints to the projection matrices {W 𝑟 } 𝑣 𝑟 =1 ∈ R 𝑑 𝑟 ×𝑙 and anchor matrices can improve the discriminative power. 𝛼 is a control parameter to tune the neighbor number around each sample. That is, adjusting the parameter 𝛼 ≥ 0 can modulate the structure of the bipartite graph, as stated in Proposition 1 below:Proposition 1: By manipulating 𝛼, a balance between two extreme bipartite graph configurations can be achieved:• Sparse bipartite graphs: each vertex is connected to only one other vertex. • Complete bipartite graphs: all vertices are interconnected with uniform edge weight 1/𝑚.In summary, Proposition 1 elucidates the continuum between sparsity and completeness in bipartite graph configuration through 𝛼 manipulation. The proof of Proposition 1 refers to [40] .Although adaptive neighbor bipartite graph learning of Eq. ( 12) can protect the local structure of data, Eq. ( 12) is sensitive to noise [11] . Thus, we seamlessly integrate IWTSN of Eq. ( 9) and Eq. ( 12) into a unified framework to enhance robustness of bipartite graph, i.e., minEQUATIONwhere 𝛾, 𝛽, and 𝛼 are the control parameters to balance the contributions of respective terms. Ψ(•) denotes tensor stacking function.In order to tackle Eq. ( 13), we introduce an auxiliary variable K to enhance the separability of the equation. Subsequently, we reformulate Eq. ( 13) as the augmented Lagrangian function outlined below. minEQUATIONThe optimization problem described by Eq. ( 14) can be effectively tackled using an alternating iterative algorithm, which is outlined as follows.▷Step-1: Solving A with B, W, Y and K fixed. Optimizing A becomes to minEQUATION) Let ∥b𝑖 𝑟 -(W 𝑟 ) ⊺ x𝑗 𝑟 ∥ 2 𝐹 = 𝑑 𝑟 𝑖 𝑗, by eliminating irrelevant variables for the 𝑟 -th view, Eq. ( 15) changes to minEQUATION)In this context, 𝑓 𝑟 𝑖 𝑗 represents an element of F 𝑟 , defined asF 𝑟 = T 𝑟 -Y 𝑟𝜇 . Subsequently, the update process for a 𝑟 is transformed into a column-wise operation as minEQUATION)where â𝑟𝑗 = 1 2 𝜇f 𝑟 𝑗 -1 2 𝛾 d 𝑟 𝑗 (𝛼+𝜇 ). Each column a 𝑗 could be optimized via the following Theorem 1. Theorem 1. With arbitrary 𝑣 vectors {â 𝑗 } 𝑣 𝑗=1 , we obtain the following closed-form solution a * 𝑗 a * 𝑗 = arg minEQUATIONwhose proof refers to [42] . ▷ Update-2: Solving W with A, B, Y and K fixed. In this case, W-subproblem of Eq. ( 14) can be written asEQUATIONwhere E 𝑟 = X 𝑟 (A 𝑟 ) ⊺ (B 𝑟 ) ⊺ . Eq. ( 19) can be solved via the Singular Value Decomposition (SVD) in Theorem 2 with complexity O(𝑣 d (𝑛𝑚 + 𝑘 2 + 𝑘𝑚)) for each iteration, where d = 𝑣 𝑝=1 𝑑 𝑟 . Theorem 2. [20] Letting the SVD of E ∈ R 𝑑 ×𝑙 be E = UGV ⊺ , where U ∈ R 𝑑 ×𝑙 , G ∈ R 𝑙 ×𝑙 and V ∈ R 𝑙 ×𝑙 , the optimal solution of maxW ⊺ W=I Tr(W ⊺ E) is W = UV ⊺ .▷ Step-3 update B: Optimizing B while keeping the irrelevant variables fixed can be understood as maxEQUATIONin which C 𝑟 = 𝛾 (W 𝑟 ) ⊺ X 𝑟 (A 𝑟 ) ⊺ . The optimal solution for optimizing B 𝑟 can be efficiently obtained using Theorem 2. ▷ Step-4 update K: Ignoring the irrelevant items w.r.t. K, updating K subproblem is minEQUATION)According to [30] , Eq. ( 21) can be solved into two steps as follows:(1) minimizing the core matrix, and (2) minimizing 𝑡-TSN.(1) Updating core matrix as minEQUATION)where regularization parameter 𝜆 = 1/(𝑚𝑎𝑥 (𝑚, 𝑣)𝑛)1 2 .And the tensor T is obtained from 𝑡-SVD on the temporary variable F, i.e., F = U * T * V.(2) Updating K as minEQUATION)With the learned low-rank core matrix 𝔓(T), we can use 𝑡product to reconstruct a tensor as L = U * 𝔓(T) -1 * V. The learned L can further produce a closed-form solution via the following Theorem 3. Theorem 3.Consider L ∈ R 𝐼 1 ×𝐼 2 ×𝐼 3 , with 𝑟 = min(𝐼 1 , 𝐼 2 ). Let L 𝑓 = U 𝑓 M 𝑓 V ⊺ 𝑓 ,then the optimization problem for weight tensor Schatten 𝑝-norm can be formulated asEQUATION)and its optimal solution isEQUATIONwhere M 𝑘 𝑓 = diag(𝜹 (M 𝑘 𝑓 )) and D𝜂, 𝝎, 𝑝 (L 𝑘 𝑓 ) = diag(𝜽 (L 𝑘 𝑓 )) are the 𝑘-th frontal slices of M 𝑓 and D𝜂, 𝝎, 𝑝 (L 𝑓 ) in the Fourier domain with respect to 𝜽 (L 𝑘 𝑓 ) = GST(𝜹 (M 𝑘 𝑓 ), 𝜂 * 𝝎 𝑘 , 𝑝). Proof: Eq. ( 24) becomes to minEQUATION)in which 𝜃 𝑗 (K 𝑘 𝑓 ) denotes the 𝑗-th singular value of K 𝑘 𝑓 , and its corresponding weight is𝜔 𝑘 𝑗 = 1 𝜃 𝑟 𝑗 (K 𝑘 𝑓 )+𝜀. Initially, each weight 𝜔 𝑘 𝑗 is set as𝜔 𝑘 𝑗 = 1 𝛿 𝑗 (M 𝑘 𝑓 )+𝜀since 𝜃 𝑗 𝑟 (K 𝑘 𝑓 ) is unavailable in the first iteration, and updated based on the previous iteration of 𝜃 𝑗 𝑟 (K 𝑘 𝑓 ). Eq. ( 26) can be solved separately for different 𝑘 as minEQUATION)Solvers are derived using the following Theorem 4 and Lemma 1. Theorem 4. Consider the singular value decomposition (SVD) of matrixT ∈ R 𝐼 1 ×𝐼 2 as T = U 𝐴 * D 𝐴 * V ⊺ 𝐴, where 𝜂 > 0, 𝑟 = min(𝐼 1 , 𝐼 2 ), and 0 ≤ 𝜔1 ≤ 𝜔 2 ≤ . . . ≤ 𝜔 𝑟 . The global optimal solution for the following weighted Schatten 𝑝-norm minimization problem, adapted from [58] , is as follows:EQUATION)As shown in [59] , the optimal solution of Eq. ( 28) is given byEQUATION)where D 𝐴 = diag(𝜹), D𝜂, 𝝎, 𝑝 (T) = diag(𝜽 ). The vector 𝜹 = 𝛿 𝑗 (T) 𝑗 = 1 𝑟 represents the singular values of T, each of which can be obtained using Lemma 1 [65] . Lemma 1. Consider the 𝑘-th subproblem of Eq. ( 28), expressed asEQUATIONWithin 𝜔 and 𝑝, soft-thresholding function 𝜂𝑝 𝐺𝑆𝑇 (𝜔 𝑗 ) is defined asEQUATIONThe minimum 𝑆𝑣 𝐺𝑆𝑇 (𝛿 𝑗 , 𝜔 𝑗 ) of Eq. ( 31) is determined by 𝑇 𝐺𝑆𝑇 𝑝 (𝛿 𝑗 , 𝜔 𝑗 ) = 0,EQUATIONIn the optimization process, we set 𝜇 = 1𝑒 -4 and 𝜇 𝑚𝑎𝑥 = 10 10 , with a computational complexity of O (𝑛). Algorithm 1 delineates the entire optimization procedure of Eq. ( 14), wherein convergence is assessed by evaluating the objective value 𝑜𝑏 𝑗 𝑡 after the 𝑡-th iteration.Input: Multi-view data {X 𝑟 } 𝑣 𝑟 =1 , cluster number 𝑐, latent space dimension 𝑘, and parameters 𝛼, 𝛾 . Initialize Q 𝑟 = I 𝑘 , and the others matrices as 0. 1: repeat 2:Update A, W, B, and K via Eq (15) , Eq. ( 19), Eq. ( 20), and Eq. ( 21), respectively; Space Complexity. Primary space consumption is to store W 𝑟 ∈ R 𝑑 𝑟 ×𝑙 , X 𝑟 ∈ R 𝑑 𝑟 ×𝑛 , B 𝑟 ∈ R 𝑙 ×𝑚 , A 𝑟 ∈ R 𝑛×𝑚 , K ∈ R 𝑚×𝑛×𝑣 , 
experiments 4.1 experimental setting: 
 Benchmark Datasets. We conducted experiments on six datasets (YTF-50, YTF-20, SUNRGB-D, ORL, MSRCv1, and Yale) to evaluate our approach. The implementations of all competing methods were obtained from their respective public repositories or directly from the authors. Our experiments for shallow methods were carried out on a computing platform consisting of a 32GB RAM and Intel Core i7 CPU, using Matlab 2021b on a 2021 Mac mini. Compared Algorithms. We employed 11 State-Of-The-Art (SOTA) competitors, including AMGL [34] , FMR [17] , PMSC [12] , BMVC [64] , LMVSC [13] , SMVSC [43] , SFMC [22] , FMCNOF [60] , FPMVS [51] , SDAFG [31] , UDBGL [8] , FastMICE [10] , MVBGC [16] , and TBGL [57] to assess the effectiveness and efficiency of our IWTSN-FMGC. These competitors were selected based on their performance in four common metrics [13, 50, 54] cited in recent literature. Experimental Results. (i) ORL: Z 3 of IWTSN-FMGC Figure 1 (a)-(c) visualize the original {X 𝑟 } 𝑣 𝑟 =1 , while (d)-(i) depict the bipartite graphs {A 𝑟 } 𝑣 𝑟 =1 generated by TBGL and our IWTSN-FMGC on the ORL dataset, respectively. TBGL and our IWTSN-FMGC exhibit improved cluster structures compared to the original data. However, TBGL shows scattered structure distributions, leading to incorrect cluster partitions. In contrast, the visualizations (g)-(i) of our IWTSN-FMGC display fewer incorrect cluster assignments than those of TBGL, consistent with the clustering results in Table 2 . Additionally, Figure 2 illustrates the evolving clustering structures with increasing iterations, indicating progressively clearer patterns. These results affirm the effectiveness and superiority of our proposed method in learning reliable bipartite graphs.To demonstrate significance, effectiveness, and robustness of tensor Schatten 𝑝-norm, we degrade our IWTSN ∥ A∥ 𝑝 I𝜔,𝑠 𝑝 to ∥ A∥ 𝑝 𝑠 𝑝 , simplifying as our w/o. By performing experiments on different noise ratios on the Synthetic dataset in Fig. 3 , we observe that: 1) our method outperforms our w/o on all evaluated metrics in Fig. 3 Algorithm 1 incorporates two parameters, 𝛼 and 𝛾, which control the influence of neighbor bipartite graph learning and the improvement of weight tensor Schatten 𝑝-norm ∥ • ∥ I𝜔,𝑆 𝑝 , respectively. These parameters are varied within the range 2 [ -5:2:5] , with 𝑚 = 2𝑐 and 𝑙 = 2𝑐 fixed. Specifically, if 𝑑 𝑚𝑖𝑛 ≤ 2𝑐, then 𝑙 = 𝑑 𝑚𝑖𝑛 , where 𝑑 𝑚𝑖𝑛 represents the minimum dimension across all views. Fig. 4 illustrates the sensitivity analysis of our Algorithm 1 using two large-scale datasets (YTF-50 and YTF-20). The results demonstrate that our IWTSN-FMGC is minimally affected by variations in 𝛼 and 𝛾. Moreover, the losses depicted in Fig. 5 consistently decrease to a stable value with increasing iterations on the YTF-20 dataset, confirming robust convergence of our IWTSN-FMGC. 
conclusion: 
 This paper proposes a novel IWTSN to better capture the low-rank property hidden local bipartite graphs. Then, we employ the developed improved weighted tensor Schatten 𝑝-norm to perform fast local multi-view clustering by integrating the designed adaptive neighbor bipartite graph learning. Meanwhile, a well-designed solution is also provided to solve improved weighted tensor Schatten 𝑝-norm. Comprehensive experiments and analysis have proved the superiority and validity of our method.  
 
 Appendix:  


Decision: The paper received four reviews by confident/very confident reviewers, all of them quite positive, with  final recommendations accept, accept, borderline accept and weak accept.

The first  two reviewers find their doubts solved by the authors in the rebuttal phase. Also the third reviewer is satisfied by the authors’ replies, and indeed increases the final recommendation (from borderline reject to borderline accept). Interestingly, reviewers’ assessment about suitability to the conference ranges between moderately suitable to definitely suitable, but this doesn’t seem to be correlate to the final recommendation.
Taking into account all the mostly positive opinion of the reviewers I recommend acceptance for oral presentation

Reviews: review: paper_summary: paper_summary: This study introduces an innovative Fast Multi-view Graph Clustering model, IWTSN-FMGC. First, an improved weighted tensor Schatten 𝑝-norm (IWTSN) is developed to enhance robustness in adaptive neighbor bipartite graph learning, which is sensitive to noise. Concretely, IWTSN has the potential to more effectively leverage low-rank structures and prior information by dynamically ranking core tensor and automatically shrinking singular values. Then, IWTSN could improve the anchor quality and refine the local manifold structure information in local graph learning process
strength_weakness: strength_weakness: - Despite a wealth of theories and proofs, the manuscript is skillfully written, with concise explanations. The precise detailing ensures concepts are communicated clearly.
-Ranking the core tensor and automatically shrinking singular values for local bipartite graphs are skillful to improve their robustness.
-Comprehensive experiments on real-world and synthetic datasets provide robust empirical evidence, confirming the superior effectiveness of the proposed IWTSN-FMGC.
limitations: limitations: -It is necessary to make a further analysis about time cost comparison on large-scale datasets, such as YTF-50 and YTF-20.
-The paper contains extensive theoretical analysis and proofs. The authors should carefully check the formulas of this manuscript. Such as “E” in Theorem 2 while be $E^r$ in line 437.  
-In line 421, superscripts should be consistently formatted.
-I'm confused that why applying less shrinkage to larger singular values for improving the robustness?
paper_summary: paper_summary: This paper proposes a multi-view clustering method called Improved Weighted Tensor Schatten 𝑝-Norm for Fast Multi-view Graph Clustering (IWTSN-FMGC). IWTSN-FMGC integrates adaptive neighbor bipartite graph learning and low-rank tensor learning into a unified framework, which can handle large-scale datasets. Additionally, an improved weighted tensor Schatten 𝑝-norm is developed to dynamically rank core tensor and automatically shrink singular values. Extensive experiments are conducted to validate the effectiveness of the proposed IWTSN-FMGC.
strength_weakness: strength_weakness: 1. This paper is well-organized, and easy to follow.
2. This paper proposes a new tensor norm with a solver to achieve the greater robustness and effectiveness than the existing tensor norm. 
3. This paper proposes a novel large-scale multi-view clustering method by employing the proposed tensor norm.
4. Comprehensive experiments show the superiority of the proposed method over existing SOTA methods.
limitations: limitations: 1. The adaptive neighbor graph learning could encode local manifold structure information by selecting the proper neighbors for samples. But for neighbor bipartite graph in this paper, how to select the neighbors.
2. Wether the prior information to enhance the robustness refer to the singular values of core tensor?  
3. There are some typos in the paper, such as the suboptimal results under the Purity metric of the SUNRGBD dataset in Table 1 is not marked.
paper_summary: paper_summary: The paper introduces a new approach named Improved Weighted Tensor Schatten 𝑝-Norm for Fast Multi-view Graph Clustering (IWTSN-FMGC). This method aims to address limitations observed in current tensor Schatten 𝑝-norm methods by proposing an enhanced weighted tensor Schatten 𝑝-norm. This enhancement involves dynamically ranking the core tensor and automatically shrinking singular values. As a result, the improved method is expected to better utilize low-rank structures and prior information, potentially improving robustness compared to existing tensor Schatten 𝑝-norm methods. Additionally, the paper proposes an adaptive neighbor bipartite graph learning strategy, offering increased flexibility and expressiveness in encoding local manifold structure information compared to conventional anchor selection approaches. The effectiveness and superiority of the proposed method are demonstrated through extensive experiments conducted on various benchmark datasets.
strength_weakness: strength_weakness: 1. The proposed method demonstrates a linear relationship with the number of samples, enabling it to accommodate large-scale datasets.

2. In the experimental section, the proposed method exhibits superior performance compared to several state-of-the-art methods.
limitations: limitations: 1. The motivation section of the manuscript's introduction lacks clarity and coherence. While the authors propose an "Improved Weighted Tensor Schatten 𝑝-norm," they fail to provide an overview of any existing works based on Tensor Schatten 𝑝-norm in the introduction. It is worth noting that there are numerous works in this field currently. Additionally, the claim that existing tensor Schatten 𝑝-norm methods cannot fully exploit the low-rank property in the core tensor space lacks substantiation. It would be beneficial for the authors to elaborate on why existing methods fail to exploit this property fully and provide examples of their performance characteristics in this regard.

2. The authors claim that the proposed method is fast, but the experimental section lacks comparisons of runtime to substantiate this assertion.

3. The writing throughout the manuscript could benefit from further refinement.
paper_summary: paper_summary: This paper propose a novel Improved Weighted Tensor Schatten 𝑝-Norm for Fast Multi-view Graph Clustering (IWTSN-FMGC). IWTSN-FMGC first develops a novel Improved Weighted Tensor Schatten 𝑝-Norm (IWTSN) with an elegant solver to better rank the core tensor and take advantage of the prior information from singular values, enhancing robustness and validity compared to the widely-used tensor Schatten 𝑝-norm. Then, IWTSN-FMGC seamlessly couples IWTSN and adaptive neighbor bipartite graph learning for fast multi-view graph clustering.
strength_weakness: strength_weakness: Innovation: IWTSN-FMGC presents a new Improved Weighted Tensor Schatten 𝑝-Norm (IWTSN), offering increased robustness and validity over the commonly used tensor Schatten 𝑝-norm. Meanwhile, an elegant solver is developed to optimize the proposed IWTSN. 
Experiments: Comprehensive experiments and analysis have proved the superiority and validity of IWTSN-FMGC compared to SOTA competitors.
Scalability: Integrating adaptive neighbor bipartite graph learning with IWTSN synergistically enhances graph quality and robustness, yielding more refined and accurate local manifold structures. Here, IWTSN also has very good scalability to the other graph learning methods.
Code: Code is available to aid reproducibility.
limitations: limitations: 1.How to enhance robustness by developing IWTSN? 
2.What is the prior information from singular values?
3.What is the difference between adaptive neighbor bipartite graph learning and anchor-based subspace learning.