Automatic and Aligned Anchor Learning Strategy for Multi-View Clustering

Venue: acmmm_org_ACMMM_2024_Conference
Published: 2024-04-07 11:02:24

Abstract: Multi-view Clustering (MVC) generally utilizes the anchor technique to decrease the computational complexity so as to tackle large-scale scenarios.  Existing researches generally are supposed to select anchors in advance to complete the next clustering task. Nevertheless, the number of anchors cannot be predetermined and must be selected as a parameter, which introduces additional time consumption for parameter search. Moreover, maintaining an identical number of anchors across each view is not reasonable, as it restricts the representational capacity of anchors in individual views. To address the above issues, we propose a view adaptive anchor multi-view clustering called Multi-view Clustering with Automatic and Aligned Anchor (3AMVC). We introduce the Hierarchical Bipartite Neighbor Clustering (HBNC) strategy to adaptively select a suitable number of representative anchors from the original samples of each view. Specifically, When the representative difference of anchors lies in a acceptable and satisfactory range, the HBNC process is halted and picks out the final anchors. In addition, in response to the varying quantities of anchors across different views, we propose an innovative anchor alignment strategy. This approach initially evaluates the quality of anchors on each view based on the intra-cluster distance criterion and then proceeds to align based on the view with the highest-quality anchors. The carefully organized experiments well validate the effectiveness and strengthens of 3AMVC.

Main paper: 
 introduction: 
 Multi-view Clustering (MVC) [13, 39, 40, 50] generally delves into the intricacies of the similarity relationships among sample pairs, and constructs a graph matrix that captures the underlying structure and relationships within the dataset, thereby enabling the accurate completion of the clustering task. MVC typically begins by meticulously constructing individual graph structures for each distinct view. Subsequently, these distinct graphs are synthesized through a fusion process designed to extract a consensus graph with enhanced quality [8, 20, 36, 42] . For example, Kang et al. [8] unified the exploration graph structure, fusion graph and spectral clustering process in one step. Wang et al. [36] introduced rank constraints to individual graphs in each view, and then integrated these high-quality graphs to obtain a clearer graph of the consensus clustering structure. Wu et al. [42] proposed a voting model for extracting neighbors across views, which can directly obtain clustering labels by skipping the traditional spectral decomposition procedure.Although a substantial number of research affirming the capability of MVC in enhancing the clustering performance, the sheer scale of data encountered in practical applications still remains a formidable challenge [4, 27] . Traditional MVC methods, which necessitate the construction of graph matrices at a minimum cost of the square of the sample size 𝑛, are rendered powerless in the face of large-scale datasets [7, 9, 48] . The anchor graph strategy [16, 28, 31, 32, 38, 43, 49, 50] can address the challenges posed by large-scale data application scenarios. Its basic idea is to approximate the overall sample using 𝑚 anchors on each view. Obviously, the consensus anchor graph has a significant impact on performance. Consequently, a multitude of studies are dedicated to exploring the selection and fusion processes of anchors, aiming to enhance the quality of the consensus anchor graph [32, 37, 44, 50] . K-means becomes the mainstream anchor generation method because of its linear complexity and the ability to preserve the inherent characteristics of the clustering structure. Traditional methods such as [44, 50] use K-means to select anchors on each view and then subsequently fuse them. Sun et al. [32] employed the direct optimization of the consensus anchor matrix, replacing the process of individual view optimization followed by the fusion of anchor matrices. Wang et al. [37] identified the Anchor-Unaligned Problem (AUP), and performed extensive experiments to demonstrate that the alignment of anchors exerts a profound influence on the performance of clustering.Despite that the aforementioned approaches have effectively improved the quality of anchor graph and improved the adaptability of multi-view clustering tasks in large-scale data scenarios, they collectively face an limitation, i.e., the selection of anchors necessitates supplementary procedures. The majority of existing approaches employ the correlation number of the cluster count 𝑘 [14, 35] or the correlation number of the sample size 𝑛 [12, 28] to judiciously determine the optimal quantity of anchors. Even if the K-means method may preserve the clustering structure to a certain degree, the method of selecting tentative values struggles to approximate the optimal anchors, let alone random sampling. Additionally, although Wang et al. has delved into establishing the correct correspondence between anchors across views [37] , it does not necessarily select the view with the highest quality anchor graph as the reference for alignment. As far as we know, there exists no universal framework capable of automatically selecting high-quality anchors and aligning them based on the optimal view. This is largely because that it is inherently a challenging problem to perform clustering with an unknown number of clusters on a single view.In this paper, we propose a general multi-view automatic anchor graph fusion framework termed Automatic and Aligned Anchor for Multi-view Clustering (3AMVC). Specifically, we propose a Hierarchical Bipartite Neighbor Clustering (HBNC) method, which can complete clustering without specifying the number of clusters, so as to automatically obtain high-quality anchors on each view. Then, we improve the alignment method of Wang et al. [37] , so that it can explore the similarity relationship of anchors between two views when the number of anchors in each view is inconsistent. In addition, we record the similarity relationship between the anchors and the samples during the process of automatically anchor selection, and use it to represent the quality of the anchor graph of the view, so as to determine the optimal view before alignment. Figure . 1 shows the difference between our method and the traditional anchor alignment framework. The traditional alignment framework fails to discern and exploit anchors of superior quality within a single view, and is more likely to perform alignment tasks based on suboptimal views because it lacks a systematic evaluation mechanism for the quality of the anchor graphs. While our method can automatically select better quality anchors on a single view and complete the alignment work based on the view with higher anchor graph quality. We summarize the contributions as follows:• We propose a novel Hierarchical Bipartite Neighbor Clustering (HBNC) algorithm to identify and select high-quality anchors on a single view. It does not require a prerequisite specification number of anchors and is capable of preserving the inherent characteristics of data structures like K-means.• We design a general multi-view automatic anchor graph fusion framework termed Multi-view Clustering with Automatic and Aligned Anchor (3AMVC), which can align and fuse anchor graphs according to the view with highest quality anchor graph when the number of anchors is inconsistent.• Extensive experiments on five benchmark datasets show the effectiveness and efficiency of the proposed method. Compared with the traditional alignment framework FMVACC, the proposed method has improved both in clustering performance and running time. 
related work 2.1 multi-view anchor graph clustering: 
 Recently, due to the superior non-linear performance of multiview graph clustering, it has garnered increasingly extensive research attention [8, 13, 25, 33, 34, 41] . However, the construction of a complete graph matrix requires a computational complexity that is at least quadratic in the number of samples 𝑛, which often proves inadequate when confronted with large-scale of data scenarios. Therefore, many studies have turned to employ representative anchors for the construction of anchor graphs instead of the complete graphs, which can effectively reduce the quadratic computational complexity to a linear correlation with the number of samples [15, 20, 30, 32, 38, 46] . Specifically, multi-view anchor graph clustering basically follows a two-step strategy of optimization-fusion. The first step is to optimize the anchor graph on each view as follows: minEQUATIONwhere Z 𝑖 ∈ R 𝑛×𝑚 and A 𝑖 ∈ R 𝑚×𝑑 𝑖 is the anchor graph and the anchor matrix of the 𝑖-th view respectively. Z 𝑖 1 𝑚 = 1 𝑛 is a sum-toone constraint that ensures that the sum of the sample similarity corresponding to the anchors equals 1. In the second step, the anchor graphs in each view are fused through:EQUATIONwhere S ∈ R 𝑛×𝑚 is the fused anchor graph, 𝜷 ∈ R 𝑣 is the weight coefficient which measures the impact of each view on the fused anchor graph, and 𝜷 ⊤ 1 𝑣 = 1 ensures that the sum of weights is 1 and satisfies constraint S1 𝑚 = 1 𝑛 . After spectral decomposition of the fusion anchor graph, Kmeans is performed on the left singular vector to obtain the final clustering result. Multi-view anchor graph clustering can reduce the time and space complexity to O (𝑣𝑚𝑛) and O (𝑛𝑚 2 ), respectively and effectively adapt to the large-scale data scenarios.Although the existing approaches can well solve the problem of large-scale multi-view clustering, they all encountered a common problem, that is, the anchors are established before the optimization process commences, hence the quality of these anchors plays a pivotal role to the performance of the clustering. Most existing methods take random sampling and K-means [9, 12] to select anchors, however, both of which require a given number of anchors. This prerequisite number of anchors then becomes a hyperparameter which requires time for searching.The concept of anchors gains traction in the realm of large-scale multi-view clustering. Wang et al. [37] have further identified a critical, yet previously overlooked issue within the selection and fusion method which is termed as Anchor-Unaligned Problem (AUP).An intuitive solution to AUP is to avoid random anchor sampling and to select the same anchor set across views as in [11] . However, this approach restricts the model flexibility and uses the same set of anchors across various views. Both intuitively and existing studies have shown that learning more representative anchors can achieve improved performance over random sampling. Another solution is to optimize the consensus anchors representing cross-view features in a latent common-shared space [32] . Liu et al. [16] added constraints to the consensus anchor graph, so that the connectivity of the anchor graph equals to the number of clusters. Nonetheless, employing projections within the common space will inevitably result in the loss of unique, view-specific information. Consequently, the strategic and flexible selection of anchors within a single view, coupled with a fusion process grounded in accurate correspondences, emerges as a superior solution to ensure clustering performance in large-scale scenarian.Wang et al. [37] proposed a Fast Multi-View Anchor-Correspondence Clustering (FMVACC) method to solve AUP. Specifically, FMVACC is to retain the feature and structure information of the anchor set for anchor matching. The optimization formulation can be expressed as:EQUATIONwhere∥Z 1 -Z 𝑖 P∥ 2 F is the feature correspondence and ∥G 1 -PG 𝑖 P∥ 2 Fis the structure correspondence. Moreover, 𝜆 is a parameter for balancing two terms, G denotes the graph structure inside the anchor set, Z is the respective anchor graph which represents the similarities between the samples and the anchors in each view, 𝑖 ∈ {2, 3, ..., 𝑣 } represents the other views expect the 1-st one.The above formulation indicates that the anchor graphs in the other views are aligned to the anchor graph in the 1-st view from the feature and structure correspondence simultaneously. While FMVACC addresses the cross-view alignment of anchors with different dimensions, it still falls short in thoroughly exploring view with superior quality, hence it simplistically adopts the first view as the baseline for alignment tasks. Should the anchor graph derived from the 1-st view fail to capture an precise representation of the underlying structure, and if the alignment between the anchor and the sample is not close enough, then there exists a risk. Specifically, it could lead to subsequent views aligning based on a flawed relationship. Furthermore, since the existing anchor graph clustering methods all use the same number of anchors across views, FM-VACC does not consider the scene where the number of anchors is different across views, so it lacks the capability to effectively align disparate anchors.At present, the prevailing array of methods inherently assume the number of clusters as a pre-defined parameter within the clustering process. For instance, the K-means and k-NN methods, which are widely applied throughout clustering procedures [5, 17-19, 26, 47] , are both derived from the premise of a predefined number of clusters 𝑘.  
the proposed method: 
 In this section, we introduce the method for selecting high-quality anchors on a single view, and elaborate the alignment strategies of these anchors in relation to the baseline view. overall framework of our Multi-view Clustering with Automatic and Aligned Anchor (3AMVC). Our method takes three steps: i) an Adaptive Anchor Selection Module to select the appropriate anchors on a single view, ii) a Cross-View Anchor Alignment Module to align the views based on the view with best quality of the anchor graph, and iii) finally performing fusion to the aligned views.The fundamental premise of our anchor concept is to effectively address the challenges posed by large-scale data so as to make the computational complexity involved in the selection of anchors within a single view affordable. Balance K-means based Hierarchical K-means (BKHK) method [52] offers a novel approach by combining the flexibility of hierarchical clustering with a relatively low computational complexity, providing us with a fresh perspective on tackling these challenges. However, BKHK imposes excessively stringent demands on the equalization performance of the dataset, and it specifies a termination condition based on the number of clusters in the hierarchical clustering process.Building upon the foundation of BKHK, we introduce an effective Hierarchical Bipartite Neighbor Clustering (HBNC) method that enable it to better accommodate datasets with uneven intracluster sample sizes and to automatically determine the optimal termination point. Figure . 3 shows the specific process of BKHK and our HBNC. BKHK classifies all samples into two categories layer by layer based on the distance between samples, and finally obtains 𝑘 clusters with balanced number of internal samples. However, real data sets often do not follow the rules of intra-cluster equilibrium, and BKHK will fail in such cases such as part (a) of Figure . 3. In comparison, our method can determine the relevant optimal threshold, so as to flexibly divide the samples inside and outside the target cluster.Intuitively, the relationships among samples within the same cluster should be markedly distinct from those between samples belonging to different clusters. As depicted in Figure 3 (c), there is a significant disparity in the distances between the target sample and the samples outside the cluster compared to those within its own cluster.Based on this principle, we design a new criterion for optimizing the optimal threshold to determine the neighbors of the target sample in the range of sample to be clustered. Take the 𝑙-th layer clustering process as an example, we first randomly select a target sample x 𝑖 from 𝑛 𝑙 samples to be clustered, and calculate the distance between the target sample and other samples denoted as d ∈ R 𝑛 𝑙 ×1 . After sorting d, we get the vector form asd = 𝑑 𝑖 [1] , 𝑑 𝑖 [2] , ..., 𝑑 𝑖 [𝑛 𝑙 ]⊤ . Our goal is to find an optimal threshold 𝑇 𝑖 related to the sample x 𝑖 , so as to divide the nearest neighbors of the target sample. The optimal threshold 𝑇 𝑖 should satisfy the following condition:EQUATIONIn this paper, we design an ingenious formula to determine the appropriate threshold 𝑇 𝑖 . Inspired by the grayscale binarization in the field of image segmentation [24] , we first introduce the concept of inter-class error. Regardless of the layer in which it is performed, the clustering task at hand involves categorizing the samples into two groups. For convenience, we consider the cluster containing the target sample as the first cluster, and all other samples as the second cluster. We represent the proportion of each type of sample within the entire samples as 𝑝 1 and 𝑝 2 , as well as the average distance between each cluster of samples and the target sample as 𝜇 1 and 𝜇 2 . Then the inter-class error can be expressed as:EQUATION)where 𝜇 is the mean value of all samples. The proportion and mean satisfy the following conditions: 𝑝 1 + 𝑝 2 = 1 and 𝑝 1 𝜇 1 + 𝑝 2 𝜇 2 = 𝜇. Based on that, we can transform Eq. ( 5) as follows:EQUATION)Our expectation for the optimal threshold is that a threshold can make the distance between samples in the cluster smaller and at the same time the inter-class error larger. Therefore, the criterion for determining the optimal threshold can be formulated as follows:EQUATIONwhere 𝑘 𝑖 is the number of samples in the first cluster with the target sample.Since the number of neighbors of the target sample is discrete and finite, constrained within {1, 2, ..., 𝑛 𝑙 }, the value of Eq. ( 7) is also discrete and finite. Consequently, by determining the 𝑘 𝑖 neighbors of the target sample x 𝑖 that can minimize the Eq. ( 7), we can obtain the optimal threshold 𝑇 𝑖 ∈ 𝑑 𝑖 [𝑘 𝑖 ] , 𝑑 𝑖 [𝑘 𝑖 +1] . In practice, we only need 𝑇 𝑖 as an exact value, so for the sake of simplicity, we let Algorithm 1 Bipartite neighbor clustering of the 𝑙-th layer Input: Samples to be clustered X l ∈ R 𝑑 ×𝑛 𝑙 .1: Randomly select a sample as the target sample x 𝑖 . 2: Calculate the distance d 𝑙 between all samples and the target sample. 3: Sort the vector d 𝑙 to get d𝑙 . 4: Calculate the neighbors and the optimal threshold of the target sample according to Eq. ( 7). Output: The neighbors and the optimal threshold of the target sample x 𝑖 .𝑇 𝑖 = 𝑑 𝑖 [𝑘 𝑖 ] . We summarize this bipartite neighbor clustering process in the 𝑙-th layer as Algorithm 1.It is widely recognized that the closer the samples within a cluster are to the clustering center, the more concentrated the distribution of samples in that cluster, and consequently, the more representative the cluster center becomes. K-means clusters according to this principle. For a specific cluster 𝑖, the problem can be expressed as: ∑︁EQUATIONwhere Θ 𝑖 denotes the 𝑖-th cluster, x 𝑗 ∈ Θ 𝑖 represents its samples, 𝜃 𝑖 denotes the center of the cluster and𝜃 𝑖 = 1 𝑐 𝑖 𝑐 𝑖 𝑗=1x 𝑗 with 𝑐 𝑖 being the number of samples in this cluster.The proposed Hierarchical Bipartite Neighbor Clustering (HBNC) algorithm introduces a new cluster through the binary classification task of each layer. Consequently, if our algorithm performs through 𝐿 layers, then similar to K-means, the intra-cluster distance of all clusters in our algorithm is expressed as:EQUATIONAlthough our HBNC algorithm operates without a predetermined number of clusters, by effectively managing the stopping criteria of the algorithm, we can halt the clustering process at an opportune moment, thereby ascertaining the appropriate number of clusters. The clustering result should lead to the centers of all clusters being highly representative. Therefore, at the beginning of the clustering task at each layer, we select the range of the sample to be clustered based on the criterion that the distance between the center and the sample within the cluster is maximized for the chosen cluster. It is imperative to avoid obtaining many clustering centers to represent outliers, and avoid huge differences in the value of Eq. ( 8) of each cluster. Here, at most one cluster center representing outliers is acceptable. Therefore, when a cluster is with a sample size of 1 again or the target value of the Eq. ( 8) of all clusters tends to average, we terminate the algorithm and obtain a more representative anchor set. The complete HBNC process ise elaborated in Algorithm 2.As for the complexity of the algorithm, the computational complexity of each layer of HBNC is to calculate the distance of all samples within the sample range of the layer, which equals to O (𝑛 𝑙 𝑑). Consequently, the complexity of the whole HBNC is O ((𝑛 1 + 𝑛 2 + ...𝑛 𝐿 )𝑑), where 𝐿 is the number of layers. Relative to the first layer, which processes the entire dataset of 𝑛 1 = 𝑛 entries, the computational complexity of the subsequent layers is significantly reduced. Update the layer 𝑙.Determine the range of samples to be clustered X 𝑙 ∈ R 𝑑 ×𝑛 𝑙 .Complete the bipartite neighbor clustering of the 𝑙-th layer according to Algorithm 1.Calculate the intra-distance of each cluster according to Eq. ( 8). 6: end while Output: Anchor quality according to Eq. ( 9) and the anchors Θ.As such, the overall complexity of our HBNC algorithm can be denoted as O (𝑛𝑑).The unequal number of anchors automatically selected on each view compels us to further explore one-to-many or many-to-one matching relationships, rather than being constrained by the strict one-to-one matching framework. In addition, matching based on the view where the higher quality anchor set is located should bring better clustering performance.In the previous section, we propose a formulation in Eq. ( 9) to simultaneously adeptly capture the relationship between the global clustering center and the samples and reflect the representative performance of the clustering center essentially. In this section, we therefore employ Eq. ( 9) as a metric of anchor representativeness to assess the quality of the anchor graph for each individual view. In the process of devising anchor quality metrics for views, we discern that the summation approach offers numerous advantages over the mean-based formulations. Since our ultimate goal is to perform spectral clustering on the fusion anchor graph, when the distribution of a certain type of samples forms a ring shape, it is preferable that the selected anchors also follow a ring pattern, rather than being at the center of the ring. For the sample-center distance in the same cluster like Eq. ( 8), sum rather than average, can avoid the ring distribution of the samples associated with the selected anchor. In addition, it is clear that under the same sample distribution, the more the number of anchors, the stronger the average representation of each anchor. Considering the variability in the number of anchors selected for each view, relying on an average-based calculation may inadvertently place views with fewer anchors at a relative disadvantage. By applying a summation technique universally, the representativeness of the anchors is effectively distributed among all samples, making it a more advantageous strategy. Ultimately, we select the view with the smallest value from Eq. ( 9), where the anchors exhibit the strongest representativeness and all samples are closer to their respective anchors, to serve as the baseline view.According to Wang et al. [37] , we can align the anchors in terms of feature and structure by summing the values of the anchors in different dimensions. In order to match the optimal view and our inconsistent number of anchors, our improved alignment method is expressed as follows:EQUATIONwhere Z 𝑏 ∈ R 𝑛×𝑚 𝑏 is the anchor graph of the baseline view, G 𝑏 and G 𝑖 represent the inner graph structures of the baseline view and the 𝑖-th view. It is noteworthy that the matching matrix P here is specific for each view and can express a one-to-many or many-to-one relationship about the baseline view.After obtaining {P} 𝑣 𝑖=1,𝑖≠𝑏 , we can obtain the fused aligned anchor graph based on the baseline view as:EQUATIONUltimately, by performing spectral decomposition on Z 𝐴𝑙𝑖𝑔𝑛𝑒𝑑 according to Eq. ( 11), and subsequently applying K-means clustering to its left singular values, the final clustering results are successfully derived.In terms of space complexity, our major memory costs are matrices{Θ 𝑖 } 𝑣 𝑖=1 ∈ R 𝑚 𝑖 ×𝑑 𝑖 , {P 𝑖 } 𝑣 𝑖=1,𝑖≠𝑏 ∈ R 𝑚 𝑖 ×𝑚 𝑏 , {Z 𝑖 } 𝑣 𝑖=1 ∈ R 𝑛×𝑚 𝑖 . The space complexity of 3AMVC is O (𝑚 𝑚𝑎𝑥 𝑑 + 𝑣 𝑖=1 𝑚 𝑖 𝑛 + 𝑚 2 𝑏 𝑣). 𝑚 𝑚𝑎𝑥 is the largest number of anchors in all views. 𝑚 𝑏 is the number of anchors in the baseline view. In large-scale scenarios, the main amount of computation is mainly the correlation term of 𝑛. Therefore, the space complexity of 3AMVC is O (𝑛).As for time complexity, which consists of three steps. As mentioned above, the time complexity of HBNC on a single view is O ((𝑛 1 + 𝑛 2 + ...𝑛 𝐿 )𝑑). In the alignment phase, updating {Z 𝑖 } 𝑣 𝑖=1 costs O (𝑛𝑚 𝑖 𝑑 𝑖 ). Updating { Θ 𝑖 } 𝑣 𝑖=1 needs O (𝑛𝑚 𝑖 𝑚 𝑏 +𝑚 𝑖 𝑚 𝑏 𝑑). Then the alignment module costs O (𝑚 2𝑏 𝑚 𝑖 ). After obtaining the aligned and fused anchor graph, SVD needs O (𝑛𝑚 2 𝑏 ). Consequently, the time complexity of our 3AMVC is approximately O (𝑛), which can better cope with large-scale data scenarios. 
experiment 4.1 experiment setting: 
 We verify the effectiveness of our algorithm on five widely used multi-view benchmark datasets: ForestTypes, Reuters, MFeat, Cal-tech256, and VGGFace2. The information of these datasets is listed in Tab. 1. We compare our method with 8 state-of-the-art multiview clustering methods: BMVC [51] , LMVSC [9] , FMCNOF [45] , FPMVS-CAG [38] , EOMSC-CA [16] , FMVACC [37] , UDBGL [3] , and fastMICE [4] . We set their parameters within the recommended range for all comparison algorithms aforementioned. In the proposed method, we adjusted 𝛽 to [10 -2 , 1, 10 2 ], 𝜆 to [0, 10 -4 , 10 -2 , 1, 10 4 ]. To assess the clustering performance, we employ three well-used criteria consisting of accuracy (ACC), normalized mutual information (NMI) and Fscore.Table . 2 reports the clustering results of other methods and our 3AMVC on five benchmark datasets. The best results are marked in red, while the second-best results are marked in blue. According to the results, we have the following observations:• Our method exceeds other algorithms in most indicators, or reaches a considerable level with other algorithms. In terms of the ACC metric, 3AMVC achieves improvements of 0.69% (ForestTypes), 3.65% (Reuters),4.97%(MFeat) 12.29%(Catech256), and 13.17%(VGGFace2) respectively, when compared to the troditional alignment algorithm FMVACC. • On other datasets, we also obtain comparable results with the suboptimal algorithm, and it is worth noting that almost all the algorithms compared require the pre-K-means process to select the anchors, while our algorithm avoids the process of searching for the optimal number of anchors.Our experimental setup involves two hyperparameters, namely, 𝛽 and 𝜆. As shown in Figure . 4, we use the ACC metric to show the influence of two parameters on the clustering effect on two datasets: ForestTypes and Caltech256. We observe that the influence of parameter changes on the clustering effect is not intensive. Relatively speaking, when 𝛽 is set as 10 2 and 𝜆 is set as 10 4 , the clustering performance will be further improved.In order to illustrate the ability of our 3AMVC in anchor selection, we employ visual representations to display the selected anchors in the single view. We run K-means and our HBNC algorithm on the two views of MFeat to select the appropriate anchors, and the results are shown in Figure . 5. The samples are represented by points, and the selected anchor points are uniformly represented by red stars. From Figure . 5, we observe that although K-means can better cluster sample points into 𝑘 clusters based on characteristics, the approach, which utilizes cluster centers as representatives for anchor points, struggles to preserve the inherent graph structure within the clusters. Our HBNC algorithm effectively secures a set of more representative anchor points without the need to predefine the number of clusters. On one hand, the samples represented by these anchors exhibit a higher degree of similarity to them, which in turn more accurately captures the graph's structure. On the other hand, this approach also circumvents potential inaccuracies that may arise from anchors being positioned in the interstitial gaps between samples. On the MFeat dataset, HBNC finally selected 54 and 64 anchors on the two views respectively.In order to illustrate the effectiveness of our automatic anchor selection strategy and baseline view alignment strategy, we conducted ablation experiments on these two parts respectively. The experimental results are shown in Table 3 . Methods that do not use either strategy are recorded as 3𝐴_𝑛𝑒𝑖𝑡ℎ𝑒𝑟 . 3𝐴_𝑤/𝑜_𝑆𝐴 means that the anchors are not automatically selected, and we use the K-means method instead. 3𝐴_𝑤/𝑜_𝐴𝐵 stands for not aligning based on the baseline view, we align based on the first view uniformly. 3𝐴𝑀𝑉𝐶 means the complete algorithm. We conduct experiments under the best parameter settings of each dataset. From the Table 3 , we find that the performance of 3𝐴_𝑤/𝑜_𝑆𝐴 is significantly improved on large datasets. This fully demonstrates the the importance of high-quality anchor selection for improving large-scale clustering performance. In addition, we found that the performance of 3𝐴_𝑤/𝑜_𝑆𝐴 is lower than 3𝐴_𝑛𝑒𝑖𝑡ℎ𝑒𝑟 on some datasets, which we believe is due to the mismatch between the anchor points selected by K-means and our view quality metrics.Our method can be well applied to large-scale scenes. We record our time comparison with other algorithms and draw Figure 6 . Compared with FMVACC, our algorithm has lower running time. We speculate that this may be due to the computational complexity of the K-means anchor selection strategy in each iteration is 𝑛, and the computational complexity of our algorithm will decrease when iterating again. 
conclusion: 
 In this paper, we introduce a novel Multi-view Clustering with Automatic and Aligned Anchor (3AMVC) method designed to address challenges of multi-view clustering in large-scale data scenarios. The algorithm initiates with the innovative Hierarchical Bipartite Neighbor Clustering (HBNC) approach, which eliminates the need for a predefined number of clusters. This method identifies and selects representative anchors within a single view, enabling the construction of a robust anchor graph without relying on prior knowledge. The subsequent phase involves the alignment of all anchor graphs to the baseline view exhibiting the highest anchor graph quality, culminating in the formation of a consolidated, highquality fused anchor graph. Our proposed method stands out in the current research landscape by offering an effective parameter-free strategy for anchor point selection. Extensive experiments verify the effectiveness and efficiency of our proposed 3AMVC.  
 
 Appendix:  


Decision: The paper initially received the following ratings: BR(C1kY), WA(VHsG), BR(XUZj), BA(Ytja). After rebuttal and discussion, the final ratings were: BA(C1kY), WA(VHsG), BA(XUZj), BA(Ytja).

Below are the details of the rebuttal:
1. Reviewer C1kY and Reviewer XUZj were almost satisfied with the response and ultimately raised their ratings.

The AC carefully reviewed the rebuttal:
1. This paper is generally well-written and well-organized.
2. Regarding the contribution of the technique, the AC agrees that 3AMVC is a novel work that selects different numbers of anchors within each view by introducing and modifying hierarchical clustering. This point may provide a good perspective to inspire the design of more effective multi-view clustering algorithms in the future. Meanwhile, the AC recommends the author on their valuable insights based on the experimental results.
3. In rebuttal, the author added a supplementary experiment to a large-scale dataset and showed a good performance. This verifies the superiority of the proposed method when applied in large-scale scenarios.

In sum, after the rebuttal, the AC believes the concerns raised by the reviewers were well-addressed and recommends acceptance of this paper. In the final version, the AC strongly encourages the authors to include all discussions from the rebuttal and improve the presentation to enhance readability for a diverse audience.

Reviews: review: paper_summary: paper_summary: The paper makes contributions to the field of multi-view clustering, particularly in handling large-scale data with varying anchor quantities across views. It successfully integrates innovative clustering techniques with practical alignment strategies, enhancing both theoretical and practical aspects of multi-view clustering. However, further exploration into the scalability, dependency on initial conditions, and potential overfitting could make the approach more robust and broadly applicable.
strength_weakness: strength_weakness: The manuscript introduces a novel approach to multi-view clustering that addresses several challenges with existing methods. Notably, the paper proposes an automatic and aligned anchor learning strategy that optimizes the selection and alignment of anchors across different views without needing predefined numbers of clusters or anchors. This approach is particularly beneficial for handling large-scale datasets where traditional methods struggle due to computational complexity. The Hierarchical Bipartite Neighbor Clustering (HBNC) method allows for adaptive selection of anchors, potentially enhancing the clustering accuracy. Extensive experiments validate the effectiveness of the approach, indicating improvements over traditional methods in both clustering performance and computational efficiency.
limitations: limitations: Despite its strengths, the paper has several limitations. The method's performance, while superior in some respects, might still depend heavily on the initial conditions and specific characteristics of the data used in the experiments. There is also a lack of detailed discussion on the scalability of the approach when applied to even larger datasets or more complex multi-view scenarios beyond the ones tested. Additionally, the complexity of the algorithm, while reduced compared to traditional methods, might still pose challenges in terms of interpretability and ease of implementation. The paper could benefit from a clearer explanation of the mathematical formulations and a more thorough theoretical analysis to support the empirical findings. Specifically, there are the following questions:

1、The introduction effectively sets up the motivation for the study. Consider highlighting the practical implications and potential applications of the framework earlier in the introduction to engage a broader audience.

2、The description of the Hierarchical Bipartite Neighbor Clustering (HBNC) method is detailed but could be improved with a flowchart or diagram that illustrates the steps visually.

3、Expand on the criteria for dataset selection and provide more detailed statistical analysis to support the robustness of the findings. Consider using box plots or error bars in graphical results to show variability and confidence intervals.

4、Deepen the discussion on how the results compare with existing methods beyond just performance metrics. Discuss any unexpected results or anomalies in the data.

5、The conclusion could better summarize the key findings and their implications. Future work should address potential limitations and suggest specific areas for further research.
paper_summary: paper_summary: This paper proposes a novel multi-view clustering method named 3AMVC, which not only skillfully combines the advantages of automatic anchor selection and baseline view alignment, but also finally obtains a high-quality consensus anchor graph for spectral clustering. There are two main innovations in this paper. First, different from traditional anchor selection methods, the framework this paper proposed 3AMVC does not require specifying the number of anchors. Second, it improves the traditional anchor alignment framework so that it can complete the fusion according to the view with the best quality of the anchor graph.
strength_weakness: strength_weakness: (1) In this paper, the author has created a delicate and clear illustration that effectively presents the central concepts introduced. 
(2) The paper thoroughly elucidates the rationale and efficacy of the proposed method.
(3) The writing is notably smooth and clear.
limitations: limitations: (1) In this paper, the anchor selection process and threshold determination process of HBNC are given in detail, but HBNC is only a module of 3AMVC, and the whole algorithm is not comprehensively summarized in this paper. It should be considered to add the algorithm.
(2) The formula 9 designed by the author is essentially used to measure the representative performance of anchors. The representativeness of the anchor does not directly correlate with the quality of the anchor. However, is it reasonable to use the representative performance of anchors as an evaluation criterion for the quality of anchor graphs in the single view?
(3) From the experimental results of running time, 3AMVC does not achieve the best results. It even has the longest running time on ForestTypes and Reuters datasets. In addition, it is worse than BMVC on large-scale datasets Caltech256 and VGGFace2.Please explain the above phenomenon regarding running time.
paper_summary: paper_summary: To avoid the additional parameter search steps caused by determining the optimal number of anchors in large-scale multi-view clustering tasks, this paper proposes a multi-view clustering method named 3AMVC that can automatically select automatic and aligned anchors. Specifically, the improved HBNC method is used to select the representative anchors on each view, and then the aligned consensus anchor graph is obtained by the fusion of all view anchor graphs according to the most representative one.
strength_weakness: strength_weakness: 1- This paper boasts a well-organized structure and coherent logic. 
2- The effectiveness of the proposed method is meticulously verified through an extensive array of experiments.
limitations: limitations: 1- In Figure 1, why cannot this method choose view 1 as the baseline view? If view 1 is selected as the baseline view, will the clustering performance be affected due to the inconsistency between the number of anchors and the number of clusters?
2- From Figures 1 and 2, it is observed that the elements in the permutation matrix P are decimals ranging from 0 to 1. However, in the final target equation (10), the matrix P's values are integers, specifically 0 or 1, please explain this phenomenon.
3- This paper proposes a novel HBNC method to select the most representative anchors. Specifically, the OTSU method is applied to the distance between samples, so as to find the optimal threshold for the target sample to complete the binary classification. However, the distance metric is greatly affected by the target sample. Changes in the target sample will lead to changes in the overall distance vector. This paper adopts a random method to select the target sample. This raises the question of whether the experimental results are significantly affected by instability.
4- It is widely acknowledged that the key to the algorithm with the unknown number of clusters is to determine the stopping condition, because the stopping time is closely linked to the final number of clusters. While Algorithm 2 provides a comprehensive depiction of HBNC's process for the specific selection of anchors, it falls short of delineating the pivotal stopping conditions. And there is no convergence proof of HBNC in this paper. Please further explain how to determine the number of clusters on a single view.
If the authors can address my concerns above, I would be happy to increase the rating.
paper_summary: paper_summary: Traditional multi-view clustering methods often suffer from scalability due to the quadratic time complexity. The anchor graph is a great strategy to address this challenge in which the quality of anchors is crucial for the performance. This paper proposes a novel method to adaptively select high-quality anchors for each view to achieve better clustering performance. The author further solves the anchor unaligned problem especially when the number of anchors in different views may not agree. Experiments on several datasets demonstrate certain superiority of 3AMVC compared to existing methods.
strength_weakness: strength_weakness: 3AMVC may be the first work to select different numbers of anchors within each view by introducing and modifying hierarchical clustering. Inter- and intra-cluster distances are employed to determine the partition border, which is impressive. The authors further modify the formula proposed by Wang et al. to handle the unaligned anchors of various numbers. Extensive ablation studies prove the effectiveness of the proposed modules.
limitations: limitations: 1.	It would be better to explain the optimization process of Eq. (10) more explicitly.
2.	The comparison with nine baseline methods on five benchmarks may not be convincing enough to demonstrate the superiority of 3AMVC. Hypothesis testing would be needed if possible.
3.	I wonder why 3AMVC takes more time on smaller datasets like ForestTypes and Reuters.