Adversarial Graph Fusion for Incomplete Multi-view Semi-supervised Learning with Tensorial Imputation
Authors: Zhangqi Jiang
Tingjin Luo
Xu Yang
Xinyan Liang
Venue: NeurIPS 2025 poster ()
Keywords: Incomplete multi-view learning
graph-based semi-supervised learning
Primary area: general_machine_learning
TL;DR: We present the first framework to tackle the Sub-Cluster Problem in incomplete GMvSSL via adversarial graph fusion and tensor-based structure recovery.

Abstract: View missing remains a significant challenge in graph-based multi-view semi-supervised learning, hindering their real-world applications. To address this issue, traditional methods introduce a missing indicator matrix and focus on mining partial structure among existing samples in each view for label propagation (LP). However, we argue that these disregarded missing samples sometimes induce discontinuous local structures, i.e., sub-clusters, breaking the fundamental smoothness assumption in LP. Consequently, such a Sub-Cluster Problem (SCP) would distort graph fusion and degrade classification performance. To alleviate SCP, we propose a novel incomplete multi-view semi-supervised learning method, termed AGF-TI. Firstly, we design an adversarial graph fusion scheme to learn a robust consensus graph against the distorted local structure through a min-max framework. By stacking all similarity matrices into a tensor, we further recover the incomplete structure from the high-order consistency information based on the low-rank tensor learning. Additionally, the anchor-based strategy is incorporated to reduce the computational complexity. An efficient alternative optimization algorithm combining a reduced gradient descent method is developed to solve the formulated objective, with theoretical convergence. Extensive experimental results on various datasets validate the superiority of our proposed AGF-TI as compared to state-of-the-art methods. Code is available at https://github.com/ZhangqiJiang07/AGF_TI.

Adversarial Graph Fusion for Incomplete Multi-view Semi-supervised Learning with Tensorial Imputation
Zhangqi Jiang'
Tingjin Luo
Xu Yang² Xinyan Liang'
National University of Defense Technology Southeast University Shanxi University
sxdxjzq0gmail. com, tingjinluo@hoomaill com, xuyang-palm@seu. edu. cn 1iangxlnyanr40116..com
Abstract
View missing remains significant challenge in graph-based multi-view semi- supervised learning, hindering their real-world applications. To address this issue, traditional methods introduce missing indicator matrix and focus on mining partial structure among existing samples each view for label propagation (LP). However, we argue that these disregarded missing samples sometimes induce discontinuous local structures, i.e., sub-clusters, breaking the fundamental smoothness assumption in IP.. Consequently, such a Sub-Cluster Problem (SCP) would distort graph fusion and degrade classification performance. To alleviate SCP, we propose a novel incomplete multi-view semi-supervised learning method, termed AGF-TI. Firstly, we design an adversarial graph fusion scheme to learn robust consensus graph against the distorted local structure through min-max framework. By stacking all similarity matrices into tensor, we further recover the incomplete structure from the high-order consistency information based the low-rank tensor learning. Additionally, the mechor-based strategy is incorporated to reduce the computational complexity. An efficient alternative optimization algorithm combining a reduced gradient descent method is developed to solve the formulated objective, with theoretical convergence. Extensive experimental results various datasets validate the superiority of our proposed AGF-TI as compared to state-of-the-ard methods. Code is available at http://gttttuccoc////aagggggag//
TI
Introduction
Multi-view data encodes complementary information from heterogeneous sources or modalities, significantly enhancing the performance of downstream tasks, such as autonomous driving [ and precision health 3 4 However, such heterogeneity between different views brings challenges for supervision signal (label) annotation, making the label scarcity problem widespread in practice. For example, diagnosing Alzheimer's Disease requires several physicians to jointly consider information from clinical records, Neuro-imaging scans, fluid biomarker readings, etc., to make informed deci- sions [5 6 Leveraging extrinsic semantic information from data geometric structure, graph- based multi-view semi-supervised learning (GMvSSL) has been studied intensively and used in various applications [7.3]. In general, GMvSSL methods describe sample relationships using graphs and rely on the smoothness assumption [14] samples sharing the same label are likely to lie on the same manifold-to "propagate" label information across the graph.
Since existing GMvSSI. approaches all focus on the extension of single-view semi-supervised learning methods to multi-view scenarios with late or carly fusion strategies, most of them typically presume that all views are available for every sample. However, multi-view data in real applications frequently suffers from the miss of certain views due to machine failure accessibility issues [1H18) Learning
Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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(A) Sy.ththheicccco SSL data
(8) incompltee SSL cata
(E)
Merge
HEVD
vacutum
region Sy
Archor-tenett biparite grapti
Simiartty
Recovery
sub-cluster
Z+
original
incomplane
Parmutsion
proddut
ahrinkage
Min-Mat Pramswork
Label
Adversarial Graph Fusion
Samphien Grach
Propagation
Fused ancher graph P
Suparision Signal Guaderca
sof lebel metriz
Figure I: (A)-(D) is an example for the Sub-Cluster Problem (SCP), and (E) shows the proposed AGF-TL. SCP: sub-clusters caused by the missing samples break the smoothness assumption in Label Propagation (LP). To address SCP, AGF-TI comprises an adversarial graph fusion operator to learn a robust fused graph, and tensor learning to recover similarity relationships of missing samples.
for such dual missing issue (i.e., missing views and scarce labels) is crucial but rarely studied. To address this, most recently, Zhuge et al. [19] proposed an incomplete multi-view semi-supervised learning (IMvSSL) method, termed AMSC. AMSC first learns multiple basic label matrices via label propagation based on partial similarity graphs, which are constructed among existing samples in individual views. Then, AMSC integrates them into a consensus label matrix for prediction. Despite adopting the p-th root strategy for adaptive view fusion, AMSC overlooks the distortions caused by the view missing issue to the graph or manifold structure.
In this paper, we argue that the view missing issue will incur unreliable neighbor relationships, thus breaking the key smoothness assumption in lahel propagation (LPP). As shown in Fig. [[A] and (B), missing samples in each view may generate multiple "vacuum regions" that fragment a complete category cluster into several sub-clusters, thereby distorting the smooth local structure on the common manifold. We term this phenomenon the Sub-Cluster Problem (SCP). Comparing Fig. [ (C) and (D), one could observe that SCP impedes the propagation of red label information its corresponding sub-cluster, erroneously making the decision boundary recede into the vacuum region. As a result, SCP would not only cause inappropriate graph fusion by misrepresenting the local structure, but also directly degrade the quality of basic label matrices in individual views. Besides, to the best of our knowledge, no existing framework has been proposed to mitigate the impact of SCP on GMvSSL.
To address SCP, we propose AGF-TI, a novel Adversarial Graph Fusion-based IMvSSL method with Tensorial Imputation, as shown in Fig. [ In essence, AGF-TI incorporates three key innovations: (1) an adversarial graph fusion operator that fully explores complementary structure information across views through a min-max framework to learn a robust consensus graph for label propagation; (2) tensorial imputation function that stacks view-specific similarity graphs into a third-order tensor recover the similarity relationships of missing samples with high-order consistent correlations across views and tensor nuclear norm regularization; and (3) an anchor-based acceleration strategy that significantly reduces the computational cost associated with min-max optimization and tensor learning. To solve the proposed max-min-men optimization problem, we design novel and efficient algorithm with theoretical convergence that incorporates the reduced gradient descent and the alternative direction method of multiplier. To verify the effectiveness of AGF-TI, we conduct empirical experiments on multiple public datasets with different view missing and label annotation ratios. Extensive experimental results show that AGF-TI outperforms existing competitors, exhibiting a more robust ability to tackle the dual missing issue. Our contributions are summarized as follows:
We identify novel challenge in GMvSSL with missing views, termed as the Sub-Cluster Problem, where missing samples may disrupt the core smoothness assumption, leading to inaccurate graph fusion and degraded performance of view-specific label prediction.
We are the first to incorporate a min-max framework into GMvSSL to learn a robust consen- sus graph against the distorted local structure, enhancing fusion performance. Unlike existing IMvSSI. methods that disregard missing samples when constructing graphs, we propose exploiting high-order consistent information to reconstruct incomplete local structures.

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We develop an efficient alternative optimization algorithm with the reduced gradient descent to solve the intractable objective of AGF-TI. Empirical results on various public datasets validate the effectiveness and efficiency of our proposed AGF-TI.
2 Preliminary and Related Work
2.1 Graph-based Multi-view Semi-supervised Learning
In general, graph-based multi-view semi-supervised learning (GMvSSL) can be viewed as the combination of a graph-based semi-supervised learning (GSSL) and a multi-view fusion strategy. Assume that X-(xI..... X.. TRRux a dimensional single-view data matrix with n samples, where the first f(< n) samples are labeled into e classes as (y)e(c)]<= For the sake of calculation, a one-hot label matrix {0, 1 jæxe is used to describe the situation of labels, Le., Yi==1 iff y==j(iE))
and 0 otherwise. Then, the GSSL can be solved by minimizing the cost function:
C(F, Ls) Tr(F" F) + ()
-Y),
where F is soft label matrix, B is diagonal regularization matrix, and Ds=Ds- S is Laplacian matrix associated with S. Here, matrix Se(0, 1] |m"=n is the symmetric similarity matrix where its element S describes the similarity between x; and Xj. The matrix Ds is diagonal degree matrix with entries d==> >, Syc. The cost function was explained with the smoothness rule and the fitting rule [14]. The left-hand term, named smoothness rule, plays a crucial role in mining extrinsic supervision signals from graph structure, forcing the soft label vectors of nearby samples to barely differ. However, the performance of the smoothness rule heavily depends on the smoothness assumption, which can be easily violated by SCP arising from incompleteness in multi-view scenarios.
Multi-view fusion strategy aims exploit the complementary information from multiple views to improve the final performance. Among them, late fusion 7 [9 approaches integrate information at the decision level by leveraging multiple base label matrices obtained via GSSL in individual views, while early fusion 9 10 (00 approaches operate graph fusion on view-specific graphs to learn a consensus one. Although most late fusion methods adopt weighting strategy balance view quality, they often overlook geometric consistencies across views, making it struggle to recognize graph distortions induced by missing samples, i.e., SCP. Therefore, in this work, we focus on the early fusion approaches and propose a novel adversarial graph fusion method using a min-max framework to explore the complementary local structure to alleviate the impact of SCP.
2.2 Accelerated GMvSSL with Bipartite Graphs
Despite the promising performance of existing GMvSSL methods, the high computational complexity prevents their application to large-scale tasks. Assume that (R. To-1 is multi-view dataset with n samples and V views. The time and space complexity of most GMvSSL are O(n²) and O(V)²) cubic and quadratic to sample numbers (9 [0]. To address this, GMvSSL accelerated by anchored or bipartite graphs has been widely studied [I]] 19 21]. These approaches effectively reduce the computational costs by constructing bipartite graphs with m(<n) anchors selected by k-means [22 23), BKHK (24), etc., instead of entire samples. After anchor selection, the bipartite graphs in each view Z can be effectively constructed by solving the following problem:
Σ" j=1 (Ixx ( a. (v) 12 (o) + 7 r(2z))))) s.l. 1,11-1,, Z,
min 2.
(2)
where x," and a represents the i-th sample and j-th anchor in the e-th view, respectively, 2
is the (i, j))th element of Z,, and 1, is an all-ones column vector with n elements. Compared to the classical Gaussian kernel method (7). once the neighbor number is given, the model in Eq. (2) enjoys a better construction performance with a parameter-free closed-form solution (25]. After obtaining the bipartite graphs, we typically construct the sample-level graphs using S. as the
input of GMvSSL 26 27 Accelerated by Woodbury matrix identity, the time and space complexity
could be reduced to O (nm?) and D(Vnm), which can be applied to large-scale datasets effectively.
2.3 Third-order Tensor for Multi-view Learning
To exploit the high-order correlation among multiple views, tensor- tensor-based multi-view learning ap- proaches have emerged [28 29]. Specifically, these methods construct a third-order tensor by stacking
3

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view-specific representation matrices and impose low-rank constraints to capture inter-view consistent structure. For example, Zhang et al. [3] apply a slice-based nuclear norm to model the high-order correlation. To better constrain the low-rankness of the multi-vies view tensor, recent works [31 32 introduce a tensor average rank with the Tensor Nuclear Norm (TNN) based on the tensor Singular Value Decomposition (I-VVD), serving as the tightest convex approximation. Aiming to alleviate SCP, this work employs the third-order tensor with the TNN to leverage the high-order correlation for recovering the local structure of missing samples the bipartite graphs. We give the definition of Tensor Nuclear Norm as follows. Basic tensor operators are defined in Appendix A
13 Given tensor ZeR X the nuclear norm of Definition 1 (Tensor Nuclear Norm) 1
the tensor defined | ni
Eminlminns (t, i), where S obtained by t-SVD of ==M+S=VT in Fourier domain through the fast Fourier transform Sy=fi(S, []3), and S is the k-th frontal slice of Sj.
Σne Dist
3 Methodology
3.1 Proposed Formulation
Assume that Tr and wa collect the index of existing and missing samples in the v-th view, respectively, Thus, the bipartite graph Z. can be divided into Zn, and Zw... where Zr, could be constructed by Eq. (2) among existing samples while Zue is initialized as equal probability matrix. Different to previous works [O I2]
that fuse the sample-level graphs S,eRaxn, we aim to directly obtain consensus anchored graph PER"Xm based on {Z,) } which accelerates subsequent label propagation and tensor learning. Rethinking Eq. ( it independently constructs the bipartite graph in each view, and unavoidably introduces the anchor-unalige
problem (34. degrading fusion performance. To tackle this issue, we introduce the permutation matrices (TT,ER"X))) J==1 to align the anchors between different views. Besides, to adaptively emphasize the contributions made by various views to the fused graph, we assign learnable weight o, (vE[V)) to each view. Finally, inspired by adversarial training, we design novel Adversarial Graph Fusion (AGF) operator based on a min-max framework, which defined as follows:
acA) PEAT min max ACF(P, (Z, (
Σ a Tr (Z.T.)) PP] s.t. T. (Vv), (3)
where In mm is an identity matrix, a=(ai...... [T and - (1m=1 03.
Remark 1 (Benefits of AGF) Compared prior early fusion methods, AGF has the following merits: (I) ACF directly learns an bipartite graph, which improves the efficiency of the algorithm: (2)
Regularization by the min-max framework of a and P makes the model less sensitive minor data fluctuations, e.g., sub-clusters, alleviating SCP and generating more robust consensus graph (35
Furthermore, capture high-order correlations across views, stack the bipartite graphs (Z,) (Z,) ==1 into a tensor - /Z...... Zv (Vv)RR>VVV X where ((-) is merging and rotating operator [36]. By optimizing Z with TNN, the similarity relationships between missing samples and anchors in each view are imputed via the cross-view consistency information. In this way, the incomplete graph structures could be recovered, thus further alleviating the effect of SCP.
Finally, to enhance the semantic information of unlabeled data, we construct a sample-level graph for label propagation by using limited labeled instances. To better leverage the local structural information across neighbors, following [11, we construct the graph among fused sample and anchor
P ER (((++m))(++m)) By introducing a soft label matrix of fused anchors
nodes as Sp-
PT
as mx the loss function Eq. can be equally rewritten into a performance gain form with the normalized Laplacian matrix Ls, Intm D Sp. SpD
Sp =I SSp:
R(P,
SpF) (T (
+2Tr (BYFT) -Tr (In+m B)F

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where F- Q] Y-[Y:0]. and B is diagonal matrix with the i-th entry being regularization parameter. Therefore, the final objective of AGF-TI can be expressed as:
max min max R R(F.ss,) ) + AAGF (Z.) p|| (Z Tule h
(5)
s.t. P. Z, E T. Im (VvE 2 Zv),
where À and p are nonnegative regularization parameters.
3.2 Optimization
To solve the intractable max-min-max model in Eq. ( the reduced gradient descent and alternative direction method of multipliers (ADMM). Inspired by ADMM, introduce auxiliary tensor variable 9 to relax 2, so the objective function of Eq. can be rewritten as the following augmented I Lagrangian function:
propose an efficient solution by combining
I ((Z... Tol F.a. a. P)
ppF) +2Tr (YFFT) -Tr (11 (In +B)F F)
(6)
+Y E. (PTTT.) - P1l (1
where 3x=A-B, W is Lagrange Multiplier, and 17 serves as a penalty parameter to control conver- gence. Then, the optimization problem of Eq. O6 can be decomposed into five subproblems, each of which optimizes its respective variables independently while keeping others fixed.
Z, Subproblem: Fixing the other variables, the problem in Eq. (6) can be disassembled into V separate subproblems w.r.f. Zwn. = 1,2,.... V:
min 7 Z l Z, -.)-Ga))aaa (Z).T.)) s.t. Zw. EA Wal
Zen,
2
The problem in Eq. (7)
can can be further rewritten as the following element-wise form:
Σ NEW Σ 1=1 (Z
G Ao, (PT 26
min Zuno EA
(8)
Noting that the problem in Eq. (8) is independent between different so individually solve the following problem in vector form for each row Z
min z 1) - ((()) ((G
s.t. Z 1m 1 2 20, iEw,,
which can be solved with closed-form solution 255.
P and a-Subproblem: By fixing the other variables, we derive a min-max optimization problem P and a, and then rewrite it with optimal value function of the inner maximization problem:
min M(a), DEAt
h(a:)= max A Σ. PEA
-T LL
(10)
Recall that F= F; Q). based the property of the normalized Laplacian matrix, the relationship Tr(FT F) Σ; m [F./Vd, /dr+71lă holds, where d; >n= Sp(i, k). Similar
the subproblem of Zu,, can obtain the optimal P of the inner maximization problem by solving the following proximal problem for each row of with a closed-form solution:
|
P. arg min P.
(ZZ.
H.)
S.t. 1
(11)
23
2
where Ž. a, is the weighted bipartite graph and H measures the distance between the soft labels of and Q with its element of Eq. (11) is a closed convex set and its objective function is strictly convex, the Hilbert projection theorem guarantees the uniqueness of the optimal solution P Then, leveraging Theorem 4.1 in 37 we have the following theorem with its proof provided in Appendix C
d /v Since the feasible region
5

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Ll + min Theorem h(a) is differentiable, and its gradient can be calculated as where P is the optimal solution of the inner maximization problem.
ah(ax)
-2a, Tr((=TTTTTT Z,
8e.
Therefore, a reduced gradient descent method can be developed to solve the optimization problem in Eq. (10). Specifically, first calculate the gradient of h(a) by Theorem I and update the along the direction of the gradient descent over the simplex constraint a with the optimal P
Consider the equality constraint of a. supposing that ar is non-zero entry of a and Vh(a) is the reduced gradient of h(a). Following [3 39), the v-th entry of the reduced gradient can be constructed as follows:
[Vh(a)] @h(aa)
ah(ar) Vugu,
V Σ v=1,0şuu
@h((x) do
Ōh(a)) da
[VM(a)
(12)
do.
8ok.
where is typically set the index of the largest entry of a. leading to better numerical stability 40 Since -/((() represents a descent direction to minimize h((x), we can directly set the descent direction g -- Vi(a). To ensure the non-negativity of we further modify the descent direction yr to zero iff =0 and [Vh(a)) >0. Then, a can be updated using the rule a+-a+êg, where is
the optimal step length calculated by linear search mechanism, e.g., Armijo's 's rule. The detailed
procedure for solving the problem Eq. (TP) is outlined Algorithm T in Appendix
F Subbproblem: Fixing the other variables leads the following problem for F,
max R(F, Ls.), s.t. F- QUER(R+RRRg (m)x
(13)
F
By setting the derivative of R(F,, Ls,.) to zero, F can be updated as =Ls +B) Since
Lgp and B are both block matrices, we can adopt the blockwise inversion to solve the first term:
-PA- - M2 - I +B.
I,+Bn
[
C
-M C
(Bs
(14)
C
where AmRmxm diagonal matrix with element Aイ=ΣE P.j. B PA
- M m.
M21. and C2=M2 M2, M
Since CR needs time complexity O(n²) to solve CT we utilize the Woodbury matrix identity to accelerate the inversion by C, -Mi "MM.+ M12 Mj MM2)) M, which reduces Onn²) Onm?²). Note that M but it is diagonal matrix and its inversion can be easily obtained with (nn). Then, leveraging the blockwise inversion in Eq. (14), the update formula for F* F*; Q*] can be divided into two parts W.E. F and Q, respectively:
M2 F = Mii M (2 M2 MMMM B, Y B,
(15)
Q = - (M2 M, M M,
G-Subproblem: When other variables are fixed, the subproblem for & is formulated as:
W 2 1 g-
-
(16)
& 7)
F
The subproblem in Eq. (6 can solved by the following theorem.
Theorem 36
Suppose G, F.RRI and T>0, the globally optimal solution
ming +|| +
-l is given by the tensor tubal-shrinkage operator, i.e.,
G=Cnar (F)=U+ (S)+VT, where = is obtained by t-SVI) and Casr (S) Herein, J is an ni Xn, Xns f-diagonal tensor whose diagonal element in the Fourier domain (i. (i,
T. Su-ppoboo:::: By fixing the other variables, T. can be independently updated by,
T = arg max Tr(T, Z P), s.t. T T. Im.
(17)
T.

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The optimal solution Tü of the problem 17 is U. where U. and V. are the left and right singular matrix of 2 P. Its detailed proof is provided in Appendix B.4
At last, the Lagrange multiplier and penalty parameter 7 min(" flma),, respectively, where 7,,>1 is used to accelerate convergence.
updated as W-W+r((-99) and
The procedure of our method is summarized in Algorithm in Appendix B.5 Besides, Appendix D.1 A
provides theoretical convergence analysis, showing that the variable sequence obtained by Algo-
rithm 2 converges to a stationary point. Its time and space complexity is discussed in Appendix D.2
4 Experiments
4.1 Experimental Settings
Datasets.
We conduct all experiments on six public datasets, including CUB, UCI-Digit, Caltech101- 20, OutScene, MNIST-USPS, and AwA. The brief information of these datasets presented in More details of them are shown in Appendix E.1
Table 1
Baselines. To validate the effectiveness of AGF- TI, we compare it with the following meth- ods. First, SLIM [46]. and AMSC [19] serve as two state-of-the-ee art IMvSSL baselines. Fur- thermore, we include numerous GMvSSL al- gorithms, such as AMMSS (7), AMGL [8]
Table 1: The description of six datasets
Datasets
Samples
Views
Classes
Anchors
CUB
600
64
UCI-Digit
32
2,000
3
10
256
MLAN (9. AMUSE 10). FMSSL [II] FM-
Calrech101-21
41
2,386
6
256
OutSoene
43
2,688
4
256
SEL [2], and CFMSC [13]. We also incorpo- rate popular regression based MvSSL methods, i... MVAR [47] and ERL-MVSC [48]. Since
USPS
5,000
2
256
AwA 45
10,158
2
512
most baselines, except SLIM and AMSC. cannot handle the incomplete data, following [9), to recover feature matrices for a fair comparison. More
adopt deep matrix factorization (DMF)
details shown in Appendix E.2
we apply the BKHK algorithm to select m anchor points among existing samples in each view. The number of selected anchors m is presented in Table [ Then we solve the problem Eq. ( to construct the bipartite graphs, i.e., { (Z,, 1V with the neighbor number l set to 7. AGF-TI has three regularization parameters X, 82. and p. In our experiments, À is fixed to V2, while p and 3x are tuned in {10', 102. 10y} and {2"...., 26,. respectively. We apply accuracy (ACC), precision (PREC), and Fi-score (FI) as the evaluation metrics. Each experiment is independently conducted ten times, and the final average results are reported.
Parameter setting. Following
4.2 Main Results
To comprehensively evaluate the effectivenesss of AGF-TI, we compare it with the baselines from two perspectives, i.e., view missing and label scarcity. For the view missing issue, we follow [19] to randomly select VMR% (view missing ratio) examples in dataset as incomplete examples, which
randomly miss 1-V-1 views. On this basis, we randomly select LAR% (label annotation ratio) examples of each class as labeled data simulate the label scarcity setting.
From view missing perspective, Table 2 presents the comparison in metrics of different methods on six datasets under multiple VMRs (30%, 50%, 70%) when LAR% % is 5%. From Table 2] we can observe that: (1) Compared to AMSC, tailored for IMvSSI, GMvSSI. methods with DMF achieve comparable even higher performance on some datasets like CUB and MNIST-USPS. This further validates our observation that missing samples disregarded by AMSC could distort local structure and mislead label propagation, thereby degrading performance. (2) AGF-TI consistently achieves the highest performance under multiple VMRs and metrics on all datasets except AwA, where it obtains two sub-optimal results at VMR=30% while showing dominant superiority with increased VMR. The results demonstrate that AGF-TI can effectively tackle the dual missing issue. (3) Compared with IMvSSL baselines, i.e., SLIM and AMSC, which disregard missing samples, the tensorial imputation strategy makes AGF-TI more robust to incomplete multi-view data. For instance, on OutScene, as VMR is increased from 30% to 70%, the ACC decline of AGF-TI is less than 6%, while the decreases of SLIM and AMSC are nearly 13%. Detailed results with standard deviations are in Appendix F.I
7

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AMINSS
ANGIL
MLAN
AMUSE
FMMSSL
FMSEEL
GFMSC
MIAR
ERL-MVSG
AMSG
SLIM
AGP-T
76
60-
ADC
70-
70-
60-
50-
50-
30
40
50-
20
Labeled Rate%
Laboled Rets% (8) UCI-
Labeled Rutiff [C] Calechh111-00
Labeled Ratie%
Labuled Redos (8) MNNUS T-USPP
Labaled Ratoss (P) AMA
(A) CUB
(D) Cuttscen
Figure 2: ACC results on six datasets with LAR varying in (1%, 2%, 4%, 8%) when VMR is 60%.
From label scarcity perspective, Fig. 2 presents the ACC results under different LARs varied in (1%, 2%, 4%, 8%) } with VMR fixed to 60%. Compared with early fusion- -bsod GMvSSL approaches, i.e., MLAN, AMUSE, FMSEL and CFMSC, AGF-TI consistently outperforms them over all cases. This result suggests that the consensus graph fused through adversarial graph fusion in AGF-TI can effectively alleviate the negative impact of structural distortions caused by missing samples on label propagation. Besides, we observe that the performance of AGF-TI when LAR is only 1 1% is on par with most baselines under higher LAR, indicating higher label utilization efficiency of our method.
4.3 Ablation Study
AGF-TI contains two main parts: adversarial graph fusion and tensorial imputation. To validate its effectiveness, we conduct the following ablation studies under different VMRs when LAR is fixed to 5%. In the adversarial graph fusion part, remove the permutation matrix for anchor alignment and the weight coefficient for view quality assessment, marked as "w/o T," and "w/o ar respectively. To evaluate the tensorial imputation part, we remove the update of Zu., from the optimization algorithm, termed as "w/o TI". The ablation results are listed in Table From the results,
Table 2: Mean results of compared methods on six datasets under different VMRs (view missing ratios) when fix LAR (label annotation ratio) to 5%.
CUB
UCI-Digtt
Method
VMR-10%
VMR-500%
VMR-775
VMR--30%
VMR-500
VMR--70%
ACC
PRBC
F1
ACC:
PRBC
F1
ACC
PREC
F1
ACC
PREC
F1
ACC
PREC
ACC
PREC
F1
AMMSS
67.33
67.33
64.57
48.44
48.
44.25
62:
74
67:74
00.93
91.46
91.46
91. 91.43
87.10
87:3 12:30
87.34
76.04
76.04
75.87
AMGL
67.11
67.11
64.64
65.77
65.
65.77
53.33
61.25
61.75
60.04
91.06
91.06
90.03
89.73
89.75
89.65
85.32
85.32
85.32
MLAN
70.09
70.09
68.35
67.75
67.75
56.04
62.25
62.25
60.68
86.13
86.13
86.27
24.82
74.32
76.31
62.43
62.43
65.09
AMUSE
64.26
64.25
61.84
63.21
63.21
51.33
62.73
62.77
60.30
98U61
90061
90.50
87:21
87:21
87.13
81.91
$1.91
$1.81
FMSSL
63.30
63.30
62.31
43.72
45.72
44.02
64.49
64.09
62.65
91.12
91.12
91.07
86.72
86.72
86.62
78.00
78000
77.72
FMSEL
70.54
70.54
09:21
60.28
67.25
56.16
62.34
62.35
00.81
92.19
92.19
92.3 22.16
98.94
88.94
88.95
83.53
83.53
43.40
CUMSC
71.63
71.83
70.81
89.19
69.19
58.02
55.23
54.23
53.13
91.48
91.48
91.43
98.14
88.1
95.08
85.47
$3.47
83.33
MIVAR
66.09
06.59
65.1
69.19
61.51
$1.51
50.20
50.15
50.18
43.15
80.51
80.51
80.43
75.46
75:45
75.11
68.01
68.01
67.78
TRL-MVSC
65.75
66.25 75
65
65.48
51.51
61.51
51.54
59.31 73
86.
$5.80
36. 76
84.61
34.61
$4.59
19.20
79.20
19.20
AMSC
68.75
68.75 75
67.60
87.00
55.05
64.
64.30
63.00
93.87
93.87
93.55 3..8
91:21
91.31
91.19
87.59
87.59
87:55
SLIM
68.30
68.30
67.36
65.12
63.12
63.08
64.04
64.04
63.42
81.93
84.93
84.30
91.41
81.41
81.42
72.55
72.55
72.43
Ours
78,33
78.33
77.03
50.23
80.23
79.10
74.35
74.25
72.30
95.95
95.98
95.97
95.23
95.33
95.25
95.16
95.16
95.13
Caltuch101-20
OutSeene
AMMSS
60:87
20:57
68
53.47
49.54
46.43
40.56
AMGL
62.15
23.50
24.06
61.60
22.56
58.28
57.71
55.5%
48.33
46.69
MLAN
39.74
09.92
33.26
63.00
26.95
57.00
42.04
AMUSE
21.87
61.85
23:27
23.95
60:43
21.40
56,80
56.1
$7.04
57
54.00
48.70
48.43
47.54
30.50
43.20
10.51
57.01
15.11
51.34
50.16
47.52
42.76
41.95
38.90
RMSEL
77.96
47.16
72.63
37.77
41.09
66.92
31.49
66.10
66.03
16
56.42
56.46
511.4
49.14
CFMSC
49.39
73.46
39.55
67.76
33.90
67.41
67
36 1.
57.6
47.59
45.67
MIAR
49.27
72.05
36.75
63.92
25.17
62-91
62:85
58.42
51.11
51.16
49.73
12.60
55.47
53.27
42.72
22.80
15
63.95
73.02
61.65
73.17
MNIST-USPS
AwA
AMMIS
00.20
84.05
81.
53.67
49
55.21
47.32
09.61
85.75
$7.74
44.52
401.09
MLAAN
$3.29
64.96
75.72
75.72
75.32
66.49
47.72
48.37
AMUSE
83.16
62.99
$1.44
34.31
FMSSL
86.36
56.36
79.72
79.72
72.58
72.58
71.76
65.60
55.79
55.32
59.34
50.30
52.24
42.66
41.57
FMSEL
90.51
$2.04
50.71
51.72
CEMSC
93.55
89.05
94.57
84.45
67.24
54.71
59.85
52.41
MVAR
79.65
71.75
66.04
66.15
69007
36.99
59.15
52.35
5192
ERL-MVSC
55.12
$5.90
K2.41
55002
50.51
48.55
49.11
41.54
AMSC
58.72
83.00
83.03
K3.60
64.34
54.74
54.06
55.00
45.76
4438
5254
45.29
42.52
SLIM
$4.61
$4.45
$1.33
19.17
79.17
73.05
64.88
$4.24
53,50
28
59.01
48.35
45.31
Ours
95.09
95.09
95.07
95.55
95.88
95.55
95.03
95.63
95.99
69.12
57.16
70.58
59.77
5862
69.11
5832
$7.59
8

---

Table 3: Ablation study of AGF-TI under LAR is 5%.
VMiR=100
NCC
CUB
F1
ACC
UCI-Digh
F1
Caleech01-30
ACC
F1
ACC
Outteene
F1
MINIS-LUPPS
ACC
Fi
ACC
APD
F1
of
CPHIE
AGP-TI
70.61
31.61
26:31
62.55
76.05
14.25
99.35
09.65
48.13
66.59
93.33
98
75.27
74.85
TT
72.92
71.87
88.96
$8.94
76.63
50.56
63.55
63.06
88.85
68.76
78.16
72.64
VMRR=1002
051
AGR-TE
72.90
98.16
95.13
72.17
19-72
63.29
95.62
98.89
89:34
73.21
T.
34.56
39.54
22:11
78.12
77.61
51.25
43.75
Figure 3: comparison CUB.
(A) CLB
(8) UCHOg!
(C) Carlech 101-20
(D) Outoonn
(E) MNIST-UPPS
(F) Avak
15-
PABC
Denution
Benddons
Figure 4: The iterative error and classification performance of AGF-TI during optimization process.
one could observe that (1) Each part plays crucial role in performance improvement, suggesting the effectiveness of our AGF-TI. (2) When the number of views is large, e.g., six for Caltech 101techh111-00 101-20 and four for OutScene, removing the permutation matrix results in large performance degradation relative to the other datasets. (3) As VMR increased, the contribution of the tensorial imputation part highlighted. For instance, removing the imputation part leads to an average performance drop of up to 17.8% when VRM is 70%, compared to a 6.2% degradation at 30% VRM in terms of ACC.
To further investigate the influence of the min-max scheme on graph fusion, we compare the evolution of the weight coefficients a from AGF-TI and a recent min-min scheme method, CFMSC. The comparison results on CUB under VMR=50% and LAR=5% are plotted in Fig. 3 Unlike the min-min framework, the proposed AGF operator alternatively magnifies the weight coefficient of each view by maxa while alleviating the structural disparity across views by minp. According [38 50). this alternative pattern could yield a robust structure against noisy perturbation by fully exploring complementary information from individual views, demonstrating the strength of AGF.
4.4 Model Analysis
In this section, we conduct model analyses under VMR=50% and LAR=5% conditions on the convergence behavior and parameter sensitivity w.r.t Bx and p. More analyses on the number of anchors m, the trade-off parameter X, and the computational costs of the TNN step are provided in
Appendix F3 and F.5
Convergence behavior. To examine the convergence behavior of AGF-TI, we calculate the error value of the soft label matrix F and present its classification performance during iteration in Fig. [ The error curves consistently exhibit early oscillations, subsequently undergoing a rapid decrease until convergence. The oscillation stage could be explained by the alternative pattern (illustrated in Fig. 3) discussed Scction 4.3 On the other hand, its performance substantially improves during this oscillation stage, further indicating the effectiveness of the proposed AGF.
Parameter sensitivity. AGF-TI introduces the regularized parameter 3A to control the connec- tivity of the fused graph and for tensorial imputation. We empirically analyze the impact of the two parameters on classification performance by tuning 33 and o in the sets {20, ......, 29) and (10%...., ...., 10%), respectively. ACC results are recorded in Fig. 5 and more results are presented Appendix F3 Compared to p. 3g has great effects on AGF-TI, indicating that appropriate connectivity of the fused graph could further enhance classification performance. Besides, we find

---

Ca
Outioon
USPP
Figure 5: Parameter sensitivity analysis of 3x and in terms of Accuracy.
that AGF-TI exhibits stability with high performance within small ranges (i.e., (8x) and 1og10 (p) 5)) across various datasets, maintaining generalizability.
Running time. To empirically evaluate the complexity of the algorithms, we compare their running times on all datasets. For baselines requiring complete multi-view data, the total running time includes both the DMF completion process and its subsequent execution. The results are recorded Table 4 and it can be observed that AGF-TI maintains acceptable computational overhead. Although AGF-TT takes slightly longer than AMSC and SL.M due additional tensor imputation and shared bipartite graph fusion, it consistently achieves advanced classification performance across all datasets. Moreover, thanks to the adopted anchor strategy, AGF-TT's running time scales efficiently with the sample size, ensuring its practicality for large-scale applications.
Table 4: Running time (in seconds) of different methods under VMR=50% and LAR=5%.
Method
CUB
UCI-Digit
Caltech101-20
OutScene
MNIST-USPS
AwA
AMMSS
4.31
7.80
86.25
15.61
48.58
621.34
AMGL
4.16
5.51
65.99
9.60
39.89
587.51
MLAN
4.68
20.29
141.65
74.99
150.06
1068.73
AMUSE
6.37
31.77
102.76
55.59
209.93
1361.76
FMSSL
4.77
5.81
82.94
24.51
74.75
808.80
FMSEL
6.45
32.70
120.18
60.12
234.50
1359.60
CEMSC
6.78
20.36
106.95
26.29
159.53
815.56
MVAR
4.26
5.40
69.84
10.36
39.30
601.27
ERL-MVSC
4.31
5.68
67.71
10.55
42.66
605.83
AMSC
0.85
2.15
11.03
4.40
3.73
33.54
SLIM
0.33
4.46
35.59
14.12
11.43
128.83
Ours
1.06
15.35
45.03
35.84
31.06
354.54
Conclusion
In this paper, we present the first study the Kub-Clusted Problem (SCP) caused by missing views in graph-based multi-view semi-supervised learning. To address SCP, novel Adversarial Graph Fusion-based model with Tensorial Imputation (AGF-TI) is proposed. Our method benefits SCP in two aspects: (1) In AGF operator, the min-max optimization paradigm makes the model less sensitive to minor data fluctuations, enabling the ability to learn a robust fused graph against SCP. (2) The recovered local structures, imputed by high-order consistency information, further alleviate the impact of distorted graphs. For real-world applications, we adopt an anchor-based strategy to accelerate AGF-TL An efficient iterative algorithm is developed to solve the proposed optimization problem. Extensive experimental results demonstrate the effectiveness of AGF-TL This work focuses on the setting where the bipartite graphs manually pre-constructed. Extending it to jointly optimize anchor selection and graph construction remains an interesting direction for future work.
10

---

Acknowledgments
We thank Yansha Jia for her comments on this paper. This work was partially supported by the National Natural Science Foundation of China under Grant No. 62376281 and the Key NSF of China under Grant No. 62136005.
References
[1] Xiaozhi Chen, Huimin Ma, Ji Wan, Bo Li, and Tian Xia. Multi-view 3d object detection network for autonomous driving. In Proceedings of the IEEE conference on Computer Vision and Pattern Recognition, pages 1907-1915, 2017.
[2] Sudeep Fadadu, Shreyash Pandey, Darshan Hegde, Yi Shi, Fang-Chieh Chou, Nemanja Djuric, and Carlos dallepi-Gonzzlez. Multi-view fusion of sensor data for improved perception and prediction in autonomous driving. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 2349-2357, 2022.
[3] Adrienne Kline, Hanyin Wang, Yikuan Li, Saya Dennis. Meghan Hutch, Zhenxing Xu, Fei Wang, Feixiong Cheng, and Yuan Luo. Multimodal machine learning in precision health: A scoping review. npj Digital Medicine, 5(1):171, 2022.
[4] Ziling Fan, Zhangqi Jiang, Hengyu Liang, and Chao Han. Pancancer survival prediction using a deep learning architecture with multimodal representation and integration. Bioinformatics Advances, 3(1 vbbd0006, 01 2023.
[5] Richard J Perrin, Anne M Fagan, and David M Holtzman. Multimodal techniques for diagnosis and prognosis of alzheimer's disease. Nature, 461(7266;)116-222 2009.
[6] Shangran Qiu, Matthew Miller, Prajakta $ Joshi, Joyce C Lee, Chonghua Xue, Yunruo Ni, Yuwei Wang, lleana De Anda-Duran, Phillip H Hwang, Justin A Cramer, et al. Multimodal deep learning for alzheimer's ss disease dementia assessment. Nature Communications, 13(1);3404, 2022.
[7] Xiao Cai, Feiping Nie, Weidong Cai, and Heng Huang. Heterogeneous image features integra- tion via multi-modal semi-supervised learning model. In Proceedings of the IEEE International Conference Computer Vision, pages 1737-1744, 2013.
[8] Feiping Nie, Jing Li, Xuclong Li, al. Parameter- free auto- wetghweighted multiple graph learning: framework for multiview clustering and semi-supervised classification. In Proceedings of the International Joint Conference Artificial Intelligence, pages 1881-1887, 2016.
[9] Feiping Nie, Guohao Cai, and Xuelong Li. Multi-view clustering and sumi-supervised classifica- tion with adaptive neighbours. In Proceedings of the AAAI Conference on Artificial Intelligence, pages 2414,,444444444 2017.
[10] Feiping Nie, Lai Tian, Rong Wang, and Xuelong Li. Multiview semi-supervised learning model for image classification. IEEE Transactions on Knowledge and Data Engineering, 32(12):2389-2400 2019.
[11] Bin Zhang, Qianyao Qiang, Fei Wang, and Feiping Nie. Fast multi-view semi-supervised learn ing with learned graph. IEEE Transactions on Knowledge and Data Engineering, 34(1):286 299, 2020.
[12] Zhongheng Li, Qianyao Qiang, Bin Zhang, Fei Wang, and Feiping Nie. Flexible multi-view semi- semi-supervised learning with unified graph. Neural Networks, 142:92-104, 2021.
[13] Bingbing Jiang, Chenglong Zhang, Yan Zhong, Yi Liu, Yingwei Zhang, Xingyu Wu, and Weiguo Sheng. Adaptive collaborative fusion for multi-view supervisd classification. Information Fusion, 96:37-50, 2023.
[14] Dengyong Zhou, Olivier Bousquer, Thomas Lal, Jason Weston, and Bernhard Schölkopf. Learning with local and global consistency. In Advances in Neural Information Processing Systems, pages 1-8, 2003.

---

[15] Yijie Lin, Yuanbiao Gou, Zitao Liu, Boyun Li, Jiancheng Lv, and Xi Peng. Completer: Incomplete multi-view clustering via contrastive prediction. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pages 11174-111888 2021.
[16] Zhenbang Wu, Anant Dadu, Nicholas Tustison, Brian Avants, Mike Nalls, Jimeng Sun, and Faraz Faghri. Multimodal patient representation learning with missing modalities and labels. In Proceedings of the International Conference on Learning Representations, pages 1-13, 2024.
[17] Jingjing Tang, Qingqing Yi, Saiji Fu, and Yingje Tian. Incomplete multi-view learning: Review, analysis, and prospects. Applied Soft Computing, 153: 11278, 2024.
[18] Yiding Lu, Yijie Lin, Mouxing Yang, Dezhong Peng, Peng Hu, and Xi Peng. Decoupled contrastive multi-view clustering with high-order random walks. In Proceedings of the AAAI Conference Artificial Intelligence, pages 14193-14201, 2024.
[19] Wenzhang Zhuge, Tingjin Luo, Ruidong Fan, Hong Tao, Chenping Hou, and Dongyun Yi. Absent multiview semisupervised classification, IEEE Transactions on Cybernetics, 54(3):1708 1721, 2023.
[20] Masayuki Karasuyama and Hiroshi Mamitsuka. Multiple graph label propagation by sparse integration. IEEE Transactions on Neural Networks and Learning Systems, (4(12):1999-2011 2013.
[21] Yuting Wang, Rong Wang, Feiping Nie, and Xuelong Li. Fast multiview eemi-supervised classification with optimal bipartite graph. IEEE Transactions on Neural Networks and Learning Systems, pages 1-12, 2024.
[22] Zhao Kang, Zhiping Lin, Xiaofeng Zhu, and Wenbo Xu. Structured graph learning for scalable subspace clustering: From single view multiview. IEEE Transactions on Cybernetics, 52(9):8976-88) 8986, 2021.
[23] Feiping Nie, Jingjing Xue, Danyang Wu, Rong Wang. Hui Li, and Xuelong Li. Coordinate descent method for k-means. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(5):2371-2385, 2021.
[24] Feiping Nie, Wei Zhu, and Xuelong Li. Unsupervised large graph embedding based on balanced and hierarchical k-means. IEEE Transactions on Knowledge and Data Engineering. 34(4):2008- 2019, 2020.
[25] Feiping Nie, Xiaoqian Wang, and Heng Huang. Clustering and projected clustering with adaptive neighbors. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, page 977-986, 2014.
[26] Wei Liu, Junfeng He, and Shih-Fu Chang. Large graph construction for scalable semi-supervised learning. In Proceedings of the International Conference on Machine Learning, pages 679-686, 2010.
[27] Feiping Nie, Yitao Song, Wei Chang. Rong Wang, and Xuelong Li. Fast sumi-super learning large graphs: An improved green-function method. IEEE Trunsactions Patlern Analysis and Machine Intelligence, 47(3):2055-200 2025.
[28] Zhen Long, Qiyuan Wang, Yazhou Ren, Yipeng Liu, and Ce Zhu. S2mvte: A simple yet efficient scalable multi-view tensor clustering. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 26213-26222, 2024.
[29] Jintian and Songhe Feng. Anchors crash tensor: Efficient and scalable tensorial multi- view subspace clustering. IEEE Trunsactions Pattern Analysis and Machine Intelligence, 2764):2660-2675, 2025.
[30] Changqing Zhang, Huazhu Fu, Si Liu, Guangcan Liu, and Xiaochun Cao. Low-rank tensor constrained multiview subspace clustering. Proceedings of the IEFF/CVF International Conference Computer Vision, pages 1582-1590, 2015.
12

---

[31] Canyi Lu, Jiashi Feng, Yudong Chen, Wei Liu, Zhouchen Lin, and Shuicheng Yan. Tensor robust principal component analysis with a new tensor nuclear norm. IEEE Transactions on Puttern Analysis and Machine Intelligence, 42(4))925-933, 2019.
[32] Pan Zhou, Canyi Lu, Jiashi Feng, Zhouchen Lin, and Shuicheng Yan. Tensor low-rank repre- sentation for data recovery and clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43(5):1718-1732 2019.
[33] Pan Zhou, Canyi Lu, Jiashi Feng, Zhouchen Lin, and Shuicheng Yan. Tensor low-rank repre- sentation for data recovery and clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 43(5):1718-1733 2021.
[34] Siwei Wang. Xinwang Liu, Suyuan Liu, Jiaqi Jin, Wenxuan Tu, Xinzhong Zhu, and En Zhu. Align then fusion: Generalized large-scale multi-view clustering with anchor matching cor- respondences. In Advances in Neural Information Processing Systems, pages 5882-5895, 2022.
[35] Ben Yang, Xuetao Zhang, Jinghan Wu, Feiping Nie, Fei Wang, and Badong Chen. Scalable min- max multi-view spectral clustering. IEEE Transactions on Knowledge and Data Engineering, 37(5):2918-293) 2025.
[36] Yuan Xie, Dacheng Tao, Wensheng Zhang, Yan Liu, Lei Zhang, and Yanyun Qu. On unifying multi-view refeeeepreeetttoonns for clustering by tensor multi-rank minimization. International Journal of Computer Vision, (26(11): 126(11):1157-119) 2018.
[37] J Frédéric Bonnans and Alexander Shapiro. Optimization problems with perturbations: A guided tour. SIAM review, 40(2):228 264, 1998.
[38] Xinwang Liu. Simplemkkm: Simple multiple kernel k-means. IEEE Transactions Pattern Analysis and Machine Intelligence, 45(4):5174-5183 2022.
[39] Yi Zhang, Siwei Wang, Jiyuan Liu, Shengju Yu, Zhibin Dong, Suyuan Liu, Xinwang Liu, and En Zhu. Dleft-mke: Dynamic late fusion multiple kernel clustering with robust tensor learning via min-max optimization. In Proceedings of the International Conference on Learning Representations, pages 2025.
[40] Alain Rakotomamonjy, Francis R. Bach, Stéphane Canu, and Yves Grandvalet. Simplemkl. Journal of Machine Learning Research, 9(83):2491-2521 2008.
[41] Zhangqi Jiang, Tingjin Luo, and Xinyan Liang. Deep incomplete multi-view learning network with insufficient label information. In Proceedings of the AAAI Conference on Artificial Intelligence, pages 12919-12927, 2024.
[42] Chenping Hou, Ling-Li Zeng, and Dewen Hu. Safe classification with augmented features. IEEE Transactions on Pattern Analysis and Machine Intelligence, 41(9):2176-2 2192,, 2019.
[43] Zhanxuan Hu, Feiping Nie, Rong Wang, and Xuelong L.i. Multi-view spectral clustering via integrating nonnegative embedding and spectral embedding. Information Fusion, 55:251-259 2020.
[44] Xi Peng, Zhenyu Huang, Jiancheng Lv, Hongyuan Zhu, and Joey Tianyi Zhou. COMIC: Multi- view clustering without parameter selection. In Proceedings of the International Conference on Machine Learning, pages 5092-5101, 2019.
[45] Changqing Zhang, Yajie Cui, Zongbo Han, Joey Tianyi Zhou Huazhu Fu, and Qinghua Hu. Deep partial multi-view learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(5):2402-2415, 2020.
[46] Yang Yang, De-Chuan Zhan, Xiang-Rong Rong Sheng, and Yuan Jiang. Semi-supervised multi-modal learning with incomplete modalities. In Proceedings of the International Joint Conference on Artificial Intelligence, pages 2998 2998-3004, 2018.
[47] Hong Tao, Chenping Hou, Feiping Nie, Jubo Zhu, and Dongyun Yi. Scalable multi-view semi-superves semi-supereised classification via adaptive regression. IEEE Transactions on Image Processing, 26(9):4283-422 2017.
13

---

[48] Aiping Huang, Zheng Wang, Yannan Zheng. Tiesong Zhao, and Chia-War Lin. Embedding regularizer learning for multi-view semi-supervised classification. IEEE Transactions on Image Processing, 30:6997-7011, 2021.
[49] Jicong Fan and Jieyu Cheng. Matrix completion by deep matrix factorization. Neural Networks, 98:34 41, 2018.
[50] Seojin Bang, Yaoliang Yu, and Wei Wu. Robust multiple kernel k-means clustering using min-max optimization. arXiv preprint arxiv:/803.022538 2018.
[51] Adrian S Lewis and Hristo S Sendov. Nonsmooth analysis of singular values. part i: Theory. Set- Valued Analysis, 13:213-241, 2005.
[52] Robert G Bartle and Donald R Sherbert. Introduction to real analysis. John Wiley & Sons, Inc., 2000.
[53] Dong Huang, Chang-Dong Wang, and Jian-Huang Lai. Fast multi-view clustering via ensembles: Towards scalability, superiority. and simplicity. IEEE Transactions on Knowledge and Data Engineering. 35(11): 1181:11181111 11402. 2023.
[54] Zhihao WU, Xincan Lin, Zhenghong Lin, Zhaoliang Chen, Yang Bai, and Shiping Wang. Interpretable graph convolutional network for multi-view semi-supervised learning. IEEE Transactions on Multimedia, 25:8593-8606, 2023.
[55] Jielong Lu, Zhihao Wu, Luying Zhong, Zhaoliang Chen, Hong Zhao, and Shiping Wang. Generative essential graph convolutional network for multi-view semi-supervised classification. IEEE Transactions on Multimedia, 26:7987-7999, 2024.
14

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
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Answer: [No]
Justification: Our method does have limitations, but due space limitations and focus on presenting the novelty of the method and the main results, we were unable to discuss these limitations in separate section within the paper. However, we have mentioned the limitations of our method in the pre-constructed graphs in Section 5 which may weaken the effectiveness of graph fusion. We hope overcome this issue by combining our method with adaptive graph learning into a unified framework, and leave this for future work. Besides, we have discussed the time and space computational costs in Appendix D.2]
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A Basic Tensor Operators
We first summarize the notations used in this paper in Table
Table Summary notations
Notations
Annotation
x, X
scalar, vector, matrix, and tensor the element in the i-th row and j-th column of X
X1,, ((i,))
X;
the i-th row vector of X
the k-th frontal slice of tensor X
Xy =f= 3)
the fast Fourier transformation (FFT) the number samples, views, anchors, labeled samples, and classes
n, V, m, e
1n [....... 11T
the all-ones column vector with n elements the dimensional identity matrix the d, dimensional feature matrix of e-th view
I ERaxn
X, ERRdd
X, E
the feature vector of i-th sample in e-th view the label set of É labeled samples
YE(0, laxe
the one-hot label matrix the symmetric similarity matrix among samples (of v-th view) the diagonal degree matrix of S with entries d,= >, S.t the Laplacian matrix of S
samples
S (S.) 1j
Ds
Ls-Ds-S
(a. (1) ERd.x x 1==1
the set of m anchors in e-th view the anchor-based bipartite matrix of v-th view the index sets of existing and missing samples in v-th view the fused/consensus anchor- based bipartite graph the permutation matrix of v-th view
Z,EAM
To, We PEAm
T,RRuxm
AE ar E(0,
the weight coefficient vector and the weight coefficient of w-th view the similarity matrix among fused n samples and anchors the soft label matrix of fused samples and anchors, where F is the soft label matrix of samples and Q for anchors
Sp (nER(ta))((tm)))
F F- F; (AER( (n+m)xe
Y=(Y;0)
the one-hot true label matrix of samples and anchors
B. (
the diagonal regularization matrix the merging and rotating operator the Frobenius norm and f2 norm the (-SVD based tensor nuclear norm { E IR" nx [1 -1,, 0}
F 1|
m
Assume that X'ER" X and YERM2xx4X0 are two third-order tensors. Then, we introduce some operators related to tensors.
Transposition of tensor X x which means that each frontal slice of the tensor is transposed.
Cyclic expansion of the tensor circ na
X1
ns
2²
X2
x
3
circ( =
(18)
:
2s
P
X1
Tensor unfolding and folding:
unfold(x*) [x .... x E Rainsxng, X fold(unfold(
(19)
t-product X' Xonxxnxxss
X * y fold(cire(XX) unfold()))).
(20)
22

---

Definition 2 (Orthogonal Tensor) The tensor X' is orthogonal if X * * =x=xa x I, where T Xn x an identity tensor whose first frontal slice I =I 1m1,, and the other frontal slices are T==0, vk 2, 2.3.... n33-
Based basic tensor operations, tensor Singular Value Decomposition (t-SVD) defined follows.
Definition 3 (t-SVD) Given lensor X'E.. Xn2xns, then the 1-SVD of X is:
X = U * S S
(21)
where MER Xn1xx and VER xna are orthogonal tensors, SeRRmaxaxn is a f-diagonal lensor.
B Optimization Details
B.1 Update Formula of Zw..
Recall that the subproblem of Zw., can be formulated as follows:
arg min _
-a; s.t. Z 1m =1, 2,0
20, iEW
(22)
(v) da (PT.)i: the Lagrangian function of problem 22 can be written
Denote E
GG(") E >| -<(z
L(Z (.4) = -
00) 1m 1a-1) Z Es
(23)
where ( and 4,20 are the Lagrange multipliers. The optimal solution z should satisfy that the derivative of Eq. (23) w.E.t Z is equal to zero, we have
z 00
G (1) 1 E(")) 1 17
51m =0
(24)
We can rewrite in the element-wise form:
2
((e)
p()) - Fij =0
(25)
ij
Note that 2 =0 according to the KKT condition. Then, we have
(0)+
( - ( +5)
(26)
Each Z iew, then be solved and update Zes..
B.2 Optimization Algorithm of P and
After obtaining the reduced gradient Vh(aa,, we set the descent direction g g-(n, (2.,99...., with the following strategy:
Vh(a) u = u. Vh(a))
9v
es a. 0, u. 0, [h(a))) > 0,
(27)
where u typically set as the index of the largest entry of a. Its main procedures updating P and are summarized in Algorithm
23

---

Algorithm Gradient Descent-based Optimization Algorithm for Updating a and P
Input: P, (Z., To Lsp, Bx.
Output: Weight coefficient and fused graph P.
1: while not converge do
2:
Calculate the soft label distance matrix H by H | Vi Annssl2
3: Calculate the fused graph P by solving Eq.
4: Calculate the reduced gradient by Theorem and Eq.
5:
Calculate the descent gradient g by Eq. (27)
6:
Update weight coefficient 04++1 < ax + 0g with the step length
7: if max( 0gl S 10
then
8: Converge.
9: end if
10: end while
B.3 Update Formula of F
P and the normalized Laplacian matrix Lsp -I,- D
Recall that Sp=
Sp
P
Sp
Sp
D,
According to the definition of the degree matrix, Dsp can be written as Dsp-
where
D, is diagonal matrix whose diagonal elements are row sums P and AgRm is a diagonal matrix whose diagonal elements are column sums of P. i.e. A A Paj. Since P1m=1,,,
have = Therefore, the normalized Laplacian matrix can be written as the following blockwise form,
L.pp Intm Ds SppDs
I
-PA
(28)
-A-3PT
I
B.4 Update Formula of To
The subproblem T, in Eq. 17 can be solved by the following theorem.
Theorem Assume that Z mm in Eq. 17) has the singular value decomposition form Z. P-U, where U., E., mm The optimization in Eq. 17 has closed-form solution as follows,
T: U.V.
(29)
Proof. Taking the equation Z. P-U, V
rewrite the Eq. 17
Tr U.E V = Tr(V. U E.).
(30)
Considering E, T. U we have E. V
V. T
T.V. -I Therefore, we can obtain:
Tr(V T. TrEE
(31)
(2) is the i-th diagonal element of Σ.. To maximize the value of Eq. the solution should where oi be given T; U, V. thus achieving the maximum by satisfying the equality condition. m
B.5 Optimization Algorithm of AGF-TI
To solve the problem in Eq. 5 the algorithm of AGF-TI is summarized in Algorithm
C Proof of Theorem 1
Proof. To prove Theorem I we first give lemma:

---

Algorithm 2 Optimization Algorithm of AGF-TI
Input: Anchor-based bipartite graphs {Z,, ERR GRRl.l/m
Ju=1: one-hot label matrix Y, the number of categories c, the regularization parameters B, X, 3, and p.
Output: Labels of unlabeled samples Y..
1: For each te [V], initialize Zu. "e.==" - | 1m T.=I, a.=
2: Initialize auxiliary variable G-0 and Lagrange multipliers W-0, p=10-2, 7,= Mmna =10
3: Calculate P by Eq. (IT) with H is set to O.
4: Calculate F and Q according to Eq. 15)
5: while not converge do
6: For each ve[V] and iew,, update the i-th row of by solving Eq. 99)
7: Update weight coefficients a and fused graph P according to Algorithm
8: Update F and Q by Eq. (15).
9:
Update G by Theorem For each rE(V]. update Tr by solving Eq. 17
11: Update = (
Update penalty parameter 1) = min(7777) 7m
13: end while
14: return classification results: Y ( maxjele) Figli-et
Lemma
35 Given a function g(x, u) where x and u belongs to compact normed spaces and
U, respectively, Assume that 9(x,-) is differentiable over X, g(x, u) and @g(x.u) are continuous X xu, then the optimal value function Sup xEx g(x, u) differentiable and @h(us) 0gix' .uo) point us if 9(x, uo) has a unique maximizer x"
du
du
Then, we construct the following function:
g(P,a) EE TT (Z,T.))- PPI Lgp
(32)
where PEA m and A.. Based the property of the normalized Laplacian matrix, the relationship d3 holds. Evidently, g(P,-) diffcrentiable
Tr(F" Ls F Σ. Σ, V.../V,,-Q,, /
over AT, and g ax) and (g(P.a) are continuous. Recall that === Raxm kIm== 2
da is compact space. Considering the optimal value function of g(P, a w.rf. a, we have:
SuppeA g(P,a) = max PEA
Σ.
(
F),
(33)
which has the exact form ((xx) Eq. TOp. According to Lemma T the differentiable property of Eq. (33). i..., h(a),, given point a depends whether we find an unique maximizer P for the inner optimization problem maxpeдm g(P, a(,,a)), i.e., problem in Eq. I1]. Since the feasible region A is a closed convex set and g(P, a²) is strictly convex, the Hilbert projection theorem guarantees the uniqueness of the maximizer for any given Following that, we conclude that 1(())) is differentiable and its gradient can be calculated by:
@h(a) 22a, Tr(P" Z, T,),
(34)
Oa,
where p* is the optimal solution of the inner optimization problem.
D Theoretical Analysis
D.I Convergence Analysis
In essence, Algorithm 2 iterates between the following two steps until convergence:
(1) Inner Step: Solve the min-max optimization problem in Eq. (1O) for fixed variables ((Zu.: T.). F, W) using a reduced gradient descent method, iee, Algorithm | to update inner variables and P.
25

---

(2) Outer Step: Update other outer variables through ADMM with augmented Lagrangian function, Le., [((u., T. T F, a², P', W), where a' and P denote the optimal inner variables for given outer variables.
To prove the convergence of Algorithm 2 we proceed in three parts. First, we prove that the solution of the Inner Step obtained by Algorithm [ is the global optimum. Second, leveraging mild assumption on the coupling between inner and outer variables, we construct a Lyapunov function based on an idealized augmented Lagrangian show that the ADMM procedure of the Outer Step produces a bounded sequence {[ (( Algorithm converges to stationary Karush-Kuhn-Tuuce (KKT) point of the original max-min-max problem in Eq. (
Je F()), g(8) W( Third, we prove that
Part I: Proof of the global convergence of Algorithm | used in Inner Step
Theorem 4 h((x) in Eq. 10) is conver w.r.t. a.
Proof. For any on and a2 E 4V. and < < 1, the following form holds:
(177)a)))))22)
max XTr PEAT
Σ (T..TT)
[Z.
-(+ -
PPI Tr(F Lss
max XTr PEAT
(raj +(1 | (Z.T.)) -(7+1-7)0
SY max XTr PEAT
Σ, af p Z. (Z, +(1-7)
max XTr
Σ o (Z.TT)) -c)
PEAT
(35)
Eq. (35 validates that h(a:) satisfies the definition of convex function.
Algorithm [ conducts the reduced gradient descent on a continuously differentiable function h(ax). which is defined on the simplex faeR Σ 0,=1, 20, Ve). Hence, it converges to the minimum of f())). According Theorem we have the following corollary.
Corollary The solution of the Inner Step obtained by Algorithm is the global optimum.
Part II: Proof of the boundedness of the sequence generated by ADMM in Outer Step
Since the Outer Step is based on the fixed inner variables and P. the following mild assumption allows us to use the idealized augmented Lagrangian function, simplifying the convergence analysis.
Assumption (Sufficient Inner Optimization) At each iteration k, the inner optimization finds sufficiently accurate solution (k) and (k) such that for some e > 0:
(+) -x"12 + [p(k) -P* S
(36)
where (o*,P*) is the exact saddle point of the min-max problem in Eq. 10 given other variables.
According to Corollary and the convergence condition adopted in Algorithm 1 this assumption is satisfiable. Besides, due to the used f norm and Frobenius norm, the angannneeTagraangaan function in Eq. (6) is strongly-convex in a and strongly-concave in P. This favorable property ensures that the optimal solution of Inner Step is Lipschitz continuous w.ntt the outer variables. Therefore, in Outer Step, the augmented Lagrangian function in Eq. (6) at each iteration can be idealized as:
(k) T() J g(k), w(() ((Z(() we
- min max T(Z We
(A) T( J (a()) ar, P.P. PM)()) m((A)
LEAL PEA"
((Z 1( fo ((()) (1) p(t) g(b), wil)))
(37)
Following 22 we can prove the following theorem.
Theorem If Assumption holds, the sequence ((Z() by the ADMM procedure of the Outer Step is bounded.
(k) ((k)) jv F(A), g(t), W(() generated
26

---

infer Proof. We first introduce a lemma.
Lemma 2
15
Suppose that F
R R
I is defined as F(X)=f
f(X)=f(o1
(x)),,, where ( is the normal SVD of matrix
X=Rm*n, r= min(m, m. n), and f(.) : R + R be differentiable and absolutely symmetric at a (X).
Then the subdifferential of F(X) at X is
(
= U-Diag((((o(()))))
(38)
0X
where (x(o (X)
(x))
8X
8x
Then, we construct the following Lyapunov function to prove the outer sequence is bounded.
(iZ!e ( (v
for F g((),W)))
V Vn (1 TOU Jos 100) g(h)) m((k) - (1) T Jur
F00, g(M) +1
=Tr =Tr ( Lsp. F(() +Ir ((F(H) TBBBF v) - -xE, a Tr (P. (z (+))
1 (WW()) (MV())) 2 -(0)))(() -( g(()))
(39)
where || 1/ is a constant. To prove the multiplier sequence (Wik) Je= is bounded. We derive the first-order optimality condition of G in the updating rule:
[lgg(t1) e
w + ( g(++1)) W(++1)
(40)
Let S V be the SVD of tensor G. According to Lemma and Definition we have
|
3 Lu * ifft 3]. * VT
2 =
(41)
1 n min(m, V)
1,
n
n
which implies @||g(k+1)||e is bounded. From Fq. 440) we can infer that the sequence (W((k) also bounded.
)
Note that all subproblems of the ADMM procedure in the Outer Step. i.e. Eq. 9 Eq. 13
Eq.
16
and Eq. (I7). are closed, proper and convex, we can infer that
V (1Z (k+11, T(k+1) Ju p(*+1), ((*+),g((+))) w(()) S Vin (1Z(2) (k)
(
g(k), m1(k))
= -Vm ( T( p(k), g(k), w 1)) 7k + 17
(I - 111
(42)
2
By summing the right-hand side of Eq. (42 from k=1 to n, we have
Vn (1+1) T(*+1) J p(k+1), g(--1), wilk) < Vno (1Z)., we? Ti Fat g2, ,W
Th + 7 [ 1E.
(43)
2m
Since
Σi=1 m-+me-1 <o0, VTDO (1Z w.
T. Fl. Gl, W9) is finite, and the sequence (M(())
2n is bounded, derive Vn, (1Z (k+1) T
Jr F (1+11, g(++1), w(k) is bounded. Recall the specific Lyapunov function in Eq. (39, except 2 p T (Z( (k+ (l.p((((t) )
(++1)
) the other terms nonnegative. Considering the constrains P, Z. , T. TI (Vwe (V). ).
T P (z) (.(+1)
<00 holds. Thus, we deduce that each term of Eq. (39 is bounded.
The boundedness of [[](++)|| I| suggests that all singular values of (A++1) are bounded. Based on the following equation
min(m, V)
( (1(()+))
1
Σ
(S)())))
(44)
27

---

the boundedness of the sequence {g I== is ensured.
Since B and Y are constants, the boundedness of Tr( - - B Y)) guarantees
the boundedness of the sequence [F(++))
1k=1
Furthermore, according to the update formulas in Eq. (66 and Eq. 17p. could be deduced that
these sequences {Z === and (T(*) =k=1 are bounded.
Part III: Proof of the convergence of Algorithm
Leveraging Corollary D and Theorem theorem.
5 the convergence of Algorithm 2 is guaranteed by the following
Theorem If Assumption holds, Algorithmm Aigorithh@will will converge to a stationary point of the problem in Eg. 5
Proof. Assume the sequence {[Z (k) T
Jos p0(), a (k), p(k) g(k), W is generated by Al- Te F(), g(h), mV() ) is guaranteed by Theorem 3 Since the feasible region of variables P and are both closed convex sets, the boundedness of (ax(k) p(k) J==1 naturally holds.
gorithm 2 during iteration. The boundedness of {iZ!e
(8) (
According to the Weierstrass- Boaesstrass-Bolzano theorem (52), there is at least one accumulation point of the sequence { (A) T(()
Ju F(A), (M),,
w R=1 we assume the one of the points as
(IZ Too F a p gov Wo)), i.e.,
Ju:
lim (17 LV F(() a p((), g(k), m((k)) = (1 Jo F , a g W.))
k-000
(45)
From the update formula of W, we have
g(++1) g(k+1) (W((+1) -w(t))/))
(46)
Leveraging the boundedness of (Wi()) Jk=1 we can obtain 200 -
Due to the first-order optimality condition of (k), we can derive W e Considering the closed-form solutions in Eq. (26), Eq. 15 Eq. 17), and Theorem ((Z T Je F g00
satisfies their corresponding KKT conditions of the problem in Eq. 6). Furthermore, the global convergence property in Corollary ensures that a and p also satisfy the KKT condition.
Thus, the accumulation point (IZ T TX Foo 00 P p0p.g
W generated by Algorithm
satisfied the KKT condition of the problem in Eq. 5
D.2 Time and Space Complexity Analysis
Time complexity analysis. For the proposed AGF-TI, the main time complexity is focused on solving for the variables Zun, P, or, F. F.G. and T.. The update of the variables (Z requires O [wol(m²
In Algorithm I calculating H, P, g, and the optimal step for each up- date needs ((nm(c+1)) O(nm(m +1)). O(mm²V), and O(V4) computational complexity, respectively, where 4 is the number of iterations. Thus, the time complexity of updating P and is O(mm ((mmm2mVV (2m++++2) +c+2) each update needs fast Fourier transformation (FFT), inverse FFT, and t-SVD operations, corre- sponding to computational complexity of (nnvV log(nV) + nmnV2). The update of (T.) requires O(nm² +2m3) for each iteration. Assume tz iterations are required to achieve conver- gence. Considering m, V,t < the overall time complexity of the optimization phase is ((22(nmV log(nV) - nm? + nmV?)).
+ Vti). For F, it requires O(nm²) for each iteration. For tensor G.
Space complexity analysis. For our proposed method, the major memory costs are various anchored matrices and tensors. According to the optimization strategy in 3.2 the space complexity of AGF-TI is O(r(mV +c)).
28

---

E Experimental Details
E.1 Detailed Description of Datasets
ineluues 11,788 samples of 200 bird species. Following 4T], we select the first 10 bird species with 1,024-d deep visual features from GoogLeNett and 300-d text features using the doc2vec model.
Set--DigaCoontaieee 2,000 images for 0 to ten digit classes, and each class has 200 data points. Following [42]. we use 76-d Fourier coefficients of the character shapes, 216-d profile correlations, and 64-d Karhunen-Love coefficients as three views.
101-0 is an image dataset for object recognition tasks, which includes 2,386 images of 20 classes. For each image, six features are extracted: 48-d Gabor, 40-d Wavelet Moments, 254-d CENTRIST, 1,984-d histogram of oriented gradient (HOG), 512-d GIST, and 928-d -d local binary patterns (L.P)).
CunSceneeconsisee of 2,688 images belonging to 8 outdoor we use the same four features: 432-d Color, 512-d GIST, 256-d HOG, and 48-d LBP.
categories. Following 43
MNIST-USPS [44] comprises 5,000 samples distributed over 10 digits, and the 784-d MNIST image and the 256-d USPS image are used as two views.
Awa (Animal with Attributes) contains 50 animals of 30,475 images. Following 45 use the subset of 10,158 images from 50 classes with two types of 4096-d deep features extracted via DECAF and VGG19, respectively.
E.2 Detailed Description of Baselines
SLIM (Semi-supervised Multi-modal Learning with Incomplete Modalities) 46 simulta- neously trains classifiers for different views and learns consensus label matrix using the view-specific similarity matrices of existing instances.
AMSC (Absent Multi-view Semi-super Classification) 19 learns view-specific label matrices and a shared label probability matrix incorporating intra-view and extra-view similarity losses with the p-th root integration strategy.
Supervise classification) 7 is a GMvSSL method that simultaneously learns consensus label matrix and view-specific weights using label propagation and the Laplacian graphs of each view.
AMMSS (Adaptive MultiModel
AMGL (Autoweighted Multiple Graph Learning) 8 automatically learns a set of weights for all the view-specific graphs without additional parameters and then integrates them into a consensus graph.
MLAN (Multi-view Learning with Adaptive Neighbors) 9 for each view and learns the optimal local structure by modifying the view-specific similarity matrix during each iteration.
simultaneously assigns weights
AMUSE (Adaptive MUltiviww SEmi-supervised model) [0 learns a fused graph through the view-speeific graphs built previously and prior structure without particular distribution assumption on the view weights.
FMSSL (Fast Multi-view Semi-Supervised Learning) learns a fused graph by using the similarity matrices of each view constructed through an anchor-based strategy.
FMSEL (Flcxible Multi-view Sumi-supee Learning) [12 utilizes a linear penalty term to adaptively assign weights across views, effectively learning a sttucttuut fused graph.
CFMSC (adaptive Collaborative Fusion for Multi-view Semi-supervised Classification) [3 simultaneously integrates multiple feature projections and similarity matrices in collab- orative fusion scheme to enhance the discriminative of the learned projection subspace, facilitating label propagation the fused graph.
http:p//wwww...ssaaa.ctcccceeoas 2011/
Image Dataset/7007177
https://cvml.ista.acccom
29

---

AMINSS
AMGI
MLAN
AMUSE
FMMOOL
FMSEEL
GFMSC
MIAR
ERL-MVSG
AMSG
AGP-TT
03
30
40
70-
22
30
50-
10
Labeled Rate%s
Laboled Rets% (8) UCI-Rgg
Labeled Rutios
Labeled Ratiol
Labuled Rudoo (8) MINIS T-USPS
Labaled Rato
(A) CUB
[C] Caleecht11-00
(8) Custscom
(P) AMA
Figure 6: Fi-score results on six datasets with LAR in (1%, 2%, 4%, 8%) when VMR 60%.
MVAR (Multi-View semi-supervised classification via Adaptive Regression) 47 classifies the multi-view data by using the regression-based loss functions with &2,1 norm.
ERL-MVSC (Embedding Regularizer Learning scheme for Multi-View Semi-supervised Classification) 48 utilizes linear regression model to obtain view-specific embedding reg- ularizer and adaptively assigns view weights. The method then simultaneously learns a fused embedding regularizer by imposing 2 norm and shared label matrix for classification.
Parameter setting for baselines. For the above comparative methods, a grid search strategy is adopted to select the optimal parameters within the recommended range, and the recommended network structures are used as their baselines. Specifically, the structure of the deep neural network of DMF is set as Σn d,-10sc-c, and the parameters 3 and À are fixed to 0.01. For SLIM, we search 10-3 10--3 ..... ..., 10 10t), }. For AMSC, the parameter search the logarithm of parameter from 0.1 to 2 with 0.4 step length and the regularization parameter from to I with an interval of 0.2. In MIAN, we randomly initialize the parameter À for the Laplacian matrix rank constraint to a positive value in the [1, 30] interval. For AMUSE, the parameter À is tuned in {10- 10-30.3.. 3 10 10*). In FMSSL, the parameter
optimal parameters A and A2 in the range of (10-1,
p is fixed to 0.5 and y is searched from 1.1 3.1 with interval of 0.4. For AMMSS,
searched in { {10-1 ... 102).
and the number of anchors m is set to 1,024 when the sample size is larger than 10,000; m-256, otherwise. In FMSEL, the three parameters 21, X2, and & are tuned in {10 1,102 100 102 10%). For CFMSC, we search the three parameters A, B. y in { {10 10-2 ..... 10 10 For MVAR, we fix the parameter r to and the weight of unlabeled samples to 102. while searching for the weight of labeled samples / in (10", 10 10%}. j. In ERL-MVSC, the smoothing factor embedding parameter 3, regularization parameter y, and fitting coefficient are set to 2, 1, and 10, respectively.
Compute resources. All the experimental environments are implemented on desktop computer with two Intel Xcon Platinum 8488C CPUs and 256GB RAM, MATLAB 2022a (64-bit).
F
Additional Results
F.1
Main Results
In the manuscript, we evaluate the proposed method from the perspectives of view missing and label scarcity. For the view missing aspect, Tables 12 13 and [4d present the detailed results corresponding to Table 2 including standard deviations. In addition, we perform the Wilcoxon rank-sum test at 0.05 significance level assess statistical reliability. The Wilcoxon rank-sum test results also validate the robustness of the proposed method in tackling the incomplete multi-vien data.
We show the comparison results for the label scarcity setting in terms of F1 and Precision in Figs. 6 and 7 As observed, our method outperforms all other comparison methods over all metrics and all datasets, demonstrating the effectiveness of the proposed approach.
F.2 Ablation Study
We present the detailed ablation results corresponding to Table including standard deviations, on the six datasets in Tables 6 and 8 in terms of Accuracy, Fl and Precision, respectively.
30

---

AMISSS
ANGLL
MLAN
AMUSE
FMMSOL
FMSEE
GFMSC
MIAR
ERL-MVSG
AMAGG
AGP-TT
70-
20
70-
50-
50-
Labeled Rate%s (A) CUB
Laboled Reto% (8) UCI-Rggt
Labeled Rutifs
Labeled Ratiol (8) CusSsonm
Labuled Rudoo (8) MINIS T-USPP
Labaled Rato (P) AMA
[C] Caleecht11-00
Figure Precision results on six datasets with LAR (1%, 2%, 4%, 8%) when VMR is 60%.
Table 6: Accuracy ablation results (mean-tstd) of AGF-TI with different VMRs when LAR% 5%.
VMR=30%
CUB
UCI-Digit
Caltech101-20
OutScene
MNIST-USPS
AwA
Avg.
AGF-TI
78.33.13.46
95.98
81.88
7.83
95.09.10.79
69.12.1081
81.79
w/o
72.1245.00
88.48-4.79
42.7119.69
27.35=7.56
82.01=7.31
12.34
64.17
w/o
76.77:1.195
90.28:16.32
69.65 1.5.73
66.41
93.24.10.81
6..49
77.14
w/o TI
72.82.12.56
88.96.12.35
76.63.1.822
63.55
88.85110.70
59.23.11.22
75.01
VMR=70%
AGF-TI
74.25++4.44
95.16=1.13
1..17=1.63
64.52+2.94
95.62:+1.35
1.241+1.26
78.52
w/o
53.40:114.97
16.58
38.5413.65
30.6214.31
78.124111.9
70.58
54.47
w/o
67.67=11.96
76.47=4.77
55.07=2.07
58.10=2.84
94,40=1.00
42.08:00.89
65.63
w/o TI
63.16.11.58
83.04
47.55.13.62
34.69
84.07
49.04.11.40
60.26
F.3 Parameter Sensitivity Analysis
Sensitivity analysis of the regularized parameters 3x and We show the parameter sensitivity results the six datasets under F1, and Precision metrics in Figs. and results also support the statement discussed Section that /8 exhibits more sensitive effects on the classification performance compared to p. Besides, the proposed approach consistently achieves stable and competitive performance within narrow parameter range across diverse datasets, demonstrating the
Table 7: Fi-score ablation results (mean:tstd) of AGF-TI with different VMRs when LAR% is 5%.
VMR=30%
CUB
UCI-Digit
Caltech101-20
OutScene
MNIST-USPS
AwA
Avg.
AGF-TI
77.03:14.39
95.97=0.82
58.02:13.91
69.56=2.53
95.07=0.880
57.16::1.26
75,47
w/o T.
70.61+6.09
88.34=4.93
21.61
26.91=7.70
81.70:77.49
19.81=
58.16
w/o av
76.0511..29
90.35-1..09
48. 48.13.123.59
66.5911.42
93. 93.14.10.85
56.0911.34
71.72
w/o
71.8743.00
88.94-2.36
50.56:12.65
63.06=1.57
88.76:00.70
50.5911..3
68.96
VMR=70%
AGF-TI
72.30-15.62
9.113=1116
39.72:13.71
63,29-3,01
95.59
17.59
70.60
w/o T.
51.19=15.30
14.56=116.0
8.29=1187
27.11=4.35
77.61
1..83:11.37
45.93
w/o au w/o
64.84.13370
75.59
22.86.12.17
57.91
94.33.11.14
35.02.11.04
58.
61.97-11.72
82.97-2.55
17.9013.62
33.02-2.56
83,9211.23
46.9911.31
54.46
Table 8: Precision ablation results (sean of AGF-TI with different VMRs when LAR% is 5%.
VMR=30%
CUB
UCI-Digit
Caltech101-20
OutScene
MNIST-USPS
AwA
Avg.
AGF-TI
78.33-23.46
95,98=0.82
56.75:13.33
69,99-2.21
95.09-10.99
58.47=0.889
75.77
w/o T.
72.12+5.00
88.48-4.79
21.35+8.43
27.41=7.85
82.01+7.31
61.43+1444
58.80
w/o a.
76.77-11.95
90.28-16.2
48.15:124.28
67.1111.41
93.24.10.81
57.90.11.22
72.24
w/o
72.82422.56
88.96-2.33
50.07-2.36
63.11=1.30
88.85-10.70
50.38:1144
69.03
VMR=70%
AGF-TI
7.42
95.16
+3.04
63.83 2,89
95.62 1.35
58.82 1.33
70.75
w/o T.
53.40+14..97
15.58:116.99
8.86+1162
29.05=4.47
18.124411.94
59.05::1.10
47.34
w/o o.,
67.67+11.96
76.47+4.77
20.97+1.97
58.09-2.79
94.40-+1..99
36.54+0.84
59.02
w/o II
63.16:11.58
83.04-2.52
15.16.33005
32.37+2.14
84.07-11.22
42.61:1.34
53.52
31

---

generalizability of our method. This observation provides useful guideline for parameter selection, enhancing its practicality in real-world applications.
IN
(Si 10130
30 Curfoom
(B) VARTURre
Figure 8: Parameter sensitivity analysis of 3x and p in of F1.
Preddon
Figure 9: Parameter sensitivity analysis of 8x and p in terms of Precision.
Sensitivity analysis of the anchor number To examine the influence of m, vary its value in the range of [2t 25, 26 20,2",277 2,,25) for CUB, and {2 2 2$ 20,210) for the other five datasets. The results for Accuracy, F1, and Precision metrics consistent trend where performance initially improves with increasing m. followed by a gradual decline, which aligns with the behavior of Bx- For lower m,, the small anchor-based bipartite graphs in each view are too coarse to capture the fine-grained geometric structures of the existing data points, limiting classification performance. Conversely, large m leads sparser graph connectivity due to fixed number of neighbors k, resulting in unstable information propagation. These results suggest that an appropriately selected m balances local structural preservation with adequate graph connectivity.
shown in Figs. TO 11 T2 We observe
Seat
Figure 10: ACC results of AGF-TI with different anchor numbers.
Figure 11: Fl results of AGF-TI with different anchor numbers.
Archor
302h- Rumber
Mecher Number
AntherPurbe Number
Kumber
100
Figure 12: Precision results of AGF-TI with different anchor numbers.
32

---

Sensitivity analysis of the trade-off parameter A. We investigate the impact of X by tuning its value within the set (V2/3, V2 V2 2V2 2V2,3V?). where the number of views. The results for Accuracy, F1 and Precision on the six datasets are displayed in Figs. 13 14 and 13 respectively. It evident that the trade-off parameter substantially affects performance, highlighting the crucial role of the graph fusion term, i.e., AGF. In addition, we find that our method consistently achieves the highest performance across all datasets under different metrics when À is V2, Based on this finding, we simply fix À to V in our experiments.
05
V9
20
Figure 13: Parameter sensitivity analysis of in terms of Accuracy.
Xicariho0003 09-20
Figure 14: Parameter sensitivity analysis of À in terms of F1.
(N XiCassdr1on3)
Figure 15: Parameter sensitivity analysis of À in terms of Precision.
F.4 Empirical Analysis of Sufficient Inner Optimization Assumption
In Appendix D.I
prove that the variable sequence obtained by Algorithm 2 converges stationary point based on the assumption [ i.e., Sufficient Inner Optimization. To validate the reasonable of this assumption, we track the error of and P between iteration steps, i.e., (k+1)
a (A1 and [[(+++] p()) during Algorithm to approximate the error between numerical and exact solutions in Eq. (36). The results for all datasets with VMR 50% and LAR 5% are shown Table g One could observe that the error of both a and P can be rapidly decreased to small number (e.g., le-5), suggesting the reasonableness of the sufficient optimization assumption.
F.5 Computational and Memory Footprint of the Tensor Nuclear-Norm Step
As analyzed in Appendix D.2 the complexity of the optimization phase is primarily determined by the tensor nuclear-norn (TNN) step for solving the G-subproblem. To further validate the scalability of the proposed AGF-TI we utilize the YTF50 dataset (-126k samples) [53] with four views and empirically analyze the computational and memory footprint of the TNN step as the number of views, anchors, and samples scale simultaneously. The results under VMR 50% and LAR 5% are shown in Table TO] As is evident, the empirical trend aligns well with our theoretical analysis. Due to the anchor strategy, the running time of the TNN step does not increase dramatically with the sample size and remains within an accepttale burden. This indicates AGF-TI is capable of practical use.
33

---

Table 9: Iterative error of a and P during Algorithm under VMR=50% and LAR-5%. ('" means converged.)
Step
1
2
3
4
5
6
CUB
ar error
1.02e-5
P error
2.83e-10
error 2.14e-6
4.34e-7
5.17c-8
UCI-Digit
P error 6.82e-5
1.31e-12
2.02c-17
enror 1.36e-4 0.87
6.71e-5
3.26e-5
1.53e-5
7.61e-6
3.22e-6
1oltech101-20
error
7.31e-9
7.56e-15
1.62e-20
1.34e-25
3.12e-30
OutScene
a error P error 1.88e-5
1.41e-4
4.67c-5
1.58c-5
5.20c-6
1.73c-6
1.8e-10
3.48e-15
8.06e-20
2.02e-24
6.33e-6
MNIST-USPS
error 1.4le-4
error 0.016
8.52e-8
1.22e-7
8.52e-8
1.22e-7
8.52e-8
AwA
P error
0.004
3.84e-4
4.50e-5
7.46e-6
1.10e-6
3.31e-7
Table 10: Computational and memory footprint of the tensor nuclear-now norm step with varying numbers of views, anchors, and samples under VMR=50% and LAR=5%.
Time (s)
Memory (GiB)
#Anchor
128
256
512
1,024
128
256
512
1,024
#Sample=10k
Viee
2
0.35
0.42
0.76
1.17
0.095
0.13
0.38
0.76
0.35
0.49
0.88
1.57
0.14
0.29
0.57
1.15
4
0.48
0.67
1.05
1.95
0.19
0.38
0.76
1.53
#Sample=70k
VView
1.93
3.02
4.56
8.89
1.17
2.18
4.28
8,56
3
2.11
3.67
6.38
14.74
1.61
3.21
6.42
12.87
2.98
4.91
9.06
23.10
2.14
4.28
8.56
17.13
#Sample=126k
VVeew
2
1.73
5.75
9.40
19.30
1.93
3.89
7.71
15.48
2.33
7.28
16.23
33.00
3.02
6.03
11.59
23.26
4
3.07
9.20
21.29
60.44
3.85
7.71
15.42
30.86
34

---

F.6 Additional Comparisons to Deep Learning Baselines
We conduct additional experiments with two deep learning-bases multi-view sumi-supervised meth- ods, i.e., IMvGCN [54] and GEGCN [55], with VMR=50% and LAR=5%. We adopt the recom- mended learning rate and network structures as their baselines. Table IT shows that our AGF-TI also outperforms them in almost all cases, further demonstrating the effectiveness of AGF-TI.
Table 11: Comparisons of the two deep learning baselines under VMR=50% and LAR=5%.
Method
CUB
UCI-Digit
Caltech101-20
OutScene
MNIST-USPS
AwA
Accuracy
IMvGCN GEGCN
63.3
82.7
68.6
53.7
79.9
65.8
49.9
90.1
67.1
55.3
91.3
58.8
Ours
80.2
95.2
80.0
69.2
95.6
70.6
Precision
IMvGCN
67.7
83.5
51.2
53.7
81.1
60.8
GEGCN
56.5
90.6
34.0
56.2
91.4
57.0
Ours
80.2
95.2
53.1
68.3
95.6
60.0
F1-score
IMvGCN
61.2
82.5
40.2
52.4
79.5
58.6
GEGCN
50.5
90.1
31.6
55.5
91.3
50.4
Ours
79.1
95.3
54.8
67.7
95.6
58.6
35

---

60.9215 00
60.04 89
60.6811.53
60.9013.55
62.6812.99 200
60.81 13.
53.13:117.35
41.1517.44 ..40
59.73122.44
63.08
63.42-12.733
72.3015.62
75.87+22.95
85.22+00.72 020
65.09+1.99
81.81
77.7242.54
83.40 14.4
83.3311.144
67.7813.55 12599 e
79.2011 .44.
87.55
72.43 11.66
15.1
F1
65
Table 12: Comparison results (mean std) of compared methods on CUB and UCI-Digit under different VMRs when fix LAR to 5%.
VMR=70%
PREC
62.741:4002
61.75
62.25 3.99
62.7742.449 49
$4.4912.344
62.3511.112
54.23+11823
50.18 5.07
59.9812.33
64.3912.15
64.04.12.65
74.2514.44
4,44
76.0413.004
85.32+0.70
62.43+2.14
81.91
78.00+2.42
83.53#1.41
83.4711.16
68.0113.56
79.20 1.44
$7.59+11. 1.01
72.55 1..5411.24-
1.24
1.13
66/0/0
test with confidence level 0.05..
74.25
62.7444.03
61.75 61.75::2.12
62.25-33999
62.77 49
44.62::28.36 64.49
62.35::3112
55.23-115226
50.1815.005
39.98122333
64.3912.15
64.0441..65
74.2514.44
76.04
85. 85.32=0..77 70
02.43+2.144
81 $1.91
78 78.00+2..22 22.422
83 33.53+1.41
83.4711.16
68.01
79.20 1..4.43
17.5911.00 1101
72.55 1.21 1.24
1.13
ACC
68.01
sun sum f//o denote that the proposed method performs significantly better/tie//worse than the basclines by the one-sided Wilcoxon rank
44.25
63.32:14.91
66.0413.711
61.3524.669 4.69
66. 06.1611.333
68. 08.0244.211
60.20-1:115
61.3413.35
65.9514.522
63.4813.36
79.1015.49
87.2410.72-
89 89.65=0.86
76.31-4..68
87. 87.1321.32
86. 86.6220.76
88.9 08.95
38.0821.199
73.11 1.85
84.5911 1.29
1910.11
81.42.1 1.00
95.2512.94
F1
UCI-Digit
CUB
VMR-50%
PREC
48 48.44166667
65.77:4.115
67.75+2.29
63 63.21+3385 63.21#3 3.5
45.72::30122
67 67.28:22.86
69. 1913370
61,51113.09
61.5111.26
67.6013.47
65. 62.12.22622
80.2311.777
87. 87.30=0..72
89 39.73+0.882
74.82+5,776 5,7
87.21=1.300
86. 76.7240.744
88 99
88. 14.11411188
73.4611.88
1.621822
84.61+1.31
91.21=00500
81.4111.04 111:11.04
95.2311..99
66/0/0
73.46
48 18.44116.67
65. 05.77:4.155
67.75:22.29
18.72::25.57
67.28 67.28:2286
69. 19113.3770
61.5113.09
61.5113.26
67.6013.47
65. 12.1..22
80.2314.77
37.30=0.722
09.73=0.82
74.82+5.76.
87.21
86. 06.7220..74
88. 99.99.9999
88. 14.1141.18
73. 73.4621.88 1.82
84.61
$1.2110.50
$1.4111.000 1.04
95.2312.99
ACC
89.73
64.57 .6
64.64 44
68.3514.11-
61.8414.60 60
62.31 5.92
6921133116
70.81
65.1913.23
63.881186
66. 66.1414.69
67.36.3..34
4.39
91.43--0.9
90.93+1.00 07
46 86.2713.50
90.50=1.188
91.07.0.733
92.1621.41
91:4411
804341
1.23 86.7811.22
14.24
93.86-10.601
84.8010.855-
90.97
F1
65.48
VMR=30%
PREC
07.33#1..82 1.82
67.1112.79 67.1112.79
70.0911330
64.2613.68 64.2613.68
3.68
65. 65.30:4.09 65.3014.09.
70.5412.73 70.5412.73
71.65 71.6511.16.
66.4912.904 06.4912.90-
65.75.1..86
68.75 68.75213.22 68.75113.22
68.3012.98 68.30.1..98
78.331.3.46
91. 91.4640.96
91.066..98
86 16.13334466
90.611111
91.1210..00
12.1211.42
91.4811.37
803111166
1.23 86.0111.23
23.8710.60
84.9310.79
15.8
66/0/0
1.33111..22
70.0911.30
65. 65.7511.866
3..33
91 91.4611.966
91.06
86.131346
90.011111
01.1220.70
12.1911.42
91.481157
80.5111,69
93.8710.600
10.799
15.9810.82
ACC
86.80
91
Win The Looss
ERL-MVSC
ERL-MVSC
Method
AMMSS
AMGL
MLAN
AMUSE
PMSSL
FMSEL
CEMSC
MVAR
AMSC
SLIM
Ours
AMMSS
AMGL
MLAN
AMUSE
FMSSL
FMSEL
CHMSS
MVAR
AMSC
SLIM
Ours
36

---

20.57 78
22.56
26.95112.01
21.40 1.89
15.11
31.491 120
33. $3.901 1.95
25. 15.17
33 33.6011.56-
26.52:11.50 1.50
37.39-22.10
39.7213.71
40.56-12.67
46.69
42,00 42.04
47.54
38.90+22445
19.1411.41
15.67
49.73 1.83
42.72+1.41
50.9311.111
52.22--0.71
63.2943.01
F1
Table 13: Comparison results (mean of compared methods on Caltech101- 20 and OutSeene under different VMRs when fix LAR to 5%.
38
49,
/ denote that the proposed methed perfoorns significantly better/tied/worse than the baselines by the one sided Wilcoxon rank sum test with confidence level 0.05.
VMR=70%
PREC
20.39
21.51++1.500
25.42.1..88 1.84
20.55
15.71=11.16
19.27-10.98
31.11
24.57+0..95
34.95 O
26.02.10.95-
34.86+11.62
36.811:3.04
42. 12.16422.03
47.88 17.44-2.92
42.90
48.43.12.20
40.95++2.02.
19.5411.73
8
46.891 19..9.22.49 2.49
51.16+1.55
43.07
8
51.93 0..6
$2.534-0.66 60
63.83+22.9
64/2/0
29.
49.
60.87+1.66 1.68
61.60 1.0011.22
1.22
1.94 63.601 1.27
60.4210.766
57.01=1.556
66.92.10.22
67.76-10.68
63.92+00.9.
55.47
64.79.10.66
65.17=0.90
72.1711.63
44.43-12.04
48.33-2..66
43.57
2.022
42.76
50.34+1.66
4..5
51.61
42.58
52.65 1..44
52.69
64.52+2.94
ACC
26.30+2.008
24.06+2.01
36.07
23.9542.72
9.8747.14
41.09.1.644
43.1542..03
40.08+2.45
42.2311.99 1.96
23.5311.500
48.48
54.7913.699
49.54.1..51
55.54+2.54
56.81
54.80.11.55
46.61+4.199
36.46:2.005-
57.96.11.76
57.36.1..566
49.04
58.8411.033
1.03
60.67 1.27
67.70
F1
56
Caltech101-24
VMR=5005
22. 1.650
33.26111.27
22.27
10.51
37.7711.33
39.55-11.58
36.78122.11
44.691-2.03
24.34110.92
44.50+1.55
53.1013.41
OutScene
51. 11.1611..26
55. 5.584-2.22 22
57. 57.00
.
54.64 24.64
47. 17.52-33.85
56. 36.1742.57
58. 1511.566
58. 38.39111.58-
49. 19.3411..24
59. 99.54:1..89 0..9
50.98-11..22 23
48.29
PREC
25. 15.59+11.55
66/0/0
144 44.66
6 .02
62.18+1.733
69. 92.66.7...6
1.685+1.61 .
43.20=14.95
72.6310.67
73.46.10.74
72.08+00.7-
65.04 65.04
.
* 73.02++0.70
79.99 1969
53.47+1.15
1.93 .93
57.11 .
55.1712.14
48.7343.244
56.4242.36 .
58.84
58.4211466
48.91
60.33
61.65 .20
29.24
ACC
62.6911.22
56.19
43.
.
63.85+1442 .
66. 16.1011.144 .
.
32.19++2..3
24.97+2.011
2..49.22.24.
24.22 1.34
30.36+14.53
51,591 .89
53.70+-2.03
53.60+2.44 244
46 46.96112.644
22.6011..19.
52.94+2.007
58.02:13.91
58.32+-448
57.04
50.16-1100 10.20
66.8411..3 1.59
62.344 .60
53.27
62.48 1.97
65. 25.16:22.22.
23.563
53.77+
F1
50.
VMR=30% 30%
PREC
30.41 1.70
22.88 12.88+1.40 1.40
1.52
21.87
30.60 00.60=4.522
47. 1.641.64-
49.3 19.39-1.66
49,2 (9.27-22.09)
19.78122.99 99
0..73:1.112 22.51::3.19
48.71+1.788
56.751...3
55.6811..2
57.71+2.58
63.8 03.8911.23
56. 16.1111.92
51.34-8.25
66.03:11002
17.07
1.85-11446
53 33.6010.96
.
63.95 1.33
65 65.92+1.644 1.64
69. 2.21
66/0/0
62.
ACC
69. 099.0==0.99
61.85=11122
74.07
60.52+1.22
69.28
77.8610.69.
88.67
78. 8181 118110.9
65.81=1.62
13.
81 1.226 1.6
58. 11.111
28.28-22.22
64.08
56.8011.93
02.88=8.00
66. 10
11.41
62.91 11336
52. 02.977
65.15
66. 66.75 1.8
70.34:1.86 186
Win /ie e/Loss 0
ERL-MVSC
ERL-MVSC
Method
AMMSS
AMGL
MLAN
AMUSE
FMSSL
FMSEL
CFMSC
MVAR
AMSC
SLIM
Ours
AMMSS AMGL
MLAN
AMUSE
FMSSL
FMSEL
CFMSC
MVAR
AMSC
SLIM
Ours
37

---

81 $1.291111 1.77
85.61 1:.3220332
75.92 3..4444-
12.9910..82
71.76=22.26
80.22
84.45=00.91
66.08133.88
02.41=0.52
83.6011.11
78.93
95.591:1.37
47-22+0.87
39.4711.66
48.37
33.511:3.19
41.57
51.72+1.60
53.31
54.92
44.54110.63
42.52
48.81
37.59+11.38
F1
Table 14: Comparison results (mean:istd)) compared methods on MNIST-USPS and AwA under different VMRs when fix LAR to 5%.
//f denote that the proposed method performs ficuntly better/iie// tiee///i///// wurse than the buselines by the one- sidesddeed Wilcoxon runk rank-sum summ test with confidencce level 0.05.
-MR=70%
PREC
81.43+1.1
1.79
85.75
75.72+4.111
16
83.16.0.777
72.58+2.12
80.3910.911
84.57-10.33
66.0413.82
.
62.4610.494 82.46+0.49
83.8311.06
79.17
95.62=1.35
46.33+:0744
40.09
41.82+0.54
34.3113.188
42.86.0.744
50.71
12.111.0.66
52.3542.37
.
45.32-10.55
43.29
49.3510.35
58.82+1.33 1.33
66/0/0
52.41
ACC
$1.43+1.775
85.7511.322-
75. 75.7244.166
13.167
72.58+2.12
80.3910.91
84.571-0.93-
66 66.0413..22
83.8311.066
79.17
95.62
55.21
47.45
47.72+-0.66
40.5113.69.
52.2440.63
57.0312.28
59.85100.66
59. 59.15112455
49.11
$2.94=:0522-
50.01
69.41+1.26
82
59
84.80+1.22 12
89.53110.75
84.88
88. 8.39910.797 38.2811 88.8810.88 10..00
79.15
85.32100.98
88.97
71.4012.00
85.85 85.85+0.966
83.0013.553
81.17
95.58110.22 95.55=0.84
49.73+1.03
44.07 11.13
52.78++0.70
38.7911.191
50.20
53.53
54.71-00.11 55.69
58.23.10788
47.63
44.83
51.28.1.766 50.73
1.67
F1
71.40
MNIST-USP?
VMR=50%
PREC
84. 93+
89.6 69.61111 1.0.44
84. 14.9611-998
79.72 79.72+1.76
85 1..40.0.77
89. 09.05+00.51
71.73
85. 05.90+:0.96 0.96
83.161
81.33+.0.77
49. 19.166+0.70
44 14.22.10.92
49. 19.44--0.73
39. 1..29
50.3 50.30
52.04+0..71
56.9 06.99100611
48.4311082
45.7 45.76110.68
1.57
65/1/0
AwA
ACC
84.93
17
89.61 10.11
84.96 .98
88.39100779
79. 79.7241.76
85. 25.4010.977
89 19.055::5511
71,73
85 25.90#0.96
83.16113.66
81.3310.77
95.5810.82
58.52+0.54
52.94
56.7410.80
46.5011.588
1.58
59.3440.39
58.3311.64
62.23
63.5740.61
12.12.10.89
55. 55.90+0..33
61. 61.40
10.58+11.15
o
0
00.13100.1
93.50110.11
88.24+1.223
93.35.10.38.
86. 86.17:00.6-
90.4710.60
93.53+0.52.
79.54
88.00:10.62
88.56:10.11
84.4511.06
95.07110.80
5386+1.144
59.2510.99
44.0611.622
55.32
60.0910.56
60.92-0.11
62.9111.08
50.5110.844
54.06 0.-66.6..
53.5010.80
1..26
F1
48.78
VMR=30%
PREC
90. 00.20=
03.54110.60
88. 08.29=11118
93.38 0.37
86.36 06.36=0.43
10.59 90.51110.59
03.55=0.52 93.55=0.522
79.65=2.11
08.02=0.62
38.72.0.87
84.61=1.022
95.09110779
53.67=0.88.
49.25
58.22+0.77
O
1..02=1.75
55. 55.79=0.68
58.86110544
59.88 0055
61.87=0.85
0
51.33
04.74=0.57
0..24
57/3/6 58 08.47+0.880 89
88
00.20==0.61-
73.54110.60
88 29:
93.3810..3 10.37
86. 16.36=00.43
79. 9..5122.10
88. 0..622-0.22
88. 05.72.10.877
1.002
95.0910..79
63. (3.48=0.81
57.7410.85
66.49 06.49-00.55
1..44+1.933
65. 55.60-00.44
65. 15.32-00.63
17.24-10.47
69.07 0
0..02
64.34 34
64.78 14.78.10366
00.11
ACC
Win T/ie e/Loss
ERL-MVSC
ERL-MVSC
Method
AMMSS
AMGL
MLAN
AMUSE
FMSSL
FMSEL
CFMSC
MVAR
AMSC
SLIM
Ours
AMMSS
AMGL
MLANN
AMUSE
FMSSL
FMSEL
CFMSC
MVAR
AMSC
SLIM
Ours
38


Reviews:
[{"review_id": "XUmO9hffkw", "reviewer": "Reviewer_wnph", "summary": "This paper introduces a novel method, Adversarial Graph Fusion with Tensorial Imputation (AGF-TI), to address the Sub-Cluster Problem (SCP) in graph-based multi-view semi-supervised learning (GMvSSL) where missing data can distort local structures and degrade classification performance. AGF-TI utilizes an adversarial min-max framework to learn a robust consensus graph that is resilient to these distortions. Simultaneously, it recovers missing structural information by stacking similarity graphs into a tensor and leveraging high-order correlations for imputation. The authors conduct extensive experiments to validate the effectiveness of AGF-TI in handling scenarios with both missing views and limited labeled data.", "questions": "1.The central formulation is a max-min-max problem. Could the authors elaborate on the intuition behind this structure? Specifically, how does the outer max over the view weights $\\alpha_v$ in conjunction with the inner min over the fused graph $P$ lead to a more robust consensus graph than a simpler min-min formulation, as seen in methods like CFMSC?\n\n2.Please clarify how the permutation matrix $T_v$ is initialized.", "limitations": "Yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1.The paper is well-written, and the motivations are clearly described. The identification and formalization of a previously unaddressed challenge in this field, SCP, provide a strong motivation and a clear target for the proposed method.\n\n2.The AGF-TI model is the first to adopt a min-max framework for graph fusion, which is a novel contribution to GMvSSL. Incorporates an anchor-based strategy to reduce the computational cost, making AGF-TI scalable to larger datasets.\n\n3.I enjoyed the significance test in Tables 5, 6, and 7 in Appendix. The provided code and sufficient information make it easy to reproduce the experimental results.\n\nWeaknesses:\n\n1.The AGF-TI model is inherently complex, involving a tri-level max-min-max optimization problem.\n\n2.The theoretical convergence relies on the sufficient optimization assumption.", "was_revised": true, "final_justification": "Thanks for the authors’ response. My concerns have been adequately addressed, and I decide to maintain my original score.", "rebuttal": "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.\n\n---\n\n### **W1/Q1: Max-Min-Max Framework and Its Benefits**\n\nThanks. 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.\n\nIn 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.\n\nIn 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.\n\n---\n\n\n\n### **W2: Sufficient Optimization Assumption**\n\nThanks. 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.\n\n\n| k                           | 1        | 2        | 3        | 4        | 5        | 6         |\n| --------------------------- | -------- | -------- | -------- | -------- | -------- | --------- |\n| **CUB**                     |          |          |          |          |          |           |\n| $\\boldsymbol{\\alpha}$ error | 1.02E-05 | \\-       | \\-       | \\-       | \\-       | \\-        |\n| $\\mathbf{P}$ error          | 2.83E-10 | \\-       | \\-       | \\-       | \\-       | \\-        |\n| **UCI-Digit**               |          |          |          |          |          |           |\n| $\\boldsymbol{\\alpha}$ error | 2.14E-06 | 4.34E-07 | 5.17E-08 | \\-       | \\-       | \\-        |\n| $\\mathbf{P}$ error          | 6.82E-05 | 1.31E-12 | 2.02E-17 | \\-       | \\-       | \\-        |\n| **Caltech101-20**           |          |          |          |          |          |           |\n| $\\boldsymbol{\\alpha}$ error | 1.36E-04 | 6.71E-05 | 3.26E-05 | 1.53E-05 | 7.61E-06 | 3.22E-06  |\n| $\\mathbf{P}$ error          | 0.87     | 7.31E-09 | 7.56E-15 | 1.62E-20 | 1.34E-25 | 3.12e -30 |\n| **OutScene**                |          |          |          |          |          |           |\n| $\\boldsymbol{\\alpha}$ error | 1.41E-04 | 4.67E-05 | 1.58E-05 | 5.20E-06 | 1.73E-06 | \\-        |\n| $\\mathbf{P}$ error          | 1.88E-05 | 1.80E-10 | 3.48E-15 | 8.06E-20 | 2.02E-24 | \\-        |\n| **MNIST-USPS**              |          |          |          |          |          |           |\n| $\\boldsymbol{\\alpha}$ error | 6.33E-06 | \\-       | \\-       | \\-       | \\-       | \\-        |\n| $\\mathbf{P}$ error          | 1.41E-04 | \\-       | \\-       | \\-       | \\-       | \\-        |\n| **AwA**                     |          |          |          |          |          |           |\n| $\\boldsymbol{\\alpha}$ error | 0.016    | 8.52E-08 | 1.22E-07 | 8.52E-08 | 1.22E-07 | 8.52E-08  |\n| $\\mathbf{P}$ error          | 0.004    | 3.84E-04 | 4.50E-05 | 7.46E-06 | 1.10E-06 | 3.31E-07  |\n\nNote: '-' means converged.\n\n---\n\n### **Q2: Initialization of $\\mathbf{T}_v$**\n\nThanks. In our experiments, the permutation matrix $\\mathbf{T}_v$ is initialized as an identity matrix. We will include this detail in the revised paper."}, {"review_id": "GcL2btEl3X", "reviewer": "Reviewer_MMoo", "summary": "The paper proposes Adversarial Graph Fusion with Tensorial Imputation (AGF-TI) for Incomplete Multi-view Semi-supervised Learning. This paper addresses a newly identified issue termed Sub-Cluster Problem (SCP). SCP arises when missing views in multi-view data disrupt the smoothness assumption of label propagation by creating disconnected sub-clusters. AGF-TI addresses SCP through i) an adversarial graph fusion scheme leveraging a min-max optimization framework to fuse view-specific graphs and 2) a tensorial imputation mechanism that recovers missing similarity information using high-order correlations. It also employ acceleration strategy to reduce the computational cost.", "questions": "If the authors could address the above weaknesses during rebuttal, I would like to keep my rating.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n\n1. Extensive experiments on six public datasets under various missing-view and label-scarcity settings show that AGF-TI consistently outperforms state-of-the-art methods in terms of accuracy, precision, and F1-score.\n2. The paper offers a comprehensive optimization algorithm combining reduced gradient descent and ADMM, supported by theoretical convergence analysis.\n\nWeaknesses\n\n1. The experimental protocol creates missing views by randomly selecting V% of the samples and randomly deleting 1 to V – 1 of their views. In Figure 1, however, the Sub-Cluster Problem is illustrated as arising when whole groups of neighboring samples are missing, leaving a \"vacuum region\" that fractures a class manifold into multiple sub-clusters and breaks the smoothness assumption of label propagation. If missing views are indeed independent and spatially random, contiguous voids, thus SCP should occur far less frequently. Could the authors quantify how often SCP actually appears under the current random-missing setup, or ii) provide experiments with more realistic, block-missing / structured-missing patterns to show that AGF-TI remains effective in scenarios where SCP is genuinely likely to occur?\n2. What is the computational and memory footprint of the tensor nuclear-norm step when the number of views, anchors, and samples all scale simultaneously? Specifically, can the authors validate them on a truly large-scale (>100 k samples) benchmark?\n3. The related-work section should discuss recent missing-view representation methods, such as \"COMPLETER: Incomplete Multi-view Clustering via Contrastive Prediction\" and \"Decoupled Contrastive Multi-view Clustering with High-order Random Walks.\"\n4. It would be better to give the full name of TNN in Definition 1.\n5. The evaluation lacks deep learning baselines. Benchmarking only against classical or shallow methods limits the paper’s relevance to modern multi-view semi-supervised learning. \n6. In Definition 1, please spell out TNN’s full name Tensor Nuclear Norm.", "was_revised": true, "final_justification": "The responses have addressed my concerns, I will increase my score.", "rebuttal": "Thank you for your acknowledgment and insightful comments! Your feedback is extremely helpful, and we are committed to addressing each question you have raised.\n\n---\n\n### **W1: Show SCP Appears under Current Random-missing Setup**\n\nThanks. 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\".\n\nFor 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.\n\n| Dataset       | #Classes | 0%   | 30%  | 50%  | 70%  |\n| ------------- | :-------: | ---- | ---- | ---- | ---- |\n| CUB           | 10       | 9    | 9    | 13   | 17   |\n| UCI-Digit     | 10       | 10   | 11   | 18   | 32   |\n| Caltech101-20 | 20       | 20   | 25   | 31   | 47   |\n| OutScene      | 8        | 8    | 7    | 10   | 18   |\n| MNIST-USPS    | 10       | 10   | 11   | 13   | 27   |\n| AwA           | 50       | 43   | 46   | 62   | 126  |\n\n\n\n---\n\n\n\n### **W2: Computational and Memory Footprint of the Tensor Nuclear-Norm Step**\n\nThanks. 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.\n\n- **#Sample = 10k**\n\n| Time (s)       |      |      |      |       | Memory (GiB)   |       |      |      |       |\n| :--------------: | ---- | ---- | ---- | ----- | :--------------: | ----- | ---- | ---- | ----- |\n| **#view\\\\#anchor** | **128**  | **256**  | **512**  | **1,024** | **#view\\\\#anchor** | **128**   | **256**  | **512**  | **1,024** |\n| 2              | 0.35 | 0.42 | 0.76 | 1.17  | 2              | 0.095 | 0.13 | 0.38 | 0.76  |\n| 3              | 0.35 | 0.49 | 0.88 | 1.57  | 3              | 0.14  | 0.29 | 0.57 | 1.15  |\n| 4              | 0.48 | 0.67 | 1.05 | 1.95  | 4              | 0.19  | 0.38 | 0.76 | 1.53  |\n\n- **#Sample = 70k**\n\n| Time (s)       |      |      |      |       | Memory (GiB)   |      |      |      |       |\n| :--------------: | ---- | ---- | ---- | ----- | :--------------: | ----- | ---- | ---- | ----- |\n| **#view\\\\#anchor** | **128**  | **256**  | **512**  | **1,024** | **#view\\\\#anchor** | **128**   | **256**  | **512**  | **1,024** |\n| 2              | 1.93 | 3.02 | 4.56 | 8.89  | 2              | 1.17 | 2.18 | 4.28 | 8.56  |\n| 3              | 2.11 | 3.67 | 6.38 | 14.74 | 3              | 1.61 | 3.21 | 6.42 | 12.87 |\n| 4              | 2.98 | 4.91 | 9.06 | 23.10  | 4              | 2.14 | 4.28 | 8.56 | 17.13 |\n\n- **#Sample = 126k**\n\n| Time (s)       |      |      |       |       | Memory (GiB)   |      |      |       |       |\n| :--------------: | ---- | ---- | ---- | ----- | :--------------: | ----- | ---- | ---- | ----- |\n| **#view\\\\#anchor** | **128**  | **256**  | **512**  | **1,024** | **#view\\\\#anchor** | **128**   | **256**  | **512**  | **1,024** |\n| 2              | 1.73 | 5.75 | 9.40   | 19.30  | 2              | 1.93 | 3.89 | 7.71  | 15.48 |\n| 3              | 2.33 | 7.28 | 16.23 | 33.00  | 3              | 3.02 | 6.03 | 11.59 | 23.26 |\n| 4              | 3.07 | 9.20  | 21.29 | 60.44 | 4              | 3.85 | 7.71 | 15.42 | 30.86 |\n\n\n\n---\n\n### **W3: References**\n\nThanks for pointing them out. We will cite and discuss these papers in the revision.\n\n---\n\n\n\n\n### **W4/6: Full Name of TNN in Definition1**\n\nThanks for pointing it out. We will revise TNN to its full name in Definition 1.\n\n---\n\n\n\n\n### **W5: Additional Deep Learning Baselines**\n\nThanks. 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.\n\n| ACC      | CUB  | UCI-Digit | Caltech101-20 | OutScene | MNIST-USPS | AwA  |\n| -------- | ---- | --------- | ------------- | -------- | ---------- | ---- |\n| IMvGCN   | 63.3 | 82.7      | 68.6          | 53.7     | 79.9       | 65.8 |\n| GEGCN    | 49.9 | 90.1      | 67.1          | 55.3     | 91.3       | 58.8 |\n| **Ours**     | **80.2** | **95.2**      | **80.0**            | **69.2**     | **95.6**       | **70.6** |\n| **PREC** |      |           |               |          |            |      |\n| IMvGCN   | 67.7 | 83.5      | 51.2          | 53.7     | 81.1       | **60.8** |\n| GEGCN    | 56.5 | 90.6      | 34.0            | 56.2     | 91.4       | 57.0   |\n| **Ours**     | **80.2** | **95.2**      | **53.1**          | **68.3**     | **95.6**       | 60.0   |\n| **F1**   |      |           |               |          |            |      |\n| IMvGCN   | 61.2 | 82.5      | 40.2          | 52.4     | 79.5       | 58.6 |\n| GEGCN    | 50.5 | 90.1      | 31.6          | 55.5     | 91.3       | 50.4 |\n| **Ours**     | **79.1** | **95.3**      | **54.8**          | **67.7**     | **95.6**       | **58.6** |\n\n\n\n**References**\n\n[R1] Ester, Martin, et al. \"A density-based algorithm for discovering clusters in large spatial databases with noise.\" kdd, 1996.\n\n[R2] Huang, Dong, et al. \"Fast multi-view clustering via ensembles: Towards scalability, superiority, and simplicity.\" IEEE TKDE, 2023.\n\n[R3] Wu, Zhihao, et al. \"Interpretable graph convolutional network for multi-view semi-supervised learning.\" IEEE TMM, 2023.\n\n[R4] Lu, Jielong, et al. \"Generative essential graph convolutional network for multi-view semi-supervised classification.\" IEEE TMM, 2024."}, {"review_id": "xLsrdhj128", "reviewer": "Reviewer_Z8gs", "summary": "This paper proposes AGF-TI, an innovative approach for incomplete multi-view semi-supervised learning, to address the Sub-Cluster Problem (SCP) caused by missing views. By integrating adversarial graph fusion (min-max optimization) and tensorial imputation (low-rank tensor learning), AGF-TI reconstructs robust consensus graphs and recovers missing structures. Combined with an anchor-based strategy for computational efficiency, AGF-TI demonstrates superior performance on various datasets, validating its effectiveness in handling both view incompleteness and label scarcity challenges.", "questions": "See the weaknesses.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. This paper identifies the Sub-Cluster Problem in graph-based multi-view semi-supervised learning, an important but previously under-explored challenge.\n2. AGF-TI innovatively combines min-max scheme with tensorial imputation, forming a robust framework to address SCP. \n3. Clear and mathematically rigorous formulation of the optimization problem, with a solid convergence analysis in the appendix.\n4. Experimental results demonstrate a substantial improvement in classification performance. The paper also includes a detailed ablation study to demonstrate the effectiveness of each component in AGF-TI.\n\nWeaknesses:\n1. Although the time complexity is discussed, it would be better to provide the running time results of the AGF-TI and baselines for comparison.\n2. What does the parameter $\\beta$ mean in Eq. (3)?\n3. How are the permutation matrices $\\mathbf{T}_v$ initialized in practice?", "was_revised": true, "final_justification": "The authors' response have addressed most of my concerns, I will maintain my score as 5.", "rebuttal": "Thank you so much for your valuable comments! We are committed to addressing each question you have raised.\n\n---\n\n### **W1: Running Time**\n\nThanks. 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.\n\n| Time (s) | CUB  | UCI-Digit | Caltech101-20 | OutScene | MNIST-USPS | AwA     |\n| -------- | ---- | --------- | ------------- | -------- | ---------- | ------- |\n| AMMSS    | 4.31 | 7.80      | 86.25         | 15.61    | 48.58      | 621.34  |\n| AMGL     | 4.16 | 5.51      | 65.99         | 9.60     | 39.89      | 587.51  |\n| MLAN     | 4.68 | 20.29     | 141.65        | 74.99    | 150.06     | 1068.73 |\n| AMUSE    | 6.37 | 31.77     | 102.76        | 55.59    | 209.93     | 1361.76 |\n| FMSSL    | 4.77 | 5.81      | 82.94         | 24.51    | 74.75      | 808.80  |\n| FMSEL    | 6.45 | 32.70     | 120.18        | 60.12    | 234.50     | 1359.60 |\n| CFMSC    | 6.78 | 20.36     | 106.95        | 26.29    | 159.53     | 815.56  |\n| MVAR     | 4.26 | 5.40      | 69.84         | 10.36    | 39.30      | 601.27  |\n| ERL-MVSC | 4.31 | 5.68      | 67.71         | 10.55    | 42.66      | 605.83  |\n| AMSC     | 0.85 | 2.15      | 11.03         | 4.40     | 3.73       | 33.54   |\n| SLIM     | 0.33 | 4.46      | 35.59         | 14.12    | 11.43      | 128.83  |\n| **Ours** | 1.06 | 15.35     | 45.03         | 35.84    | 31.06      | 354.54  |\n\n\n---\n\n\n\n### **W2: $\\beta$ in Eq. (3)**\n\nThanks 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.\n\n---\n\n\n\n### **W3: Initialization of $\\mathbf{T}_v$**\n\nThanks. In our experiments, the permutation matrix $\\mathbf{T}_v$ is initialized as an identity matrix.\n\n**Last**\n\nThank you again for your suggestion. We will include these details in the revised paper."}, {"review_id": "fxfsQfGKm7", "reviewer": "Reviewer_dtsk", "summary": "This paper tackles the challenge of missing views in multi-view semi-supervised learning, which often disrupts local data structures and impairs classification accuracy. The authors introduce AGF-TI, a novel framework that combats structural discontinuities by generating a unified graph through adversarial learning. It leverages high-order consistency via tensor-based structure recovery and incorporates anchor points to enhance efficiency. Extensive experiments confirm that AGF-TI consistently outperforms existing methods across multiple benchmarks.", "questions": "Please refer to the weaknesses.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The writing and presentation are good, then the readers can quickly grasp the core idea of the paper.\n2. The experiments are sufficient to verify the effectiveness of the proposed method.\n\nWeaknesses:\n1. The anchor learning and low-rank tensor decomposition have been widely studies in multi-view clustering, I cannot see some insightful observations and interesting idea.\n2. The authors demonstrate the Sub-Cluster Problem on a Synthetic dataset. However, this could be artificially crafted. Hence, the such verification should be conducted on real-world datasets. \n3. The most used multi-view datasets are too small, it’s suggested to add more large-scale datasets.\n4. The author claims that introducing anchors can decrease the computational burden, then the running time should be compared.", "was_revised": true, "final_justification": "The authors basically addressed my concerns, so I decide to raise my score.", "rebuttal": "Thank you so much for the valuable comments! We are committed to addressing each question you have raised.\n\n---\n\n### **W1: Novelty of the Proposed AGF-TI**\n\nWe 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:\n\n- **Anchor graph fusion paradigm**\n\nMost 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.\n\n- **Adversarial min-max framework**\n\nWe 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.\n\nIn addition to the proposed AGF, we also present a severe yet neglected problem, the Sub-Cluster Problem, when encountering view missing scenarios.\n\n**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**.\n\n**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.\n\n---\n\n### **W2: SCP in Real-world Datasets**\n\nThanks. 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.\n\nTo 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.\n\n| WebKB         | #class | 0%   | 30%  | 50%  | 70%  |\n| ------------- | ------ | ---- | ---- | ---- | ---- |\n| page view     | 2      | 2    | 3    | 4    | 9    |\n| link view     | 2      | 11   | 15   | 19   | 27   |\n| **Wikipedia** |        |      |      |      |      |\n| image view    | 10     | 7    | 9    | 11   | 16   |\n| text view     | 10     | 10   | 11   | 15   | 22   |\n\n---\n\n### **W3: Additional Large-scale Datasets**\n\nThanks. 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.\n\n| NoisyMNIST |       | 30K   |       |       | 70K   |       |\n| ---------- | ----- | ----- | ----- | ----- | ----- | ----- |\n| **Method**     | **ACC**   | **PREC**  | **F1**    | **ACC**   | **PREC**  | **F1**    |\n| AMMSS      | 60.72 | 59.74 | 62.59 | -     | -     | -     |\n| AMGL       | 51.8  | 51.02 | 52.11 | -     | -     | -     |\n| MLAN       | -     | -     | -     | -     | -     | -     |\n| AMUSE      | 60.29 | 59.63 | 59.5  | -     | -     | -     |\n| FMSSL      | 78.12 | 77.44 | 77.11 | -     | -     | -     |\n| FMSEL      | 52.66 | 51.94 | 51.51 | -     | -     | -     |\n| CFMSC      | 72.67 | 72.11 | 71.67 | -     | -     | -     |\n| MVAR       | 61.81 | 61.15 | 60.42 | 68.21 | 67.39 | 67.35 |\n| ERL-MVSC   | 77.31 | 77.04 | 76.83 | -     | -     | -     |\n| AMSC       | 72.04 | 71.34 | 73.09 | 71.45 | 70.88 | 72.75 |\n| SLIM       | 66.74 | 66.18 | 65.63 | -     | -     | -     |\n| **Ours**       | **81.48** | **81.01** | **81.08** | **77.34** | **77.78** | **76.75** |\n\nNote: Some results were not reported due to out-of-memory errors or because the computations had not been completed within the allotted time.\n\n---\n\n\n### **W4: Running Time**\n\nThanks. 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.\n\n\n\n| Time (s) | CUB  | UCI-Digit | Caltech101-20 | OutScene | MNIST-USPS | AwA     |\n| -------- | ---- | --------- | ------------- | -------- | ---------- | ------- |\n| AMMSS    | 4.31 | 7.80      | 86.25         | 15.61    | 48.58      | 621.34  |\n| AMGL     | 4.16 | 5.51      | 65.99         | 9.60     | 39.89      | 587.51  |\n| MLAN     | 4.68 | 20.29     | 141.65        | 74.99    | 150.06     | 1068.73 |\n| AMUSE    | 6.37 | 31.77     | 102.76        | 55.59    | 209.93     | 1361.76 |\n| FMSSL    | 4.77 | 5.81      | 82.94         | 24.51    | 74.75      | 808.80  |\n| FMSEL    | 6.45 | 32.70     | 120.18        | 60.12    | 234.50     | 1359.60 |\n| CFMSC    | 6.78 | 20.36     | 106.95        | 26.29    | 159.53     | 815.56  |\n| MVAR     | 4.26 | 5.40      | 69.84         | 10.36    | 39.30      | 601.27  |\n| ERL-MVSC | 4.31 | 5.68      | 67.71         | 10.55    | 42.66      | 605.83  |\n| AMSC     | 0.85 | 2.15      | 11.03         | 4.40     | 3.73       | 33.54   |\n| SLIM     | 0.33 | 4.46      | 35.59         | 14.12    | 11.43      | 128.83  |\n| **Ours** | 1.06 | 15.35     | 45.03         | 35.84    | 31.06      | 354.54  |\n\n\n**References**\n\n\n\n[R1] Blum, Avrim, and Tom Mitchell. \"Combining labeled and unlabeled data with co-training.\" Proc. Conf. Learn. Theory, 1998.\n\n[R2] Wang, Hao, et al. \"Multi-view clustering via concept factorization with local manifold regularization.\" IEEE ICDM, 2016.\n\n[R3] Ester, Martin, et al. \"A density-based algorithm for discovering clusters in large spatial databases with noise.\" kdd, 1996.\n\n[R4] Yang, Mouxing, et al. \"Robust multi-view clustering with incomplete information.\" IEEE TPAMI, 2022."}]

Author rebuttal:

Meta-review: After the rebuttal, all reviewers acknowledged that their concerns had been addressed and were consistently positive about this submission. I recommend accepting it. The authors are suggested to improve the camera-ready version according to the discussion.
Decision: Accept (poster)