Add files using upload-large-folder tool

paper_markdowns/bamboo-03963.md

@@ -57,7 +57,7 @@ We give a formal definition of federated multi-view clustering. Suppose there is
 
 $K$ clusters. Each client has only one view-specific data $X ^ { m } = \{ x _ { 1 } ^ { m } , x _ { 2 } ^ { m } , . . . , x _ { N _ { m } } ^ { m } \} \in \mathbb { R } ^ { N _ { m } \times D _ { m } }$ , where $D _ { m }$ denotes the dimensionality of the samples in view $m$ . Due to incomplete samples in the clients, $N _ { m } \leq N$ . In order to facilitate the processing, for those missing samples, we use the zero vector 0 to fill in, i.e., $X ^ { m } \in \mathbb { R } ^ { N \times D _ { m } }$ . The graph structure of the raw data is represented by adjacency matrices $A = \{ A ^ { 1 } , A ^ { 2 } , . . . , A ^ { M } \}$ , and $A ^ { m } \in \{ 0 , 1 \} ^ { N \times N }$ $A _ { i j } ^ { m } = 1$ or 0 denotes the presence or absence of edges between $x _ { i } ^ { m }$ and $x _ { j } ^ { m }$ . The adjacency matrix of the fused graph from the server is denoted as $\overline { { A } } \in \{ 0 , 1 \} ^ { N \times N }$ .
 
-Our model architecture consists of $M$ clients and one server. Each client utilizes its private data for local view training. The high-level features $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ are obtained through two level feature extraction guided by the globally fused graph $\overline { { A } }$ . Then clustering is performed based on them. Based on the clustering results, we compute the weights $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ m}Mm=1 by highand re. After that, the server receives from the clients and performs fe $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ graph fusion and global clustering. The overview illustration of the model is shown in Figure 1.
+Our model architecture consists of $M$ clients and one server. Each client utilizes its private data for local view training. The high-level features $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ are obtained through two level feature extraction guided by the globally fused graph $\overline { { A } }$ . Then clustering is performed based on them. Based on the clustering results, we compute the weights $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ } m=1 by high-level feature. After that, the server receives $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ and {wm}Mm= $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ from the clients and performs feature fusion, graph fusion and global clustering. The overview illustration of the model is shown in Figure 1.
 
 # 3.2. Client Local Training with Global Guidance
 
@@ -76,7 +76,7 @@ where $S _ { i j } ^ { m } \in [ 0 , 1 ]$ denotes the similarity between $x _ {
 To handle the incomplete data problem, global graph structure migration is proposed. Under the GCN encoder on clients, each sample estimates single-view features according to its attribute and neighboring nodes’ attributes. Since
 
 ![](images/cccdfb0d451f91aff35e40deff95ad330f386d863691ec339f850c29c2b2284a.jpg)  
-Figure 1. Overview of the FIMCFG framework. It contains $M$ clients and a server. (1) Client: $M$ -th client consists of two processes: feature extraction and clustering. With the guidance of fused graph $\overline { { A } }$ , client extracts underlying features $\widetilde { H } ^ { M }$ by dual-head graph encoder and high-level features $H ^ { M }$ by fusion module. Under the supervision of pseudo-labels $P$ , the clustering centers $u ^ { M }$ are optimized by reducing the KL loss between soft distribution $Q ^ { M }$ and $P$ . Based on clustered $H ^ { M }$ , the silhouette coefficients $w ^ { M }$ are calculated as aggregation weights. (2) Server: the server performs feature fusion with $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ m}M to obtain global features $H$ and graph fusion with latent graph {Aˆm}Mm= $\{ \hat { A } ^ { m } \} _ { m = 1 } ^ { M }$ 1 of {Hm}Mm= $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ to obtain fused graph $\overline { { A } }$ . Then the server perform K-means algorithm on $H$ to get global clustering centers $U$ and pseudo labels $P$ .
+Figure 1. Overview of the FIMCFG framework. It contains $M$ clients and a server. (1) Client: $M$ -th client consists of two processes: feature extraction and clustering. With the guidance of fused graph $\overline { { A } }$ , client extracts underlying features $\widetilde { H } ^ { M }$ by dual-head graph encoder and high-level features $H ^ { M }$ by fusion module. Under the supervision of pseudo-labels $P$ e, the clustering centers $u ^ { M }$ are optimized by reducing the KL loss between soft distribution $Q ^ { M }$ and $P$ . Based on clustered $H ^ { M }$ , the silhouette coefficients $w ^ { M }$ are calculated as aggregation weights. (2) Server: the server performs feature fusion with $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ M to obtain global features $H$ and graph fusion with latent graph {Aˆm}Mm= $\{ \hat { A } ^ { m } \} _ { m = 1 } ^ { M }$ 1 of {Hm}Mm= $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ to obtain fused graph $\overline { { A } }$ . Then the server perform K-means algorithm on $H$ to get global clustering centers $U$ and pseudo labels $P$ .
 
 missing samples are represented by zero vectors, the encoder automatically ignores them during computation. However, when using similarity-based calculations for the local adjacency matrix, missing samples represented as zero vectors are isolated from complete samples, making their feature computation infeasible. To address this issue, we propose a global graph structure migration technique, which replaces rows corresponding to missing samples in the local adjacency matrix with corresponding rows from the global adjacency matrix. This approach enriches the local adjacency matrix with global structural information, enabling the missing samples to estimate features based on adjacent complete samples.
 
@@ -86,13 +86,13 @@ As demonstrated in Figure 1, the client’s feature extraction consists of three
 
 Dual-Head Graph Encoder. Dual-head Graph Encoder contains two parallel graph encoders. The samples $X ^ { m }$ ,
 
-local graph $A ^ { m }$ and fused graph $\overline { { A } }$ are input to the dualhead graph encoder to extract two different underlying features $\sum _ { \tilde { l } } \tilde { m } _ { l } ^ { m }$ and ${ \widetilde { H } } _ { g } ^ { m }$ . Specifically, stacked GCN layers are used as encoders and the global graph encoder $f _ { \theta _ { g } ^ { m } } ^ { m }$ and local graph encoder $f _ { \theta _ { l } ^ { m } } ^ { m }$ are used as two heads. Their mapping functions are : $f _ { \theta _ { g } ^ { m } } ^ { \dot { m } } ( X ^ { m } , \overline { { A } } ) \longmapsto \widetilde { H } _ { g } ^ { m } \in \mathbb { R } ^ { N \times \widetilde { d } _ { m } }$ and g $f _ { \theta _ { l } ^ { m } } ^ { m } ( X ^ { m } , A ^ { m } ) \longmapsto { \widetilde { H } } _ { l } ^ { m } \in \mathbb { R } ^ { N \times { \widetilde { d } _ { m } } }$ , where ${ \widetilde { \cal H } } _ { g } ^ { m }$ and $\widetilde { H } _ { l } ^ { m }$ are the underlying features containing global and local view information, respectively. The computation process of the $t$ -th layer of the local graph encoder is denoted as:
+local graph $A ^ { m }$ and fused graph $\overline { { A } }$ are input to the dualhead graph encoder to extract two different underlying features $\sum _ { \tilde { l } } \tilde { m } _ { l } ^ { m }$ and ${ \widetilde { H } } _ { g } ^ { m }$ . Specifically, stacked GCN layers are used as encoders and the global graph encoder $f _ { \theta _ { g } ^ { m } } ^ { m }$ and local graph encoder $f _ { \theta _ { l } ^ { m } } ^ { m }$ are used as two heads. Their mapping functions are : $f _ { \theta _ { g } ^ { m } } ^ { \dot { m } } ( X ^ { m } , \overline { { A } } ) \longmapsto \widetilde { H } _ { g } ^ { m } \in \mathbb { R } ^ { N \times \widetilde { d } _ { m } }$ and $f _ { \theta _ { l } ^ { m } } ^ { m } ( X ^ { m } , A ^ { m } ) \longmapsto { \widetilde { H } } _ { l } ^ { m } \in \mathbb { R } ^ { N \times { \widetilde { d } _ { m } } }$ , where ${ \widetilde { \cal H } } _ { g } ^ { m }$ and $\widetilde { H } _ { l } ^ { m }$ are the underlying features containing global and local view information, respectively. The computation process of the $t$ -th layer of the local graph encoder is denoted as:
 
 $$
 \begin{array}{l} \widetilde {H} _ {l} ^ {m} _ {(t)} = \sigma \left(\widetilde {D} ^ {m - \frac {1}{2}} \widetilde {A} ^ {m} \widetilde {D} ^ {m - \frac {1}{2}} \widetilde {H} _ {l} ^ {m} _ {(t - 1)} W _ {(t)} ^ {m} + b _ {(t)} ^ {m}\right) \tag {2} \\ + \widetilde {H} _ {l} ^ {m} _ {(t - 1)}, \\ \end{array}
 $$
 
-where $\sigma ( \cdot )$ is activation function. $\tilde { A } ^ { m } = A ^ { m } + I$ and $\begin{array} { r } { \widetilde { D } _ { i i } ^ { m } = \sum _ { j } \widetilde { A } _ { i j } ^ { m } } \end{array}$ . $I$ is the unit matrix, and $W _ { ( t ) } ^ { m }$ and $b _ { ( t ) } ^ { m }$ are the trainable parameters of the $t$ -th layer. $\mathring { A } ^ { m }$ is the fixed local graph after global graph structure migration. Skip connections are introduced to prevent model degradation. The global graph encoder is computed in the same way with local graph encoder.
+where $\sigma ( \cdot )$ is activation function. $\tilde { A } ^ { m } = A ^ { m } + I$ and $\begin{array} { r } { \widetilde { D } _ { i i } ^ { m } = \sum _ { j } \widetilde { A } _ { i j } ^ { m } } \end{array}$ 一 Am. $I$ is the unit matrix, and $W _ { ( t ) } ^ { m }$ and $b _ { ( t ) } ^ { m }$ are the trainable parameters of the $t$ -th layer. $\mathring { A } ^ { m }$ is the fixed local graph after global graph structure migration. Skip connections are introduced to prevent model degradation. The global graph encoder is computed in the same way with local graph encoder.
 
 The dual-head graph encoder is identified by optimizing
 
@@ -126,7 +126,7 @@ During the communication phase, the client uploads highlevel features to the ser
 
 # 3.2.3. PSEUDO-LABEL-GUIDED CLUSTERING TRAINING
 
-To obtain the clustering assignment, we construct a clustering layer $c _ { u ^ { m } } ^ { m }$ with trainable parameters $\{ u _ { j } ^ { m } \in \mathbb { R } ^ { d _ { m } } \} _ { j = 1 } ^ { K }$ , where $K$ represents the number of clustering and $u _ { j } ^ { m }$ represents the clustering center of the $j$ -th cluster on client $m$ . The soft distribution $Q ^ { m } \in \mathbb { R } ^ { N \times K }$ is computed as follows:
+To obtain the clustering assignment, we construct a clustering layer $c _ { u ^ { m } } ^ { m }$ with trainable parameters $\{ u _ { j } ^ { m } \in \mathbb { R } ^ { d _ { m } } \} _ { j = 1 } ^ { K }$ , where $K$ represents the number of clustering and $u _ { j } ^ { m }$ represents the clustering center of the $j$ -th cluster on client $m$ The soft distribution $Q ^ { m } \in \mathbb { R } ^ { N \times K }$ is computed as follows:
 
 $$
 q _ {i j} ^ {m} = \frac {\left(1 + \left\| h _ {i} ^ {m} - u _ {j} ^ {m} \right\| _ {2} ^ {2}\right) ^ {- 1}}{\sum_ {j = 1} ^ {K} \left(1 + \left\| h _ {i} ^ {m} - u _ {j} ^ {m} \right\| _ {2} ^ {2}\right) ^ {- 1}}, \tag {7}
@@ -176,9 +176,9 @@ where $\hat { A } _ { l } ^ { m } ~ \in ~ [ 0 , 1 ] ^ { N \times N }$ indicates
 
 # 3.3. Server Global Aggregation
 
-The server receives the high-level features $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ }Mm=1 and weights {wm}Mm= $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ m=1 from the clients and performs graph fusion, feature fusion and global clustering.
+The server receives the high-level features $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ } m=1 and weights {wm}Mm= $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ from the clients and performs graph fusion, feature fusion and global clustering.
 
-Graph Fusion. To fuse the graphs, the server extracts the latent graph $\{ \hat { A } ^ { m } \in [ 0 , 1 ] ^ { \bar { N } \times \bar { N } } \} _ { m = 1 } ^ { M }$ }m=1 from {Hm}M $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ according to by Eq. (1). After that, the server fuses the latent graph $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ . Specifically, it fuses the latent graphs as follows:
+Graph Fusion. To fuse the graphs, the server extracts the latent graph $\{ \hat { A } ^ { m } \in [ 0 , 1 ] ^ { \bar { N } \times \bar { N } } \} _ { m = 1 } ^ { M }$ } m=1 from {Hm}M $\{ H ^ { m } \} _ { m = 1 } ^ { M }$ by Eq. (1). After that, the server fuses the latent graph according to {wm}Mm= $\{ w ^ { m } \} _ { m = 1 } ^ { M }$ . Specifically, it fuses the latent graphs as follows:
 
 $$
 \bar {A} = f _ {k} \left(\frac {1}{M} \sum_ {m = 1} ^ {M} \left(W ^ {m} \odot \hat {A} ^ {m}\right)\right), \tag {13}
@@ -192,9 +192,9 @@ $$
 H = \left[ \bar {w} ^ {1} ^ {\prime} H ^ {1}, \bar {w} ^ {2} ^ {\prime} H ^ {2}, \dots , \bar {w} ^ {M} ^ {\prime} H ^ {M} \right], \tag {14}
 $$
 
-where $\textstyle H \in \mathbb { R } ^ { N \times \sum _ { m = 1 } ^ { M } d _ { m } }$ denotes the global feature, and $[ \cdot , \cdot \cdot , \cdot ]$ stands for matrix concatenation operation. $\overline { { w } } ^ { m ^ { \prime } } =$ $\begin{array} { r } { \dot { 1 } + \log \dot { ( 1 + \frac { \overline { { w } } ^ { m } } { \sum _ { m = 1 } ^ { M } \left| \overline { { w } } ^ { m } \right| } ) } } \end{array}$ wm , where $\overline { { w } } ^ { m }$ is the view silhouette coefficient and is calculated by $\begin{array} { r } { \overline { { w } } ^ { m } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } w _ { i } ^ { m } } \end{array}$ .
+where $\textstyle H \in \mathbb { R } ^ { N \times \sum _ { m = 1 } ^ { M } d _ { m } }$ denotes the global feature, and $[ \cdot , \cdot \cdot , \cdot ]$ stands for matrix concatenation operation. $\overline { { w } } ^ { m ^ { \prime } } =$ $\begin{array} { r } { \dot { 1 } + \log \dot { ( 1 + \frac { \overline { { w } } ^ { m } } { \sum _ { m = 1 } ^ { M } \left| \overline { { w } } ^ { m } \right| } ) } } \end{array}$ wm , where $\overline { { w } } ^ { m }$ is the view silhouette P m= |wm| coefficient and is calculated by $\begin{array} { r } { \overline { { w } } ^ { m } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } w _ { i } ^ { m } } \end{array}$ .
 
-Global Clustering. K-means algorithm is used to cluster $H$ to obtain the global clustering center $U \ =$ $[ \overline { { w } } ^ { 1 ^ { ' } } U ^ { 1 } , \overline { { w } } ^ { 2 ^ { ' } } U ^ { 2 } , \cdot \cdot \cdot , \overline { { w } } ^ { M ^ { ' } } U ^ { M } ]$ , where $U ^ { m }$ denotes the global clustering center of the $m$ -th view. One point worth noting is that $\{ U ^ { m } \} _ { m = 1 } ^ { M }$ is aligned in classes. Based on the Student-t distribution, the soft distribution for each sample is:
+Global Clustering. K-means algorithm is used to cluster $H$ to obtain the global clustering center $U \ =$ $[ \overline { { w } } ^ { 1 ^ { ' } } U ^ { 1 } , \overline { { w } } ^ { 2 ^ { ' } } U ^ { 2 } , \cdot \cdot \cdot , \overline { { w } } ^ { M ^ { ' } } U ^ { M } ]$ , where $U ^ { m }$ denotes the global clustering center of the $m$ -th view. One point worth noting is that {U m}Mm= $\{ U ^ { m } \} _ { m = 1 } ^ { M }$ 1 is aligned in classes. Based on the Student-t distribution, the soft distribution for each sample is:
 
 $$
 s _ {i j} = \frac {\left(1 + \left\| h _ {i} - U _ {j} \right\| _ {2} ^ {2}\right) ^ {- 1}}{\sum_ {j = 1} ^ {K} \left(1 + \left\| h _ {i} - U _ {j} \right\| _ {2} ^ {2}\right) ^ {- 1}}, \tag {15}
@@ -230,7 +230,7 @@ Table 1. Experimental results on the four datasets. The best results in each col
 
 # Algorithm 1 Optimization algorithm for FIMCFG
 
-Require: Data with $M$ views $\{ X ^ { m } \} _ { m = 1 } ^ { M }$ }m=1 distributed across $M$ clients, number of clusters $K$ , number of communication rounds $T$ , number of training rounds $E$ .
+Require: Data with $M$ views $\{ X ^ { m } \} _ { m = 1 } ^ { M }$ distributed across $M$ clients, number of clusters $K$ , number of communication rounds $T$ , number of training rounds $E$ .
 
 Ensure: Clustering results $Y = \{ y _ { 1 } , y _ { 2 } , \cdot \cdot \cdot , y _ { N } \}$
 
@@ -252,7 +252,7 @@ Ensure: Clustering results $Y = \{ y _ { 1 } , y _ { 2 } , \cdot \cdot \cdot , y
 16: Update $H$ by Eq. (14).   
 17: Obtain $\{ U ^ { m } \} _ { m = 1 } ^ { M }$ by K-means.   
 18: Obtain $P$ by Eq. (16).   
-$\overline { { A } }$ , $\{ U ^ { m } \} _ { m = 1 } ^ { M }$ , $P$   
+19: Distribute $\overline { { A } }$ , $\{ U ^ { m } \} _ { m = 1 } ^ { M }$ , $P$ to clients.   
 20: end while   
 21: Calculate the clustering label by Eq. (17).
 
@@ -340,7 +340,7 @@ Chao, G., Xu, K., Xie, X., and Chen, Y. Global graph propagation with hierarchic
 Chen, M.-S., Lin, J.-Q., Li, X.-L., Liu, B.-Y., Wang, C.-D., Huang, D., and Lai, J.-H. Representation learning in multi-view clustering: A literature review. Data Science and Engineering, 7(3):225–241, 2022.   
 Chen, X., Xu, J., Ren, Y., Pu, X., Zhu, C., Zhu, X., Hao, Z., and He, L. Federated deep multi-view clustering with global self-supervision. In Proceedings of the 31st ACM International Conference on Multimedia, MM ’23, pp. 3498–3506, New York, NY, USA, 2023. Association for Computing Machinery. ISBN 9798400701085. doi: 10.1145/3581783.3612027.   
 Chen, X., Ren, Y., Xu, J., Lin, F., Pu, X., and Yang, Y. Bridging gaps: Federated multi-view clustering in heterogeneous hybrid views. arXiv preprint arXiv:2410.09484, 2024.   
-Du, G., Zhou, L., Li, Z., Wang, L., and Lu, K. Neighbor- ¨ aware deep multi-view clustering via graph convolutional network. Information Fusion, 93:330–343, 2023. ISSN 1566-2535. doi: https://doi.org/10.1016/j.inffus.2023.01. 001.   
+Du, G., Zhou, L., Li, Z., Wang, L., and Lu, K. Neighbor-¨ aware deep multi-view clustering via graph convolutional network. Information Fusion, 93:330–343, 2023. ISSN 1566-2535. doi: https://doi.org/10.1016/j.inffus.2023.01. 001.   
 Fei-Fei, L. and Perona, P. A bayesian hierarchical model for learning natural scene categories. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern
 
 Recognition (CVPR’05), volume 2, pp. 524–531 vol. 2, 2005. doi: 10.1109/CVPR.2005.16.   
@@ -404,7 +404,8 @@ Figure 7. Parameter sensitivity analysis of $\gamma _ { 1 }$ when $\delta = 0 .
 ![](images/907f6853f7556792c86c8e23aee78ae62e181f7c78ddc7e0a198fd1d2d629f4f.jpg)  
 (a)
 
-![](images/35d8fbf2beba405f0bf1ebf0ed6c6989497ba3a8c30d07cdcc4723ba80555718.jpg)
+![](images/35d8fbf2beba405f0bf1ebf0ed6c6989497ba3a8c30d07cdcc4723ba80555718.jpg)  
+(b)
 
 ![](images/25894efb3b784533ea8f9f6bcf3ad46b8abb8d0ed70b02bbef37cb9e3d746934.jpg)  
 (c)