Adversarial Graph Fusion for Incomplete Multi-view Semi-supervised Learning with Tensorial Imputation

Thank you so much for the valuable comments! We are committed to addressing each question you have raised.

W1: Novelty of the Proposed AGF-TI

We acknowledge that both anchor learning and low-rank tensor decomposition are existing techniques in the multi-view clustering field. Actually, in this paper, the anchor strategy and tensor method are utilized to enhance efficiency and unify the framework. Beyond combining these techniques, our work mainly focuses on introducing a novel adversarial graph fusion (AGF) operator based on anchors to learn a robust consensus graph for semi-supervised classification, specifically designed to address the sub-cluster problem. The proposed AGF offers the following advantages:

• Anchor graph fusion paradigm

Most existing multi-view semi-supervised learning (MvSSL) methods [10, 12] directly fuse the sample-level graphs $\mathbf{S}_v\in \mathbb{R}^{n \times n}$. In contrast, AGF-TI operates on anchor-based bipartite graphs of size $n \times m$, enabling fusion at a lower dimensional level. Besides, the subsequent tensor learning can also be conducted on an $m\times V\times n$ tensor instead of $n\times V\times n$. This paradigm significantly enhances the computational efficiency for both graph fusion and tensor imputation.

• Adversarial min-max framework

We are the first to introduce a min-max optimization framework into graph-based MvSSL, enabling the learning of a robust consensus graph. This novel approach mitigates the impact of distorted local structures and improves the robustness of graph fusion. Unlike conventional min-min paradigms, our adversarial min-max cycle avoids collapse into a single dominant view and achieves a better exploration–exploitation balance, ultimately leading to improved classification performance.

In addition to the proposed AGF, we also present a severe yet neglected problem, the Sub-Cluster Problem, when encountering view missing scenarios.

Sub-cluster observation. We identify a novel challenge in graph-based MvSSL with missing views, called Sub-Cluster Problem (SCP). Specifically, missing samples in individual views can violate the core smoothness assumption underlying label propagation, resulting in inaccurate graph fusion and poor label prediction. Moreover, we validate the presence of the SCP in real-world webpage and document classification datasets in the next W2.

Other contributions. We develop an effective optimization algorithm that combines alternating direction method of multipliers and reduced gradient descent method to tackle the challenges posed by the proposed objective function. Furthermore, we provide a theoretical convergence guarantee under a mild assumption, highlighting the soundness of our approach.

W2: SCP in Real-world Datasets

Thanks. To address the reviewer’s concern, we validate the SCP using two real-world datasets: WebKB [R1] and Wikipedia [R2], commonly used in webpage and document classification tasks. Specifically, WebKB comprises 1,051 webpages across 2 classes, 230 for course and 821 for no-course, with page and link views. Wikipedia contains 693 multimedia documents from 10 categories, represented by image and text views.

To quantify the number of category clusters in each view, we employ the density-based clustering algorithm DBSCAN [R3]. For each view, we calibrated DBSCAN at VMR=0% by setting $minpts=5$ and tuning $\epsilon \in [0.1, 1]$ (step size 0.01) to approximate the real number of classes. To account for the reduction in sample size at higher VMRs, we decrement $minpts$ by one for each increase in VMR. The results are shown below, and one can observe that the number of detected clusters by DBSCAN rapidly grows with VMR, indicating that class manifolds fragment into smaller sub-clusters. Notably, in the ''link'' view of WebKB, DBSCAN identifies 11 clusters even at VMR=0% (more than 2 classes), which suggests that SCP may not only be widely present in incomplete multi-view data but also arise in complete data scenarios. These findings support the validity and generality of SCP in real-world applications.

W3: Additional Large-scale Datasets

Thanks. Actually, the datasets in our experiments are commonly used in the multi-view learning field, exhibiting considerable variation in sample sizes (600 to 10,158). To address the reviewer's concern, we conduct additional experiments on the NoisyMNIST datasets (30k/70k samples, 2 views, 10 classes) [R4] to evaluate the classification performance of compared algorithms on larger-scale datasets. With VMR and LAR set to 50% and 5%, respectively, the results below demonstrate that AGF-TI continues to achieve superior performance, confirming its scalability and effectiveness.

Note: Some results were not reported due to out-of-memory errors or because the computations had not been completed within the allotted time.

W4: Running Time

Thanks. To address the reviewer's concern, we compared algorithm running times on all datasets, setting VMR=50% and LAR=5%. For baselines requiring complete multi-view data, the total running time includes both the DMF completion process and their subsequent execution. The results are recorded below, and one can observe that AGF-TI maintains an acceptable running time. Although AGF-TI takes slightly longer than AMSC and SLIM due to its missing data imputation and shared bipartite graph learning, it consistently achieves significantly better performance across all datasets. Moreover, due to its anchor strategy, AGF-TI's running time scales efficiently with the sample size, ensuring its practicality for large-scale applications.

References

[R1] Blum, Avrim, and Tom Mitchell. "Combining labeled and unlabeled data with co-training." Proc. Conf. Learn. Theory, 1998.

[R2] Wang, Hao, et al. "Multi-view clustering via concept factorization with local manifold regularization." IEEE ICDM, 2016.

[R3] Ester, Martin, et al. "A density-based algorithm for discovering clusters in large spatial databases with noise." kdd, 1996.

[R4] Yang, Mouxing, et al. "Robust multi-view clustering with incomplete information." IEEE TPAMI, 2022.

Thank you for taking the time to review our rebuttal and raising the score. We sincerely appreciate your support and endorsement!

Thank you so much for your valuable comments! We are committed to addressing each question you have raised.

W1: Running Time

Thanks. To address the reviewer's concern, we compared algorithm running times on all datasets, setting VMR=50% and LAR=5%. For baselines requiring complete multi-view data, the total running time includes both the DMF completion process and their subsequent execution. The results are recorded below, and it can be observed that AGF-TI maintains an acceptable running time. Although AGF-TI takes slightly longer than AMSC and SLIM due to its missing data imputation and shared bipartite graph learning, it consistently achieves significantly better performance across all datasets. Moreover, due to its anchor strategy, AGF-TI's running time scales efficiently with the sample size, ensuring its practicality for large-scale applications.

W2: $\beta$ in Eq. (3)

Thanks for pointing it out. In Eq. (3), $\beta$ is the trade-off parameter of the regularization term $||\mathbf{P}||_ F^2$. In our final objective, it is multiplied by $\lambda$ and written as $\beta_\lambda$. We will clarify this in the revision.

W3: Initialization of $\mathbf{T}_v$

Thanks. In our experiments, the permutation matrix $\mathbf{T}_v$ is initialized as an identity matrix.

Last

Thank you again for your suggestion. We will include these details in the revised paper.

Thank you for taking the time to review our rebuttal. We're glad our clarifications addressed your concerns, and we sincerely appreciate your support and endorsement! We will carefully revise the paper to incorporate the discussed points.

The authors' response have addressed most of my concerns, I will maintain my score as 5.

Thank you for your acknowledgment and insightful comments! Your feedback is extremely helpful, and we are committed to addressing each question you have raised.

W1: Show SCP Appears under Current Random-missing Setup

Thanks. Our experimental setup replicates that of previous work [17] to ensure a fair comparison. To address the reviewer's concern, we adopt a density-based adaptive clustering method [R1], DBSCAN, to validate that "current random-missing setup effectively mimics SCP as VMR increases".

For validation, we calibrated DBSCAN on the first view of each dataset at VMR=0% by setting $minpts=5$ and tuning $\epsilon \in [0.1, 1]$ (step size 0.01) to approximate the real number of classes. To control for the sample size reduction accompanying higher VMRs, we decrement $minpts$ by one for each increase in VMR. The results are shown below, and one can observe that the number of detected clusters by DBSCAN rapidly grows with VMR, indicating that class manifolds fragment into smaller sub-clusters. This confirms our experimental setup can effectively simulate SCP and validates the effectiveness of our proposed AGF-TI for addressing SCP.

W2: Computational and Memory Footprint of the Tensor Nuclear-Norm Step

Thanks. Theoretically, we have discussed the time complexity of the tensor nuclear-norm (TNN) step in Appendix D.2, which is $\mathcal{O}(nmV\text{log}(nV)+nmV^2)$. To address the reviewer's concern, we adopt the YTF50 dataset (~126k) [R2] with four views and empirically analyze the computational and memory footprint of the TNN step when view number, anchor number, and sample size scale simultaneously. The results under VMR=50% and LAR=5% are shown below. As is evident, the empirical trend aligns well with our theoretical analysis. Due to the anchor strategy, the running time of TNN step does not increase dramatically with the sample size and remains within an acceptable burden. This indicates AGF-TI is capable for practical use.

• #Sample = 10k

• #Sample = 70k

• #Sample = 126k

W3: References

Thanks for pointing them out. We will cite and discuss these papers in the revision.

W4/6: Full Name of TNN in Definition1

Thanks for pointing it out. We will revise TNN to its full name in Definition 1.

W5: Additional Deep Learning Baselines

Thanks. To address the reviewer's concern, we conduct additional experiments with two deep learning-based multi-view semi-supervised methods, i.e., IMvGCN [R3] and GEGCN [R4], on all datasets with VMR=50% and LAR=5%. We adopt the recommended learning rate and network structures as their baselines. The table below shows that our AGF-TI outperforms them in almost all cases, further demonstrating the effectiveness of AGF-TI.

References

[R1] Ester, Martin, et al. "A density-based algorithm for discovering clusters in large spatial databases with noise." kdd, 1996.

[R2] Huang, Dong, et al. "Fast multi-view clustering via ensembles: Towards scalability, superiority, and simplicity." IEEE TKDE, 2023.

[R3] Wu, Zhihao, et al. "Interpretable graph convolutional network for multi-view semi-supervised learning." IEEE TMM, 2023.

[R4] Lu, Jielong, et al. "Generative essential graph convolutional network for multi-view semi-supervised classification." IEEE TMM, 2024.

Thank you for taking the time to review our rebuttal and raising the score. We're glad our clarifications addressed your concerns, and we sincerely appreciate your support and endorsement! We will carefully revise the paper to incorporate the discussed points.

Thank you so much for the valuable comments! We are delighted that you found the strengths of AGF-TI. Below, we address each of your questions in detail.

W1/Q1: Max-Min-Max Framework and Its Benefits

Thanks. Compared to the traditional minimization paradigm, the proposed AGF-TI introduces an adversarial learning method to optimize the shared bipartite graph, thus formulated as a max-min-max framework. As discussed in Subsection 4.3, such adversarial learning enhances the performance and robustness of the approach. On the other hand, although our formulation looks intractable, it actually can be solved by an efficient optimization algorithm through ADMM and the reduced gradient descent method.

In the traditional min-min paradigm, if a particular view temporarily achieves better performance, the algorithm tends to increase its weight, thereby diminishing the influence of other views (please see Fig. 3). This can lead to the algorithm becoming overly reliant on a single view, which may compromise its final performance.

In contrast, our max-min-max framework creates an adversarial dynamic between maximizing the overall objective and minimizing the view weights $\boldsymbol{\alpha}$. This min-max cycle effectively prevents the algorithm from collapsing to a single dominant view, achieving exploration–exploitation balance. This mechanism enables the proposed AGF-TI to fully integrate complementary information from individual views to learn a stable and reliable fused bipartite graph and improve the final performance. Furthermore, as supported by recent studies [33, 36, 48], this adversarial property across views also enhances the model's robustness to noise when facing noisy views or datapoints.

W2: Sufficient Optimization Assumption

Thanks. According to Corollary 1 in Appendix D.1, Algorithm 1 will converge to the global optimum of the Inner Step, hence the sufficient optimization assumption is actually a mild assumption and can be quickly satisfied in practice. To validate the reasonable of this assumption, we track the error of both $\boldsymbol{\alpha}$ and $\mathbf{P}$ between iteration steps, i.e., $||\boldsymbol{\alpha}^{(k+1)}-\boldsymbol{\alpha}^{(k)}||_2^2$ and $||\mathbf{P}^{(k+1)}-\mathbf{P}^{(k)}||_F^2$ during Algorithm 1 to approximate the error between numerical and exact solutions in Eq. (36). The results on all datasets with VMR=50% and LAR=5% are shown below. These results show that the error of both $\boldsymbol{\alpha}$ and $\mathbf{P}$ can be rapidly decreased to a small number (e.g., $1e-5$), suggesting the reasonableness of the sufficient optimization assumption.

Note: '-' means converged.

Q2: Initialization of $\mathbf{T}_v$

Thanks. In our experiments, the permutation matrix $\mathbf{T}_v$ is initialized as an identity matrix. We will include this detail in the revised paper.

Thank you for taking the time to review our paper and rebuttal. We're glad our clarifications addressed your concerns, and we sincerely appreciate your support and endorsement!