Efficient Federated Incomplete Multi-View Clustering

We sincerely thank Reviewer NJFx for the thorough and constructive review. We provide point-by-point responses to the questions raised as follows:

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Q1: Most anchor-based MVC methods enforce orthogonality constraints on anchor matrices to ensure diversity. The authors remove this constraint without justification. What performance changes would occur if orthogonality constraints were reinstated?

A1: We sincerely appreciate your question. While many anchor-based methods enforce orthogonality to promote anchor diversity, we intentionally relax this constraint for two key reasons:

We aim to learn anchors that are representative of each category. However, in practice, the number of anchors typically exceeds the number of categories. Under such circumstances, it is unnecessary to enforce distinctness among all anchors - anchors from the same category may exhibit similarity. Imposing strict orthogonality constraints would be overly restrictive for this purpose.

Besides, our ablation studies as follows show that removing orthogonality constraints actually improves clustering accuracy across all datasets, likely because it allows for more flexible anchor representations that better capture relationships in partial observation scenarios.

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Q2: Previous methods typically apply k-means directly on learned embeddings. The choice to cluster left singular vectors of $\mathbf{Z}$ in Algorithm 1 lacks theoretical justification. Why not use standard spectral clustering?

A2: We sincerely thank you for the problem. Standard spectral clustering can only operate on a complete $n\times n$ similarity matrix and cannot directly process the anchor graph $\mathbf{Z}$. According to [1], $\mathbf{Z}^\top \mathbf{Z}$ can be interpreted as reconstructing the full similarity matrix from the anchor graph $\mathbf{Z}$. Notably, performing $k$-means on the right singular vectors of $\mathbf{Z}$ is theoretically equivalent to applying standard spectral clustering to $\mathbf{Z}^\top \mathbf{Z}$.

[1] Kang et al. Large-scale multi-view subspace clustering in linear time. AAAI 2020.

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Q3: The huge performance gaps between the first four methods and the remaining methods in Table 2 suggest unfair experimental settings. Were all methods given equal hyperparameter tuning efforts?

A3: We sincerely thank you for the question. The first four methods evaluated in Table 2 are centralized incomplete multi-view clustering algorithms. Compared to the subsequent four federated MVC methods, these centralized approaches hold inherent advantages in terms of full data accessibility and built-in mechanisms for handling missing values, thereby achieving superior clustering performance on incomplete multi-view datasets.

In contrast, our proposed method not only processes distributed multi-view data but also incorporates a dedicated incomplete data handling mechanism. As a result, it achieves comparable performance to state-of-the-art centralized incomplete MVC algorithms while significantly outperforming existing federated MVC methods.

Besides, for all baseline methods, we strictly followed the parameter configurations reported in their original papers. Notably, for federated MVC methods, all missing values were imputed with zeros to ensure a fair comparison. We will add these implementation details to the final version.

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Q4: Existing MVC methods optimize client/server variables under a unified objective to ensure convergence. How does EFIMVC guarantee convergence with separate client/server objectives?

A4: We sincerely appreciate the comment. In contrast to conventional approaches that optimize server and client variables under a unified objective function, we propose a decoupled optimization framework that separates these two components. This innovation effectively reduces the communication overhead caused by frequent interactions. We further provide both theoretical and experimental convergence analyses for the decoupled objectives relative to the global objective function.

Specifically, Figure 3 illustrates the monotonic decrease of objective values (for server-side, client-side, and global objectives) with increasing iterations, demonstrating their eventual convergence to stable values. Additionally, in Appendix A.2, we present a rigorous theoretical analysis of convergence properties for each component.

We sincerely thank Reviewer 3S7F for the thorough and constructive review. We provide point-by-point responses to the questions raised as follows:

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Q1: The authors claim their method solves three key challenges (communication overhead, privacy preservation, incomplete views) but fail to explicitly demonstrate how each component of the framework addresses these issues in the method part.

A1: We sincerely appreciate your question. Our framework explicitly addresses the three challenges through dedicated components:

1. Communication Overhead: By decoupling optimization variables into server-side and client-side, we significantly reduce the variable transmission costs inherent in conventional frameworks, where optimization is performed alternately between the server and clients.
2. Privacy Preservation: The proposed framework employs anchor graphs as communication variables, avoiding the transmission of sensitive embeddings or sample-level similarity matrices, thereby reducing privacy risks.
3. Missing-View Handling: We learn partial anchor graphs with available data on each client via Eq. (2), and integrates them into a consistent full anchor graph on server-side via Eq. (5).

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Q2: The clustering performance does not demonstrate superiority. As shown in Table 2, EFIMVC underperforms state-of-the-art methods on ProteinFold, WebKB, CCV and MNIST datasets.

A2: We sincerely thank you for the critical perspective. While centralized SOTA methods (FIMVC, SCBGL, DVSAI, DAQINT) achieve marginally higher accuracy on some datasets, they fundamentally violate federated constraints. In contrast, our federated framework achieves comparable or superior performance to these centralized SOTA methods while operating under strict distributed data storage protocols. This demonstrates our method’s unique capability to balance clustering accuracy with federated learning requirements. We will emphasize this distinction in the final version.

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Q3: In experiments, the federated baselines perform poorly compared to centralized methods. Were these baselines adapted fairly to handle missing view?

A3: We sincerely appreciate the comment. First, federated multi-view clustering methods inherently underperform centralized approaches because they cannot simultaneously access all view data. Second, since existing federated MVC methods lack the capability to handle missing multi-view data, we filled all missing values with zeros in our experiments. This also leads to slightly inferior performance compared to specialized missing MVC algorithms with built-in imputation mechanisms. Notably, as the first federated incomplete MVC method proposed, our approach ensures a fair experimental comparison by benchmarking against both state-of-the-art centralized missing MVC methods and existing federated MVC methods.

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Q4: The excessive mathematical notation severely impacts readability. A notation table should be added.

A4: We sincerely appreciate the constructive suggestion. We will add a notation table in the final version. Specially, the main used notations through our paper is summarized as follows:

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Q5: What motivated the specific construction strategy for anchor similarity matrix in Eq. (3)? Would alternative similarity measures like cosine distance yield different results?

A5: We sincerely thank you for the insightful question. The anchor similarity matrix quantifies pairwise distances between all anchors, serving as the foundation for subsequent server-side anchor structure alignment. In Eq. (3), we compute the distance between each anchor pair using the ℓ2-norm (Euclidean distance), though this metric can be substituted with alternative methods (e.g., cosine similarity). To evaluate the impact of distance metrics, we replace the L2-norm with cosine similarity and compare the results with existing approaches, as shown in the table below.

From the experimental results, ℓ2-norm performs slightly better than cosine similarity. We will include this ablation experiment in the final version.

We sincerely thank Reviewer 9Soy for the thorough and constructive review. We provide point-by-point responses to the questions raised as follows:

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Q1: The paper uses excessive mathematical symbols without proper definitions. For instance, $\mathbf{L}$ in Eq. (13) is never explained.

A1: We sincerely appreciate your feedback. The symbol $\mathbf{L}^{(v)}$ in Eq. (13) denotes the Laplacian matrix derived from the anchor similarity matrix $\mathbf{S}^{(v)}$ of the $v$-th view, where $\mathbf{L}^{(v)} = \mathbf{D}^{(v)} - \mathbf{S}^{(v)}$ and $\mathbf{D}^{(v)}$ is the degree matrix. We acknowledge the oversight in explicitly defining this notation and will comprehensively explain all mathematical symbols in Appendix to ensure full clarity in the final version.

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Q2: The "dual-anchor alignment" concept remains vague. How does it differ from single-level alignment in [1]?

[1] Align then Fusion: Generalized Large-scale Multi-view Clustering with Anchor Matching Correspondences. NeurIPS 2022.

A2: We sincerely thank you for this insightful comparison. While [1] achieves alignment through sequential feature alignment and structural alignment, our dual-anchor alignment unifies these two objectives into a single optimization framework in Eq. (5). Specifically:

• Feature alignment (anchor embedding matching): $\left\|\left\|\mathbf{P}^{(v)}\mathbf{Z}\mathbf{G}^{(v)} - \mathbf{Z}^{(v)}\mathbf{G}^{(v)} \right\|\right\|_{\mathbf{F}}^2$
• Structural alignment (anchor graph topology preservation): $\operatorname{Tr}\left((\mathbf{P}^{(v)})^\top\mathbf{L}^{(v)}\mathbf{P}^{(v)}(\mathbf{Z}\mathbf{Z}^\top)\right)$

Besides, our method introduces missing-view robustness through the indicator matrix $\mathbf{G}^{(v)}$, enabling alignment even when partial data observations are absent across views. This differs fundamentally from [1], which assumes complete data availability.

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Q3: The missing view definition is ambiguous. Is it sample-level missing (some samples lack views) or feature-level missing (entire view channels unavailable)?

A3: We sincerely thank you for highlighting the problem. Our work addresses sample-level missing views, where individual samples may lack specific view channels (e.g., a patient missing MRI data but having CT scans). Formally, given the indicator vector $r^{(i)} \in \mathbb{R}^{n_v}$ containing the index for $n_v$ existing samples for $v$-th view in sort, the indicator matrix $\mathbf{G}^{(v)} \in \lbrace 0,1 \rbrace^{n \times n_v}$ satisfies:

$$\mathbf{G}^{(v)}{i,j} = \begin{cases} 1, & \text{if the entry } r^{(i)}{j}=i, \\\\ 0, & \text{otherwise.} \end{cases}$$

where $\mathbf{X}^{(v)}\mathbf{G}^{(v)}$ denotes the sorted complete data matrix in the $v$-th view.

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Q4: Table 2 shows empty results for FMVC on four datasets. Is this due to implementation errors or intentional omission?

A4: We sincerely thank you for the question. The empty entries of FMVC in Table 2 stem from its prohibitive memory demands when handling large datasets. Specifically, FMVC constructs Laplacian matrices with size of $n \times n$, which resulted in an out-of-memory error when running large-scale datasets. We will give further explanation in the final version.

We sincerely thank Reviewer JeXy for the thorough and constructive review. We provide point-by-point responses to the questions raised as follows:

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Q1: It is recommended that all illustrations use vector graphics.

A1: We sincerely appreciate your constructive suggestion. In the final version, we will convert all figures to vector graphics (e.g., PDF formats) to ensure optimal resolution and scalability.

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Q2: Some of the data sets listed are not the highest results. Can you add relevant explanations?

A2: We sincerely thank you for raising this critical point. We clarify that the first four methods in Table 2 (FIMVC, SCBGL, DVSAI, DAQINT) are state-of-the-art centralized incomplete multi-view clustering methods. While they achieve marginally higher performance on specific datasets, they fundamentally address centralized scenarios and cannot handle distributed multi-view data with privacy constraints. In contrast, our federated framework achieves comparable or superior performance to these centralized SOTA methods while operating under strict distributed data storage protocols. This demonstrates our method’s unique capability to balance clustering accuracy with federated learning requirements. We will emphasize this distinction in the final version.

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Q3: Although the dual anchor point alignment mechanism improves the quality of graph fusion, its computational complexity is not analyzed in detail. If there are more anchor points, the computational overhead may increase. It is recommended to provide theoretical analysis or experimental results.

A3: We sincerely appreciate the important technical concern. The computational complexity of the dual anchor alignment mechanism comes from optimizating the permutation matrix $\mathbf{P}^{(v)}$ on the server, which is $\mathcal{O}(m^3 + m^2n)$. Specifically, constructing matrix $\mathbf{Q}$ requires $\mathcal{O}(m^3 + m^2n)$ to process matrix multiplications. Solving $\mathbf{P}^{(v)}$ by performing SVD decomposition on $\mathbf{Q}$ requires $\mathcal{O}(m^3)$. While increasing anchor points $m$ raises time costs, our method fundamentally avoids the prohibitive space complexity ($\mathcal{O}(n^2)$) and time complexity ($\mathcal{O}(n^3)$) of full-graph approaches. We acknowledge the reviewer's concern and will explore hierarchical anchor optimization to further enhance efficiency while maintaining performance.

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Q4: The data distribution from different perspectives may be quite different, and the selection of anchor points may directly affect the final clustering effect. The paper does not discuss how to ensure the robustness of anchor point selection.

A4: We sincerely thank you for the insightful comment. To evaluate anchor robustness, we conducted ablation studies using three strategies:

1. $k$-means (default)
2. Random selection
3. Density-based sampling

The comparison results of different strategies are shown in below:

While $k$-means achieves the most stable performance, we acknowledge that advanced anchor initialization could further improve robustness. We will discuss this limitation and cite relevant techniques [1] in the final version.

[1] Liu et al. Learn from View Correlation: An Anchor Enhancement Strategy for Multi-view Clustering. CVPR 2024.