Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
Authors: Shengju Yu
Yiu-ming Cheung
Siwei Wang
Xinwang Liu
En Zhu
Venue: ICML 2025 poster (2025)
Keywords: Incomplete Mulit-view Clustering
Primary area: general_machine_learning

Abstract: Despite remarkable advances, existing incomplete multi-view clustering (IMC) methods typically  leverage  either perspective-shared or perspective-specific determinants to encode cluster representations. To address this limitation, we introduce a BACDL algorithm designed to explicitly capture both concurrently, thereby exploiting heterogeneous data more effectively. It chooses to bifurcate feature clusters and further alienate them  to enlarge the discrimination. With distribution learning, it successfully couples view guidance into feature clusters to alleviate  dimension inconsistency. Then, building on the principle  that samples in one common cluster own similar marginal distribution and conditional distribution, it unifies the association between feature clusters and sample clusters to bridge all views. Thereafter, all incomplete sample clusters are reordered and mapped to a common one to formulate  clustering embedding. Last, the overall linear overhead endows it with a resource-efficient characteristic.

Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
Shengju Yu?
Yiu-ming Cheung
Siwei Wang? Xinwang Liu En Zhu
Abstract
need of any labels known in advance, is generally perceived as an encouraging method to tackle these data, and has been widely deployed in fraud detection, personalized medicine, social network analysis, etc, (Zhang et al., 2021; Yu et al., 2023; Zhang et al.; Liu et al., 2024). The prerequisite for MVC algorithms'' proper execution is that all views are re- quired to be complete (Yu et al., 2024d; Ma et al., 2024a; Gu et al., 2024a; Wan et al., 2024). Due to equipment defects or collector faults, however, in real-life it inevitably causes cer- tain views having missing samples, inducing the incomplete multi-view clustering (IMC) issue (Yu et al.; Wang et al., 202 la; Liu et al., 2023a; Yu et al., 2024a; Gu & Feng).
Despite remarkable advances, existing incomplete multi- view clustering (IMC) methods typically leverage either perspective-sharee or perspective- specific determinants to encode cluster represen- tations. To address this limitation, we introduce a BACDL algorithm designed to explicitly cap- ture both concurrently. thereby exploiting hetero- geneous data more effectively. It chooses to bi- furcate feature clusters and further alienate them to enlarge the discrimination. With distribution learning, it successfully couples view guidance into feature clusters to alleviate dimension incon- sistency. Then, building on the principle that sam- ples in one common cluster own similar marginal distribution and conditional distribution, it uni- fies the association between feature clusters and sample clusters to bridge all views. Thereafter, all incomplete sample clusters are reordered and mapped to a common one to formulate cluster- ing embedding. Last, the overall linear overhead endows with resource-efficien characteristic.
To cope with IMC tasks, recently, great deal of promising methods have been carefully presented (Xu et al., 2022; Wang et al., 202 lb; Yu et al., 2024c; Tang & Liu, 2022). For instance, Wang et al. (2022) adopt consensus bipartite affin- ity to characterize arbitrary views and jointly construct an- chors using an unified learning mechanism. Li et al. (2023b) decrease the disturbance of superfluous properties through projecting operations and employ low-rank tensor regulariz- ers leverage high-order representation inside samples. Gu et al. (2024b) utilize dictionary learning strategy to recover missing parts and integrate Gaussian error rank into Lapla- cian manifold optimization to explore local correlations. Motivated by prototype advances, Li et al. (2024) build up the conjunction between prototypes and observed instances to avoid the generation of full-sized similarity and directly formulate the overall graph structure without additional hyper-paramen parameter searching. These methods enhance the clustering quality from various aspects, nevertheless, they investigate either perspective-share or perspective-specinf determinants to encode cluster representation. This single paradigm could not sufficiently exploit the interrelations among data features, restricting the model's performance.
1. Introduction
In the era of information, heterogeneous data that are com- monly gathered from various channels and modalities of the same one object are growing more prevalent (Ma et al., 2024b; Wang et al., 2023; Yu et al., 2025: Zhang et al., 2025). Accordingly, how to effectively excavate out valuable po- tential patterns from this type of data is grasping increasing attention (Li et al., 2025; Yu et al.,, 2024b; Liang et al., 2024). Multi-view clustering (MVC) technology, in virtue of the ability to seamlessly integrate multi- information and powerfully partition samples into distinct sets without the
To get rid of this limitation, in the manuscript propose IMC algorithm named BACDL, and its overall pipeline is described in Fig. 1. Concretely, inspired by non-negative matrix factorization, we choose to bifurcate feature clus- ters and each bifurcation is explicitly interconnected to a type of determinants. Through mutual exclusion learning, we alienate them to strengthen their discrimination. Subse- quently, in virtue of distribution learning, we couple view
Department of Computer Science, Hong Kong Baptist Univer- sity. 'intelligent Game and Decision Lab. "National University of Defense Technology. yu-shengju @ foxmail.com. Correspon- dence to: Yiu-ming Cheung <yme& @ comp.okbu.cdee hdb..cdu.hk>..
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
G2
View Data
PCD
IV1SC
AW
Space
D
Space
View Data
VG
PDD
IVRSC
CSC
LE Space
ME
Cp
Embedding
Pg
VG
PCD
IVVSC
View Data
Feature Clusters
Association
Sample Clusters
Results
Figure 1. Pipeline of proposed BACDL. firstly bifurcates feature clusters each incomplete view and magnifies the discrimination between determinants via mutual exclusion learning. Then, it couples view guidance into feature clusters through distribution learning alleviate the dimension inconsistency. Subsequently, unifies the association between feature clusters and sample clusters to bridge all views. Further, all sample clusters are reordered in latent subspace and, following adaptive weighting, mapped to a common one to constitute clustering embedding. VG: view guidance; PDD: perspective-shares determinant; PCD: perspective-spechie determinant; ME: mutual exclusion; D: association between feature clusters and sample clusters: AW: adaptive weighting; IVISC: incomplete view I sample clusters: IVVSC: incomplete view sample clusters; CSC: common sample clusters.
sample clusters; IVRSC: incomplete
guidance into feature clusters to climinate the inconsistent dimensions. Further, relying on the fact that samples one common cluster are with similar marginal distribution and unify the association between
multancously enlarging mutual information and shrinking mean discrepancy, Ren al. (2024) advance imputation fidelity in unsupervised manner and conduct data match- ing across clients by utilizing the sample homogeneity and view multi-functionalityy Inspired by contrastive learning, Yang et al. (2022) utilize observed pairs as positives and randomly-selected cross-view samples as negatives to climi- nate the partial unalignment and alleviate the noisy impact. Zhao al. (2023) introduce consistency constraint main- tain the similarity between graphs on different views and utilize the tensor means to extract graph correlations in manifold space. Orthogonal to them, He al. (2023) learn asymmetric similarities based on structural anchors to replace distance-based weighting and extend late fusion to general scenarios for affinity construction. Sun et al. (2023) generate uniform probability representation by relax- ing standard spectrum to increase the stability and integrate balance constraint adaptively exploit intra-view features. Li et al. (2023a) design dual-stream mechanism to model prototype-sample similarity and utilize the relationship be- tween available views to conduct sample recovery. With the idea of neighbor group, Wong et al. (2023) produce neighbor sets for each sample pair to partition structure embedding and encode view-missing position to guide the fusion of individual graph. Zhang al. (2024) introduce kernelized subspace to extract intrinsic structure between views and perform low-rank learning and affinity construction jointly.
conditional distribution, feature clusters and sample clusters to bridge all views. Af- terwards, we reorder sample clusters by space rotation and map them after adaptive weighting onto a common one to form clustering embedding. Then, for efficiently minimiz- ing the formulated objective function, we give nine-step updating rule with overall linear overhead and theoretical convergence. For verifying the effectiveness of BACDL, organize extensive comparison experiments under multiple missing ratios. To sum up, in this manuscript we (1) propose a new IMC learning paradigm, which achieves the simulta- determinants and perspective-speciff determinants via coupled distribu- tion learning: (2) design a updating rule with overall linear overhead, enhancing the practicality (3) conduct compre- hensive experiments, demonstrating the effectiveness and merits of presented BACDL. method from multiple aspects.
neous exploration of both perspective-shee
2. Related Works
To effectively handle IMC problems, in recent years re- searchers have proposed many prominent algorithms. Xu al. (2023) impose adaptive projection in feature space to evade imputation, and formulate cluster structures by si-
2

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
3. Preliminary
needs satisfying the constraint [P,CIC,] 14. 12k- So, the loss can be defined as
Non-negative matrix factorization (Ding et al., 2006; Long al., 2012: Ding et al., 2010) is usually deemed pow- erful means tackle multi-view data, The basic idea is formulated
- - P,C G.
min C. E.. 1=1
s.t. C2 O, 2 0 D. 2 E, 2 0.P, 20, [P,C T 1a, 12x-
e
Σ X, C,D,E, (1)
I
min C,
(2)
Further, in conjunction with the probability distribution char- acteristic of sample clusters, we have that it needs satisfying E, 1,, On the other hand, in light of the fact that samples in one common cluster are with similar marginal unify the as- sociation between feature clusters and sample clusters to bridge all views. Besides, considering that views could have different levels of contributions, we assign weight for each view to automatically balance them. Consequently, the loss can be designed as
where X, RR.xx denotes the (complete) data matrix view r. d, is the feature dimension on view and is the number of samples. C, E Rd, x denotes w feature clusters. Each column of C, represents probability distri- bution on d, features. t" = arg max; (Cr),,t indicates that feature s is affiliated feature cluster 1" where (C,)s,, is the element in the s-th row and t-th column of C,. The ma- trix E, n denotes 2 sample clusters. Each row of E, represents a probability distribution. 1" arg max, s.,),.. indicates that sample is affiliated to sample cluster t". D, E R=x= denotes the association matrix between C, and E,.. The element (Dr)e,t expresses the association probability between feature cluster s and sample cluster t.
distribution and conditional distribution,
Co = min X. - P.CCI,, G
s.t. C C, 2.D 2 0, E, 2 O,P, 2 0, 2 0,
Besides, the formula minc,.D..E. X - C,D,EE, can equivalently deemed as mitlc,.., |X, C,
Σ a, = 1, T 1s, = 12k. G. E, = 1
and minD,,,E. ||,,- D.E..1 1| C, maps the original data X, to the potential space Z, Correspondingly, the marginal distribution P(x,)) is transformed to P(z,). C, learns the marginal distribution. Similarly, D, maps the potential space Z, to the sample clusters E,. Correspondingly, the conditional distribution ( is transformed to P(y (.)).
(3)
a.). [+]1] denotes the matrix where e = C, concatenation operation.
Thereafter, to enhance the discrimination between double determinants, we make P,C and C, alienated from each other. Specially, by point-to-point alienation, we utilize the matrix inner product (P,C, C,) to render them dissimi- lar possible. So, we have the following loss,
D, learns the conditional distribution.
4. Proposed Model
L1 = min (P,C, C.C.P..
(4)
Firstly, for incomplete dataset (X, 1==1- in order to work with the incompleteness, we introduce the index matrix G, E n where (Gr)s. = 1 (w,)) = i otherwise (Gr), = 0 for any j [1,2,... ny). W, E R=,x1 is the indicator vector. n, is the number of samples observed on view Accordingly, X,G, E dn. denotes the available samples the r-th view.
s.t. C 0, C, 2 2, P, ld, -12--
Subsequently, sample clusters on different views could be misregistered due to the unsupervised property, and we associate transformation space for the sample clusters each view to reorder them. Further, sample clusters to common one so as to formulate full clustering embedding. Consequently. the loss is defined as
map reordered
Then, to achieve double-determinant exploration, we bifur- cate the feature clusters. Specifically, we attempt to jointly utilize C E k and C, E d to exploit perspective- shared determinants and perspective-speeific determinants respectively where & is the number of clusters. However, due to the feature dimension inconsistency, it is hard to embed C into feature clusters. To alleviate this dilemma, we introduce view guidance P, e xk to assist C ex- ploring inter-view characteristics. Accordingly, the feature clusters are coupled as [P,C Additionally, given the probability distribution characteristic of feature clusters, it
L2 max Tr E,F,
Σ by F,
(5)
s.t. E.E
It
Σ 82 1, 14.200 0,
where 6, is a weight variable.
Therefore, the loss of proposed BACDL is formulated as
L = Co + >L1 - BL2.
(6)
3

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
5. Updating Rule
Then, we can get the following updating rule,
When updating E,, the loss equivalently becomes
Er a X,K,
(C)ss - (C)ss
i.
min a G, A,E, G. -bb Tr EF,
o E, K, + 2 M
E,
f. (11)
s.t. E, 2 0, E, E, I
(7) where K, G,G.E.D,,. D, DIsk.:. M
where A, P.C
E r= P. C,. Further, we have the following theorem,
Theorem 3. With allumn-normalized P., P,C satisfies column normalization only if C is also
We can derive the following updating rule for E,,
1/2
[B,X,A, b, poos
When updating Cr, the loss is transformed as
(Er)ss (Er)a.
(B,E,A,A A, b, (Cr)nes)
min at X,G. A,E. G. + (CT..C.P.C.. C.
(8)
where B, = G,G,G,G Č = E.. (.)pe and (:)eg denote the positive part and negative part respectively.
s.t. C, 2 0, C, 1d, 1s.
(12)
We have the following updating rule,
Denote the loss (7) as ((E,). We give the auxiliary function definition of (E,,).
X (x,,E,,D.) 1
(C+) + (Cr)ss
Definition 1. F(E,, E,) is an auxiliary function of ((E,) when for any E, and E, satisfies F(E,, E,) 2 (E,,) and F E,, E,) = (EE (Ding et al., 2010).
A.,.BB..DDJ + PPP
(13)
where D. DR41226.-
Then, we have the following theorem holds, Theorem 1. Eq. (9) is an auxiliary function of ((Ex) and also convex.
When updating D, the loss can formulated as
a X, IP,CC,, G, ||?
min
F(E.: Ex) = at E.G.G,EEAAAA.....
(14)
s.t. D 0.
2 2 Σ [G,,) A. )is (Dr)ss
We can obtain the following updating rule for D,
-b,, ((Er)ss ((C))
1. (rr)ss
aL
(9)
(E,), 1n, 1 )., (Dr)ss
Di. Das
E LL, G.
EGG,O, E. ),, (s
(15)
Σ (CCr)ne)) (Jr)is) + 2
bb,
where L, = [.,
1.5
When updating Pr, the loss can formulated as
where (Dy)..s 1 + log (Er)a./ (E.)..) (Hr)ss
min a² X. -AA, G. +ATI (AT.CCP.,.C.
(Iriss = (E,)), +(E,) (Er))..
P.
S.l. P. 2, C 1
On the basis of Theorem 1, we further have the following theorem holds,
(16)
We have the following updating rule,
Theorem 2. Under the rule Eq. (8), the loss Eq. (6) is monotonically decreasing.
(C.X,K.C)) (aCA,E,K,CC + CACCTT
1
(Pr) - (Pr)es
1.
When updating C, the loss can be simplified as
2 + + C.)
min
Σ
2 X,G, - A,E, A G,
When updating E, the loss equivalently becomes
F
8.t. C20,C 1d, =
min
Tr
E
Σ b, E,
s.t. E E I (18)
(10)
E
4

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Its optimal solution can be obtained by setting E the product of U and V where U and VT are the singular matrices of the term Dr Fr.
Algorithm Proposed BACDL Algorithm
Input: Data matrix X,, indicator vector W,.
syper-parameters À and $:
1: Constructing index matrix Gy;
When updating Fr, the loss equivalently becomes
2: Doing distribution normalization on X,G,;
4 2 le - 3 do 4: Updating the feature clusters E, by (8):
3: while (g² (g"--"+1)//0 //*1)//²*
min- F.
Tr
E 1 E, s.t. F.F. = I (19)
5: Doing row- now-normalization on E,;
6: Updating the perspective-sharee matrix C by (11); Doing celumn-nommalzzationn on C; Updating the perspective-speeifie matrix C, by (13);
The optimal solution is UVT. U and V are the singular matrices of E, E.
When updating ar. the loss can be simplified as
9: Doing
C,;;
10: Updating the association D by (15);
min Gr
af
X. A,,,, G.
[| s.t. Σ 0r 1, 2 0.
11: Updating the view guidance P, by (17);
12: Doing golumn-normalizatioon on P,;
(20)
13:
Updating the common sample clusters E by (18);
14: Updating the space rotation matrix F, by (19);
The optimal solution is
15:
Updating the view weight a, by (21);
16:
Updating the sample cluster weight b, by (23);
G, A.B.G.
0
(21)
17:
h h= h 1;
A.G.AA.E.GG. G.
18:
end while
Output: Spectral clustering on E;
Remark 1. The value of of| - A,E, Gr equals to that of G. - G. 1/z. The former needs
Table 1. Details of Multi Datasets Used in Experiments
at least ((n,)) computing overhead while the latter can be calculated within (n) overhead.
Dataset
Samples
Views
Clusters
Dimensions
FLOEVEN
1360
7
17
1377612399512257766655 77777777777777
When updating by, the loss can be simplified as
SYNTHREED
601
3
3/2003/3
DEOLOG
358
22/12
YALTHRER
165
3
18
$804449696750
BGFTA
2500
3
1000/500/50
min by
Tr
E
E,E,F,
s.t.
62 1, b, 2 O.
AWTEN
5814
6
10
2000/2688/20000 200
HDIGTWO
10000
10
256/784
(22)
YOUFOURV
38654
10
512844/576/140
The optimal solution is
6. Experiments and Analysis
M,
b
(23)
6.1. Benchmark and Baseline
Σー= [M,)
The following IMC algorithms are utilized as the baselines: Sparse Consensus Affinity Construction (LRTL (Chen et al., 2023)), Interview Graph Connectivity Learning (TCIMC (Xia et al., 2022)), Between-view Inferring and Within-view Preservation (AGCIM (Wen al., 2021a)), Local Graph Embedding Generation and Unified Repre- sentation Learning (LSIMV (Liu et al., 2023b)), Local Geometric Similarity Conservation (GIMC (Wen et al., 2021b)), Individual Consensus Structure Set Exploration (IMVCI (Tang et al., 2024)). Projective Graph Regular- ization Balance Learning (PIMVC (Deng et al., 2024)), Similar Neighho- Confidence Representation Gener- ation (HCCGL (Wen et al., 2023a)). Multiple Spectral Con- nection Relationship Fusion (USETL (Chen et al., 2024)), Multi-mate calti-mmtrix-ffane faatri--fcttoizzation
where M, = Tr E.F..
Algorithm 1 summarizes the overall procedure of proposed BACDL where gå denotes the loss value at the h-th iteration.
For the lower bound, we have the following theorem holds, Theorem 4. The loss value is lower-bounded by - /unk.
On the basis of Theorem 2 and Theorem 4, we further have Theorem 5. The proposed BACDL algorithm is convergent.
About the complexity of BACDL, we have Theorem 6. BACDL is with the time and space overhead linear to the sample size n.
Learning (LBIMV (Wen et al., 2023b)), Feature Inferring and Cross- view Correlation Guidance (UIMC (Lin et al., 2024)).
Remark 2 In virtue of the overall linear overhead, BACDL scalable to large-scale scenarios.
5

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Table 2. Clustering Results of Different IMC Algorithms
DATASET
FLOEVEN
SYNTHREED
RATIO
0.2
0.5
0.8
0.2
0.5
0.8
METRICC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
F3C
PUR
ACC
FSC
PUR
ACC
FSC
LRTI
11.08
9.27
7.85
10.85
10.85
7.600
11.35
9.41
7.21
30.87
40.08
31.14
38.67
40.67
51.47
35.83
39.83
31.88
TCIМC
10.93
10.35
8.21
10.63
9.63
6.87
0.58
9.48
5.69
57.57
36.92
30.76
37.53
55.72
31.23
36.83
36.25
29.87
AGCIM
11.02
10.07
8.43
9.34
8.38
7.22
10.67
10.32
7.19
33 30.50
33.50
27.79
34.00
34.00
28.75
34.17
34.17
29.15
ESIMV
8.74
9.47
8.21
9.67
6.44
7.58
57.3
39.57
12.21
36.14
35.13
34.78
37.89
GIMC
8.32
9.38
5.97
8.63
7.94
6.34
7.26
7.91
5.25
36 89
35.72
32.67
35.89
33.73
34.01
33.84
32.75
33.84
IMVCI
10.37
10.86
10.95
10.21
6.31
11:35
10.17
5.08
40.38
32.80
39,77
40.68
32.29
38.93
33.53
30,41
PIMVC
11.11
9.28
4,32
11.28
9.36
7.32
10.78
8.56
6.35
40.27
30.67
30.89
39.50
39.50
$1.04
37.002
35.02
30.82
ROCGL
7.13
6.68
6.91
7.08
40
39.67
34.51
37.01
USETL
9.99
9.43
7.22
10.19
9.56
7.667
9.30
8,90
7.25
35. 96
38.96
26, 86
36.15
35.51
29 19.56
36.28
36.28
29.08
LBIMV
10.56
10.31
5.09
11.07
9.85
5.18
10.01
9.06
9.23
27
40.5
27.45
40.33
40.07
50.16
35.67
35.67
27.04
UMC
10:88
9.22
5.80
11.12
9.96
5.99
10.51
9.22
5.83
39
38.33
32.13
36,17
40.17
33.26
38.83
37.50
33.86
OURS
11.17
10.80
5.88
11.25
10.53
5.88
11.54
10.74
5.88
42. 50
42.12
35.21
42.83
42.83
35.24
41.04
41.04
34.59
DEOLOG
YALTHREE
L.TI.
31.43
21.59
15.29
30.71
19.87
16.54
31.37
20.90
15.95
21
21.52
8.59
18.88
20.17
19.30
18.85
6.61
TCIMC
30.86
19:13
16.43
30.73
19.26
15.74
335.47
20.30
15.88
21:35
17.98
7.79
21.51
17.87
6.98
20.21
17.79
5.89
AGCIM
30:45
21.45
18.64
29.55
22.44
17.88
32.01
22.89
17.32
22.12
18.12
7.88
18.18
17.97
5.87
18.48
17.57
6.42
ESMMV
26
18.43
14:37
15.32
32.94
18.74
16.
16.42
6.73
6.26
16.34
16,43
GIMC
31.61
19.32
17.13
32.64
17.36
15.63
31.53
18.57
16,43
19.46
16.72
6,84
17.83
17.96
6.72
19.43
17.43
8.82
IMVCI
30.98
21.58
17.43
30,93
21.93
17.98
30,82
24.52
18.25
21.15
21.74
7.12
19.15
20.35
7.42
20.36
20.17
6.78
PIMVC
31.33
22.04
17.32
29,36
20.24
17.54
51.62
23.03
18.97
20 64
17.24
7.34
18.12
15.52
6.26
18.36
15.42
7.56
RCCGL
20.05
20.05
17.57
30.58
19:72
17.16
20.32
24.20
17.77
21. 91
21.71
7.13
21:15
30.18
7.16
20.61
20.36
9:23
USETL
30.84
18.99
18.13
22.61
17.68
30.94
22.22
21.85
8.21
17.33
6.31
20.85
19.88
LBIMV
50.09
20.00
17.71
30.93
22.98
18.07
20.27
21:42
17.07
22.
19.12
6.29
20.94
19.91
6.18
21.15
19.75
5.51
UIC
30.83
19.89
16:23
20.82
16.57
20.03
19.5
6.39
18.16
17.58
17.82
7.30
OURS
31.84
22.32
18:45
31.84
23.74
18.41
31.28
24.25
19.56
24.944
23.55
7.89
21.73
20.61
6.20
21.55
20.70
6.70
BCFFFA
AWTRN
L.TI.
17:37
20.21
18.16
18.27
20.62
19.01
18.22
19.23
18:07
17.13
9.67
8.43
18.03
11.28
6.42
19.53
9.40
8.38
TCIMC
19:63
20.46
18:26
19.43
20.88
18:33
20.16
19.60
17.93
16.78
10.71
10.63
16.67
11:48
9.37
17.84
9.28
9.13
AGCIM
20.56
20.43
16.32
21.28
20.20
16.13
21.36
20.32
15.33
13.56
10.28
10.56
14.59
10.32
9.47
13:47
10.73
9.89
LSIMV
19.73
18:24
18.05
18.26
18.48
10.37
9.72
17.39
9.56
17,09
10.57
GIMC
21.26
19:26
19.42
18.29
17.97
19.17
18.97
18.57
18.32
20.47
10:95
9.83
20.89
10.24
9.16
18:36
9.43
8.74
IMVCI
19.58
20.48
16.32
21,02
20:02
16.73
18:42
21.92
20:21
9.69
11.54
20.29
9.56
19.55
3.07
9.35
PIMVC
22.58
20.52
16.28
20,66
20.87
17.25
20.76
20.57
15.73
20:82
10.64
10.85
20.44
9:87
10:24
19.87
9.19
9.27
HCCGL
30
18.46
21.32
19.32
19:32
20.60
17.36
18.43
20.26
10.18
7.83
20.21
9.64
8.33
20.13
9.67
7.38
USETL
21.32
18.01
16.38
18.56
16.89
20.21
18.00
7,31
8.89
20.01
8.88
LBIMV
21.08
20 11
18:21
22.18
20.75
16.16
21.30
20.52
16.11
20.43
8:95
8.03
20.12
9.42
8.59
20.07
9.23
8.69
UIM
21.53
18.60
18.75
19.
19:33
10:25
9.43
19.85
10.23
OURS
22.00 08
22.08
20.08
22.64
22,40
20.16
22:48
22.28
20.20
20:74
12:24
11.01
20.17
12.01
10.97
20.18
12.96
10.94
LIDIGTWO
YOUFOURV
L.TII
11.62
10.02
8.43
11.62
10.61
8.92
9.78
10.48
8:87
TCICC
11.48
10.62
9.42
10.32
10.63
8.62
9.57
9.52
7.63
AGCIM
11.82
10.28
7.88
11.59
10.43
9.39
10.68
10.49
8.13
LSIMV
10.73
9.36
12.67
8.93
3.56
GIMC
10:85
8:33
7.84
11.32
8.42
9.32
9.63
6.89
12.56
9.45
9.28
11.36
8.09
5,66
11:32
8.79
8.76
IMVCI
10:83
10:35
8.50
11.51
11.05
9.36
11.18
10.65
9.09
PIMVC
10.93
8.44
8.60
11.75
9.15
9.28
10.73
8.94
7.43
15.67
9.77
8.87
14.12
9.79
9.33
15 36
9.94
8.62
HCCGL
10:42
11:27
11.16
8:52
USETL
10.28
9.22
8.52
9.74
9.73
7.98
8.37
9.92
7.09
11.32
8.94
9.68
10:43
9.35
9:12
10.42
8.75
8.47
LBIMY
9.77
10.71
8.10
9.35
9.35
9.26
9.16
10.33
9.34
15.32
10.54
8.99
16.03
10.36
9.16
14.32
10.21
9.14
UIC
10.74
11:32
8.49
9.73
10.56
9.33
8.42
10.42
8.52
OURS
11.73
11.51
10.00
11.75
11.60
10.11
11.99
11.70
10.22
15.99
10.91
10.48
15.95
10.85
10.47
15:95
10.80
10.47
All algorithms are evaluated on the following eight multi- view benchmark datasets: FLOEVEN. SYNTHREED, DE-
YOUFOURV, we are consistently in Top-2; On DEOLOG and AWTEN, there are only two sub-optimal results totally. This suggests that BACDL is effective to handle IMC issues.
OLOG, YALTHREE, BGFEA, AWTEN, HDIGTWO, and YOUFOURV. Table presents their details.
(2) Algorithms LRTL, TCIMC, AGCIM, LSIMV, IMVCI, HCCGL and UIMC are unable to deal with the large-scale dataset YOUFOURV due to the complexity limitation. Or- thogonal to them, not only do we normally work but pro- vide competitive results, which illustrates that the proposed BACDL is more widely applicable.
6.2. Results and Discussions
We test the clustering performance under diverse missing ratios, i.e., 0.2, 0.5 and 0.8 respectively. The results are reported in Table 2. We can draw that (1) Our BACDI makes more favorable results than multiple competitors. For example, on SYNTHREED, we receive the most desirable results; On BGFEA, HDIGTWO and
(3) There are some inferior results on FLOEVEN and YALTHREF, possibly because we generate the cluster la-
6

---

Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
15
8-
10
(a) FLOEVEN
(b) SYNTHREED
(c) DEOLOG
(d) YALTHREE
1.CL
ROCL
25-
USETL
25
JIVC
20
20
15
5
(e) BGFEA
( AWTEN
HDIGTWO
(h) YOUFOURV
Figure Running Time of Different IMC Algorithms
Table Memory Overhead (GB) of Different IMC Algorithms
Table 4. Ablation for Perspective-shared Determinants
Method
DATI
DAT2
DAT3
DAT4
DATS
DAT6
DAT7
DATS
FLOEVEN
LRTL
1.61
0.12
0.05
0.07
1.85
19.49
19.59
A/B
0.2
0.5
0.8
TCTMC
2,56
0.23
0.04
0.11
3.45
22.73
37.70
ACC
FSC
PUR
AGCIM
0.77
0.06
0.08
0.06
0.84
7.13
14.65
OPDD
9.86
9,53
4.52
9.57
9.36
4.46
9.81
9.47
4.62
1.60
10.80
10.53
11.54
GIMC
1.22
0.09
0.03
0.41
0.50
4.63
5.47
98.86
IMVCI
0.99
0.07
0.03
0.07
5.96
11.41
SYNTHREED
PIMVC
3,89
0.05
0.05
3.66
0.45
4.14
4.77
80.51
OPDD
36.14
35.84
33.21
37.17
37.17
33.39
37.25
37.25
33:43
HCCGL.
1.32
0.10
0.05
0.07
1.66
13.02
21.46
WPDD
42.12
USETL
0.72
0.04
0.08
0.07
1.05
5.85
15.27
84.66
DEOLOG
0.04
3.94
OPDI
29.53
20.04
16.13
29.02
20.41
16.34
29.02
21.64
17.58
UIMC
3,81
0.13
0,04
0.15
2.68
28.29
28.34
WPDD
22.32
18.45
31.84
23.74
31.28
24.25
19.56
OURS
1.07
0.03
0.04
0.11
0.13
1.82
0.35
3.37
YALTHREE
OPDD
21.27
21.27
6.93
21.36
18.97
5.87
19.03
19.26
6.06
WPDD
24.94
23.55
7,89
21.73
20.61
6.30
21.55
20.70
6.70
BGFEA
bels from formed spectrum rather than directly from original data. This could, to certain extent, degenerate the diversity of view data, weakening the clustering performance.
17.57
18.
18.90
WPDD
22,08
22.08
20.08
22.64
22.40
20.16
22.48
22.28
20.20
AWTEN
10.11
WPDD
20.74
12.24
11.01
20.17
12.01
10.97
20.18
12.06
10.94
6.3. Time and Space Overhead
HDIGTWO
Time Overhead: To validate the efficiency, we record the running time of each algorithm, as suggested in Fig. (The y-axis denotes (-) seconds plus constant.) According to this figure, one can conclude that
OPDD
9.29
8.07
9.31
9.32
8.08
WPDD
11.73
11.51
10.03
11.75
11.60
10.11
11.79
11.70
10.22
YOUFOURV
OPDD
13.48
8.30
7.99
13.47
8.33
8.01
13.51
8.27
8.01
WPDD
15.99
10.91
10.48
15.95
10.85
10.47
15.95
10.80
10.47
(1) Our BACDL requires relatively lower time overhead than multiple competitors. For instance, takes the least time SYNTHREED, DEOLOG, HDIGTWO and YOUFOURV. On FLOEVEN, YALTHREE, BGFEA and AWTEN, its running speed is still comparable. This demonstrates that the proposed BACDL is computationally-effiiiennn
factors and integrates low-dimensiioal embedding into local graphs to form uniform representation, PIMVC utilizes group of projections to leverage view diversities in consen- sus low-dimensional manifold space, HCCGL learns only one affinity across views using a small-sized confidence graph and refines structures directly from original similar- ity. In spite of the time-saving benefits, they typically do
(2) LSIMV, PIMVC and HCCGL run faster than BACDL in some cases, potentially because LSIMV introduces sparse

---

Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Table 5. Ablation for Perspective-sppecffc Determinants
Table 6. Ablation for the Alienating Action
FLOEVIN
FLOEVEN
0.2
0.8
0.2
0.5
0.8
A/B
A/B
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
OPCD
9.43
9.07
4.32
9.80
9.30
4.43
9.26
9.07
4.42
OALG
9.95
9.57
4.96
10.16
9.56
5.03
10.56
9.62
4.98
WPCD
11.17
10.80
5,89
11.25
10.53
5.88
11.54
10.74
5883
WALG
11.17
10.80
5.89
11.25
10.53
5.88
11.54
10.74
5.83
SYNTHREED
SYNTHREED
OPCID
36.75
35.71
32.57
37, 57.50
37,42
33.27
36.77
36.57
32.49
OALG
35,26
34,93
33.
35.17
35,83
35.33
33.27
WPCD
42.50
42.12
35.21
42.83
42,83
35.24
41.04
41.04
34,59
WALG
42,50
42.12
35.21
42,83
42.83
35.24
41.04
41,04
34.59
DEOLOG
DEOLOG
OPCD
28.93
15.07
14.85
59
17.89
21.39
20.12
30.41
22.23
17.61
WPCD
31.84
22.32
18.45
31.84
23.74
18.41
31.28
24.25
19,56
WALG
31,84
22.32
18.45
31,84
23.74
18.41
31.28
24.25
19.56
YALTHREE
YALTHREE
OPCD
20.76
4.48
20.64
4.41
18.13
OALG
22.92
21.92
6.58
19.01
19.80
19.65
5.36
WPCD
24.94
23.55
7.89
21.73
20.61
6.30
21.55
20.70
6.70
WALG
24.94
23.55
7.89
21.73
20.61
6.30
21.55
20.70
6.70
BGFEA
BGREA
OPCID
20.12
19.54
17.32
42
19,34
34
17.31
19:42
19.14
17.36
OALG
21.29
19:23
21.73
49
19:36 36
22.65
21.41
19:35
WPCD
22.08
22.08
20.08
22.64
22.40
20.16
22.48
22.28
20.20
WALG
22.08
22.08
20.08
22.64
22,40
20.16
22.45
22.28
20.20
AWTEN
AWTEN
OPCD
18.44
10.24
9.03
18.34
10.18
9.32
18.44
10.17
9.29
OALG
18.72
10.82
9.63
18.72
10.60
9.58
18.72
10.58
9.54
WPCD
20.74
12.24
11.01
20.17
12.01
10.97
20.18
12.06
10.94
WALG
20.74
12.24
11.01
20.17
12.01
10.97
20.18
12.06
10.94
BIIIGWO
HDIGTWO
OPCD
9.29
9.20
7.99
9.13
8.95
7.95
9.18
9.06
7.95
OALG
10.03
9.84
8.40
10.09
9.91
8.46
10.10
10.03
8.53
WPCD
11.73
11.51
10.03
11.75
11.60
10.11
11.79
11.70
10.22
WALG
11.73
11.51
10.03
11.75
11.60
10.11
11.79
11.70
10.22
YOUFOURV
YOUFOURV
OPCD
13.15
7,89
7.87
13.16
8.32
7.68
13.18
8.42
7.69
OALG
13,90
8.85
8.44
13.86
8.81
8.44
13,82
8.75
8.45
WPCD>
15.99
10.48
15.95
10.85
10.47
15.95
10.80
WALG
15,99
10.91
10.48
15.95
10.85
10.47
15.95
10.80
10.47
According to Table 4 where OPDD and WPDD represent the results without/with perspective-shared determinants, one can observe that WPDD consistently surpasses OPDD, confirming that the perspective- determinants are ben- eficial for performance increment.
not explore double- dettrmintts producing sub-optimal clustering outcomes.
To demonstrate the space-friendly char- acteristic. we count the memory consumption, as presented in Table 3 where DATI~DATS are the alternative names of datasets in Table 1. One can observe that
Space Overhead:
Similarly, Table 5 where OPCD and WPCD represent the results without/with perspective-speeific determinants sug- gests that the perspective-specific determinants indeed help improve the clustering results.
(1) Our BACDL consumes the least amount of memory on SYNTHREED, BGFEA, AWTEN, HDIGTWO and YOUFOURV. Especially, YOUFOURV, it is clearly the lowest. On FLOEVEN, DEOLOG and YALTHREE, it still compares favorably with the optimal one. This gives evi- dence that BACDL is memory-efficient.
Both Table 4 and 5 illustrate that our dual-determinant paradigm can facilitate the clustering performance.
Besides, we alienate them enhance the discrimination. Ta- ble 6 where OALG and WALG are the results without/with alienation indicates that our alienating action is functional.
(2) USETL, IMVCI and AGCIM achieve slightly less over- head in some cases. The reasons could be that USETL constructs low-rank spectrum and decreases the information redundancy by multi-level partition, IMVCI generates con- sensus structure and skips the procedure of seeking eigen- vector, AGCIM integrates the graph restoration into clus- ter structure and jointly conducts uniform representation construction and similarity completion. Despite resource- saving, they usually do not take into account view-specific characteristics, harming the representation diversities.
Due to space limit, other ablations are located in Section L..
7. Concluding Remarks
In the manuscript, devise dual--determinant exploration paradigm for IMC issues. It bifurcates features clusters and alienates them via mutual exclusion mechanism. To- gether with coupled distribution learning, it effectively al- leviates the dimension inconsistency. Then, it bridges all views through unified association. All sample clusters are re- ordered and mapped to formulate full clustering embedding. The overall linear overhead further enlarges its applicability.
6.4. Ablation
Table 4, and 6 summarize relevant ablation (A/B) results.
8

---

Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Impact Statement
Gu, Z.,, Li, Z.., and Feng, S. Edison: Enhanced dictionary- induced tensorized incomplete multi-view clustering with gaussian error rank minimization. In Forty-first Interna- tional Conference on Machine Learning, 2024b.
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
He, W., Zhang. Z., Chen, Y., and Wen, J. Structured anchor- inferred graph learning for universal incomplete multi- view clustering. World Wide Web, 26(1):375-399, 2023,
Acknowledgments
This work was supported in part by the Research Grants Council (RGC) Joint Research Scheme under grant: N H.KBU214/21, the General Research Fund of RGC un- der the grants: 12202622 and 12202924, the RGC Senior Research Fellow Scheme under grant: SRFS2324-2SDD and the National Natural Science Foundation of China un- der Grant No. 62406329, 62476280, 62441618, 62325604, 62276271.
Li, H., Li, Y., Yang, M., Hu, P., Peng, D., and Peng, X. Incomplete multi-view clustering via prototype-basee based imputation. In Proceedings of the Thirty-Second Inter- national Joint Conference on Artificial Intelligence, pp. 3911-3919, 2023a.
Li, M., Wang, S., Liu, X., and Liu, S. Parameter- and scalable incomplete multiview clustering with prototype graph. IEEE Transactions on Neural Networks and Learn- ing Systems, 35(1):300-310, 2024.
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
Appendix
A. Solving Procedure
split the entire original optimization problem Eq. (6) into nine sub-problems, and manage to provide a solving scheme for each sub-problem respectively.
A.1. E, Sub-problem
Under given C, C,,, D, P,, E, F,. ar and by. the original optimization problem equivalently transformed
min
2
X,G, P.CIC,,DEE G, 1 TTH
E Σ E,
(24)
s.t. E, 2 0, E, n
Owing to the fact that E, on each view is independent of each other, we equivalently have the following optimization problem,
min E,
a G, - DE Bb, Tr E E,F,
(25)
s.t. E, 2 0, E. 1m.
To optimize this problem, we firstly design its Lagrange function as
C(E,, 2,) = a G, C,,CCC, Grlle Bb, Ir E,F,) +Tr Φ, GG, E, 1m,) G. -1n.) 1
(26)
where 4, E R is the Lagrange multiplier matrix.
Expanding F-norm and removing irrelevant items to E,, we can equivalently get
C(E,, = a (G, E, "PC." DE, G, - 2G, X DE G.)
(27)
-5b, Tr E,F.) +h E G, - - E,
Combined with the trace eyelic property and transposition invariance property, we further have
L(E,, k.) = Tr G.G.G.,..D T
T P.CCIC.
(P.CC, -F, E. G. G. - 2G, 1, E
(28)
Let 8C DE
and we have
R.C.D///O.PP.......-///CCCCCCC.. EF GG G. G, 2G, 1n, 1 0.
(29)
Given that D, is a diagonal matrix, we further have
a² G, E,D P,C.C, - C
PPCCC,
(30)
2 D. + G. G. G. - G, =0
In conjunction with complementary slackness condition for non-negative E,, we can get
a P,C.C, [P,CC, dGG,G. ", _..EF.. (Er)i.)
is
(31)
+ (G, E,111 - 1 (Er)..
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Then, eliminate the multiplier matrix 4,, do normalization the of G, E, E R each step optimization. Accordingly, we can obtain that Σ)= (G, E,),, holds for = 1, ... ny where the notation (-),, represents the element located in the i-th row and j-th column of the matrix. That is, G. E, In, holds. Thus, we have
2, G, G,0,1.11.
(32)
Combining Eq. (31) and Eq. (32) yields
- P, E., c P.G.EEDDPP..C.....CC.DC.CCCC..
(E,) D.
(33)
Further, noticed that E and F, orthogonal matrices, and accordingly there exist negative elements in the product of E and
F,, In response, partition EF, into two parts, (EF?) pos and (EF?) neg" That is, EF, (EF,) pos (EF neg where (EF, pos and (EF, neg represent the positive part and negative part of EF, respectively. Then, we have
E. PICC,, G.
(34)
B by
br EF
by
EE
(E, (Er).s 0.
2
ney
in
Consequently, get the following updating rule about E,,,
[,.CGGG///PPCCCC.DD- bb, (EF, pos) in
(Eriss f- (Er)is
(35)
D.C.G.E..D f (EF )eg
A.2. C Sub-problem
Under given E,, Cr, D,. E, Fr, a and by, the original optimization problem is equivalently transformed as
min
- DE, G, A (CT
(36)
s.t. 0. = =
Due to the coupled property, separate out C, first denote D,, Disk,: and D. D Then, equivalently transform Eq. (36) the following optimization problem,
H, - P. C.D.D, E.D.E.G. G, (CT.CC
min
(37)
C... 1t,
where H. = -.D.G.
After removing irrelevant items and looping matrices, we can equivalently simplify Eq. (37) as
min T Σ G.G.C.D.E -2 Σ 2P. &.P.HH.. H.H.H.G.E.D.J CT +Xc C.TCC.) ) (38)
s.t. C C20.C =
Subsequently, its Lagrange function can be designed as
CC, v) Tr Σ P.C.P.C 2LC + M Σv. v. (C 1a. 11) (C 1a. -1a) 1x) 1.) (39)
13

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
where J, multiplier matrix.
E; E
and M Σ.= C.C,. v, E BR kxk is the Lagrange
Combined with the zero partial derivative, we have
d (P P. - 2L + AM + Σ P. 1a IT.. - 2P. 1a, 1,, 0.
(40)
Note that J and , are symmetric matrices, we can equivalently obtain
C.P.PCC, 2L + AM+ Σ
-P
2P, 1a, 11.1, v = 0.
(41)
Combined with the complementary slackness property, we further have
"
Σ
a
L M EP. I,C,VV - -P, 1 V.
C.. = 0.
(42)
ig
Therefore, we can obtain the following updating rule for C,
(L P 1a, 1 1. 4r)
(C), + (C)as
(43)
(E PPP.P.CJ A Er= P. 10, 1 ,,
Afterwards, to eliminate ,, we perform normalization on the columns of P,C so as to make Σ (P,C), equal to I for any E [1,2, .. k.. That is, C I. = 14 holds. Then, we have
1a. P. = 1
(44)
In conjunction with Eq. (42), we further have
(L) (E--1 PPP.P.J, M i.
(C),.) + (C)b.
(45)
Subsequently, based on the facts that
P H. = Σ
(46)
p x, G. X.G.G.GG..DDD Σ E.C.D.DEEGGGG.....
and that
Σ a CJ, + E C.D. E.D. = Σ o PP G.G.
1=1
D.C.D.DEE.GGGGG.......
(47)
DE. D.C.
and Eqs. (42), (45), we can obtain the following rule,
X.G.G.GGG.G........ a
(C)s - (C),
(48)
E P.P DE
M
where M Σi= P.C. C.C.
14

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
A.3. C, Sub-problem
Under given E,, C, Pr, Fr, Op and by, the original optimization problem is equivalently transformed
X,G, DE, G, + C.C
min
(49)
s.t. C, 2, C. = 11.
According to the fact that [+ are mutually independent, we can equivalently transform the above optimization problem
as
min a |, -P,CC G, G + ATT C.
(50)
s.t. C, 2 0, C, la, = 1
Splitting D into D, and Di, D = -irk;; and D = 112kk,, we can get
min a G, P,CD, E G, - E, Grll + (TT(C C,
(51)
s.t. C, 2 0, O. 1d, = 1*-
Unfolding F-norm via trace operation and deleting unrelated items yield
min Tr C,
X,, C.
C, + C.CTP.C.)
s.t. C, 2 0, C, = 1.
(52)
The Lagrange function can written as
C(C,, 2,) = Tr (a, C, J.C. D.X.D.DCCCC........EE.....CC
(53)
C.D,E, G.G.E.D. C. +.C.P. (C, 12, 11))).
where D2 R is the multiplier matrix.
Thus, we have
Ct G.G.D.E...GG.DDD... G.G.E.D. 2,
(54)
2a P. C.D.D,,,GGGG + AP,C 1,,1 C, + 1a, 17, C, 2 1a, 1 =0. 0.
Further, combined with the complementary condition, we can obtain
X
2P,c) (Cr)ss 2 ij
C., G,C.D.EE,GG,EEDD, G.G. + o
(55)
-,,,, (Cr)as + (1a, 17, C, 1a, 12), (C,)is=0)
To eliminate 22, we conduct normalization the columns of C, each step optimization as make C. 1a, 1 hold. Based on this, we can obtain 1a, d, C, 1a, I. Further, we have
A AP.C) D,C.D.E,, G.G. + HP, E. G.G.G.G.. G. E.D.
(C,)ss= = G, E,,.E,DD), (C+)i..-
ij
(56)
Thus, have the following updating rule for C,,
(2X.GGGGEE..)
(Cr)., (Cr)ss (,C.DDEE,GGG,EDDD C.C.DDD,....CD..CCCCCC.CC. +PP.C
(57)
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
Combined with the fact that D + P,CD, [P,C|C,]D, we can equivalently have
1
(, DE + PPP
(Cr)sg + (Cr)ass
(58)
A.4. D Sub-problem Under given E,, C, C, P,, E, Fr, Or and by, the original optimization problem is equivalently transformed
min - [P,C G.
(59)
s.t. 0.
After norm unfolding and unrelated item removing, we can equivalently obtain the following optimization problem,
min (PPCIC, G. G. E.D P.C.C, 2x, G, E.
(60)
s.t. D 0.
By means of the trace cyclic characteristic, we can further transform the above optimization problem as
min
a
(61)
s.t. 2 0.
Let the partial derivation objective function equal zero, we have
Σ a
", X, G,G,G,E, =
(62)
Further, conjunction with the complementary condition, we can obtain
Σ "
" P.I...... G,G, Dig = Σ S.P.CJC.Jx.GGGGI
Dig-
(63)
ij
is
Therefore, we have the following updating rule for the variable D.
Dij + Dij
(64)
E P.C.C G. 1.
A.5. P, Sub-problem
Under given E,,, C, Cy,, D, E, F,, a, and by, the original optimization problem equivalently transformed as
min
X.G. D.C G, + P.CCC
(65)
s.t. P. P, P. 1
Based on the fact that guidance matrices (Pr)"= on different views are independent of each other, we can equivalently obtain the following optimization problem.
min a. X. (P,CIC, G, | (C.P...
(66)
s.t. P, P. =
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
After removing irrelevant terms and combining trace cyclie property, equivalently have
min Tr (d,P, D. -2a [X, C, G.) G, C. +.P. P.
(67)
P. 2, 1d, 1k:
where the last item based on the fact that Tr (CTP..,.) is equal to Tr (P.
Its Lagrange function can be formulated
((,,,I,)) IIr (a P.D, C. - (X, -,,, C.E.D.
(68)
CP +T,(C P. P. P. 1x))),
where 1,, E g& denotes the multiplier matrix.
Given the zero partial derivation, we can get
E.P.CDD,E,GGG, E, - 2a -,D, G.) E,
(69)
+xC,C + la, P,C - la, T, =0.
Combined with the symmetry of T, and complementarity of Pr, we have
(o,,,CD,E,G.G, E. a (X, C,D G.) C. C, +1,,1,, - I.NTI, in
(70)
(Pr)as=0)
Further, we conduct normalization operation on the columns of P,C. to ensure obtain
C C.P.Iaa 1a.
1a hold. Accordingly, we can
a C.C.D.E,C..CEEDD.CC
D, C, CT
+ 2
-C,
A (Pr)ss=
is
(71)
(cX.GGGG,E.D, (Pr)ss
Kindly note that P, E.C.DEE, G.G.G.G,GG,.D,CCCC C.D.E.G therefore obtain the following updating rule for P..
P, CDDEEGGCEEA..oo-PP...ooC......C.
CT, and
(. JJ DEP. + C
(Pr)s (Pr)ss
(72)
A.6. E Sub-problem
Under given E,, C, Cr, D, P,, Fr, and br, the original optimization problem is equivalently transformed as
max Tr
E
b, E,
E
(73)
s. E= I
Let b, equal to UΣVT where U, Σ and V denote its left singular matrix, singular value matrix, right singular matrix, respectively. Then, the optimal solution of E can be acquired by setting E = UVT
A.7. F, Sub-problem
Under given E,, C, Cr, D. Pr, E, o, and b, the original optimization problem is equivalently transformed as
max Tr
E
b
E,F,
F.
(74)
r=1
s.t. F.F. I
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Owing (F,)i=1 being independent of each other, further transform the above optimization problem
max Tr (E F,
F,
(75)
S.t. F.F. Ls.
Denote S as Fr, then, equivalently have the following problem,
max Tr (s E' E,)
S.
(76)
s.t. S, = Ig--
Therefore, the optimal solution of variable F, is UVT where U and V represent the left and right singular matrices of E, respectively.
A.8. or Sub-problem
Under given E,,, C, Cr, P, E. F, and b,, the original optimization problem is equivalently transformed as
2 X, -P.
min
(77)
a, = 0r 2. s.L.
To solve it, we formulate its Lagrange function as
" C(ar, ( = Σ -.KCG,-P.CIC,DDEE G, G +
Ea,-1
(78)
Then, we have
2a, -
(P.CIC.
1 G. <= = 0,r 1,2,..,,, + + ag 0.
(79)
Thus, we can further obtain
Or = J G, [P, [P, I
(80)
Combined with + a2 + ... 1 = 0, we can get
< s= Σ
(81)
D.
Plugging Eq. (81) into Eq. (80), consequently, we have
Eis x, DE, C.
X. G. P.C DE, G.
0r
(82)
X,G, DE.
X DB
A.9. b, Sub-problem Under given E,, C, C,, D. P,, E, F, and o,, the original optimization problem is equivalently transformed as
by Tr E F.F.))
max b,
(83)
s.t.
12 1, b, 20
y=1
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
We provide solving solutions for it. Using Lagrange function method, can get
[(br, n1) Σ by Tr E E,F. +1
Σ b2 -
(84)
Then, have
Tr (E' ( E,F,,) 2nb, 0,
Σ 6 11 0.
(85)
Tr(E Tr F.). 2n
Accordingly, we can obtain by
In conjunction with Σr=1 b, -
0, we have
=
# EE Tr E ())
Further, combined with the non-negative constraint, we have
Tr ( E,),F
br
(86)
E (Tr E EFF))
Using Cauchy inequality (E 2:yii? & (Ei 2²?)
(Ein vf) we have
2
Σ by Tr (T
s Σ Tr E T E,),)
(87)
where the equality holds if and only TT(E E. F.) THE F2F2)
Ty( F.)
by Let THE F,F,)) where is constant, and then we have b, = cr E E, F.). Further, according Σ b2 = 1,
we can obtain Σr=1 d² (Tr (E T E.F.)) 1. So, c=
Accordingly, the optimal solution b,
v ViL (Tr(E E.F.))
Tr(E.,,F.) EE (T(ETTE
B. Proof of Theorem 1
Proof. During optimizing E,,, on the basis of Eq. (28), for the term Tr (EF, E., and F,. the elements of EF, could not non-negative. view of this, we split EF, into (EF, pas and (EF) )negs i.e., EF, (EF,))m pos- (EF, neg where (EF,) pas and (EF.) Jneg denote the positive part and negative part of EF respectively. Therefore, have Tr EF.. E, Tr (EF.) E,) -Tr (EF, E.).
due to the orthogonal properties of
ppos E, combined with the non-negative property of E,,, have For the term Tr (EFF)) pos
1 log (E, )a.
Tr (EE pos = Σ (EE)),,) (EF
N Σ Er)as
(EF))
poss)
(88)
pos i.j
(E, )a.5
where the inequality holds based on the fact that 2 + log x which x requires be non-negative. Then, for the term Tr ((EF.) Ineg
) E.T.), via element expanding, we have
EE )3, + (Er) (E,)
Tr (EFF)) E,
Σ EEr)ss ((EF)) neg)
S 2
[(EF,)
Σ EFF
(89)
neg ij
Therefore, for the term Tr [EF E..)) we have
Tr E.. > Σ E.) (EF)) poz))
EE)is
Σ [(EP.) )ee) i.
1E,) + (Er)
1 log
(90)
E
)is
Eg)is
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
Subsequently, for the term Tr GG, P., in Eq. (28), in conjunction with the non-negative property of G, P.C/C. obtain
Er
Tr GG
2 G.
1+ log (E,)..s
(91)
For the matrix G, n. 1 E,, likewise, we have
(E))is
Tr (G, 1n, E. Σ E Das (E In. 1n.11)) 12ss
(1+lgg
(92)
(Er)is)
Further, for the term Tr P.CCC,..DPPCCI...PPccCc..// by utilizing element unfolding and matrix symmetry, we can get
Ir E.D P. P.CC
[PC) Σ E GG. (B,s. (D') P.C|C,] P..CCCI....P.CC..D........... P.C. (Er)i,
EE (E,) PPCC, C.C.C...........................
(93)
Suppose the elements of E, are related to that of E, with certain scale factors, i... (E,)..s = 2i, (Er)es where 2nd is a constant, then we have
(E,)i,
G.G
S
(94)
+ 2i.s)
and
(95)
(2ig z..)
Further, using element folding, we can get
P.C.
(96)
(E,)a.t
and
(97)
Σ
(E,)ss
Additionally, for (.. D.C.GGCC..........P............. in virtue of element transfer and
20

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
element folding, can get
Dac.c........................... D.G.C..............R..C........D (Er),
(98)
(E,)i
In conjunction with the element relationship between E, and E,, we can further have
(Br),,
(Er)?
P.CCC.
(E,).,
(99)
E ad E, )i.
P.C.C.E..
Similarly, for the term D.GGE we can get
E )
(100)
GG. E.D. P.C.C P.CCC.......................
(.CC,,
(E,)...
Combined with Eqs. (93), (94), (95), (96), (97), (99) and (100), we have
Tr (G,
G. E ]. IP,
(2 + 22,5) +
(22, + 22,2)
(101)
(E,)
I.G.E...D...00///P..CC.....
(Er)i
(E,)
P.C.
EE,),t
Consequently, we can obtain
(G..E..DT...CC///RRCCCC.......
Tr (G..G.E.D P.CCC./
Σ
(102)
(E,)as
For the term Tr (G, d G. E, E.) in Eq. (28), we have
(Er)
Tr D. C. = G.
$ G.G
E )i.j (103)
21

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead Combined with Eqs. (90), (91), (92), (102) and (103), therefore, have
(E
F(E,, E,) u D.G.
(E,)a,
EE ).. (E, (..)
2a
E,
log
Erjij
3b, Σ (Er)
(EF))
1 + log
pos 1.j
E Err),
(104)
84 +
E 3r, + (Er)i
Σ
(EF))
2
neg
1.
Bijis
(Er)
EG, E.1
(Er)is
E, Ji.
2 > (m, 1 log
E )a.j
According the bound of each branch, have that F(E,, E,) 2 (E,)) and F(E,. E,) ((E,). Therefore, an auxiliary function of ((E,). Further, its Hessian matrix is semi-positivee and consequently is convex. The global solution can be acquired by setting its derivative equaling zero. That is,
//G.C/ Go, EF pos + 1.1
(Er)is
1.
(105)
[ECC.//E.DTPP..C.TTP.CCCC. b, (EF neg G G, E, E,
After eliminating 2,, we can equivalently obtain the updating rule Eq. (8).
C. Proof of Theorem 2
Proof. Let E (h+1) = arg ming, F(E,, E..)) and then we have F(E! ( ( s F(E E(A) In conjunction with
Theorem 1, have that F(E,, E,a)) convex and the global optimum be achieved. Further, combined with the definition of auxiliary function, have
TTE,,,,,,M L(B(())
(106)
and
F(E(++1), E((A)) 2 ((E(A+1)).
(107)
Therefore, we can obtain
((E(++1) < L(E."),
(108)
which demonstrates that is steadily decreasing.
Accordingly, we have the following inequality holds,
I E (1+1), ( C.N) D (K) (h)
E (h) F( a,h)) BBh) IJ (E,)) (()) c(.)
D (h)
((h)
E
(())))))))))))))
a.))) B(h))
(109)
where J is the objective value of Eq. (6).
22

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
D. Proof of Theorem 3
Proof. For the j-th column of P,C, we have
k
[P.OO
Σ
Σ
(Pr)a Ci3
(110)
i=1
1=1
After exchanging the order of summation, we can obtain
(P Σ
Σ (Pr).t) Cij.
(111)
Since P, is column-normalived, we can get that Di=1 Pr)it = 1 holds for any I (1,2,..,, ,k]. Therefore, we have
Σ
(PPC) Σ Cij-
(112)
1=1
1=1
So, to ensure that the sum of the elements in each column of P,C is equal to 1, under column-normalized P,, we only need
Σ
Ct. = 1, j= 11,2. .
(113)
1=1
That is, C also needs to be column-normalized.
E. Proof of Theorem 4
Proof. For the objective function, according to F-norm characteristic, we have |, D, G. 2 0. Combined with the non-negative property C and C, as well as Pr. we can obtain C,, 2 0. Therefore, we have that the objective function is lower-bounded by -3 Tr (E" Σ F,).
Finding the lower bound of Tr E To obtain one upper bound of Tr rrll b, F firstly, based on the linearity property of trace operator, we can obtain
b, F.) is equivalently to find the upper bound of Tr ( E
b F,).
Tr E
b E.F,
Σ by Tr E E,F,)
(114)
Then, utilizing the cyclic property of trace, can get
Tr (E E,F,F Tr (F,E EFF.EE)))FFEEE..)))
(115)
The trace of a product of matrices is maximized when the matrices are aligned a way that maximizes the sum of their diagonal elements. Combined with the fact that and E, are orthogonal, therefore, the maximum value Tr (F, E,) occurs when E, is aligned with E and F,..
To demonstrate this point from theory, firstly decompose E, UΣVT where U and V are the left singular matrix and right singular matrix respectively, and Σ is the singular value matrix with elements greater than or equal to zero. Then we have Tr F, E E,) = Tr [F UEVT). Further, via cyclic operation, we have
(F,E E.) = Tr (V F.E.
(116)
Kindly note that V. F,., E and are all orthogonal matrices, accordingly, V F,E also an orthogonal matrix. Therefore, we have
Tr (F. < Tr (>),
(117)
where the equality holds if and only V WVT.FFF F,E U equals an identity matrix.
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
In conjunction with Tr (>) Σ mmin(a.k) (E,)) where o, denotes the i-th singular value of E,, we have
ii=1
min (n.k)
Tr <
min ki
of (E,)
(118)
Then, given the fact that of (E,) is equal to AE, E,) where AE, E,) (E, E,) is equal to Tr(E, E,), we have
denotes the i-th eigenvalue of E, E, and that
min k)
TrEE, E,) = (E,))
Σ 1=1
(119)
Further, considering that || equal to Tr(E, E,) and that E, is normnazmalized and non-negative, we have that C) is less than or equal to I for any i = [1,2, ... .n). Consequently, we can obtain
Tr(E, E,) S n,
(120)
where the equality holds and only if each row of E, is a one-hot vector. can be further derived that at this point E, is (one-sided) orthogonal matrix.
In conjunction with Eqs. (117), (118), (119) and (120), consequently, we can obtain
Tr (F,E E,) & minnfn, k)n.
(121)
Since the sample size n is largely greater than the cluster number k we have
Tr (F,E E,) S Vnk.
(122)
Combining Eqs. (114), (115) and (122) yields
Tr
E T b, E,F,
S Vik Σ by.
(123)
Afterwards, using Cauchy inequality, we can get
Σ
br
Σ b2
Σ 12
(124)
Accordingly, combining Eqs. (123) and (124) yields
Tr
E
b E,F,F S Vonk.
(125)
Therefore, the objective value is lower-bounded by --Bennk. That is.
Σ 2 -.GC..P.CCC.DE.. G, C,P,C,, 8TH 1
E Σ b, E,E,F, 2 2-BVunk:
(126)
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
F. Proof of Theorem 5
Proof.
During optimizing C, based on Eq. (39), we can equivalently rewrite its Lagrange function as
Σ a C.P.P.CCDDEEGG.....,,..
- -2' G. + A.EP.CCC
v) L(C,
Ir
2 P. C. C² +
- P. V.
r=1
(127)
For the matrix Σ E, combined with the property of each matrix, can get that it is element- wisely non-negative. Meanwhile, unfolding the trace by element, we have
e Tr
Σ C.D.J.CGCC,,DD,CCC
=Σ
Σ X.
G.
C₁.
ij
(128)
Cif 1og
Σ
Σ P. E.D.
Či.
1 log
Ci.
bd
For the term Tr(P. 1a, 1 C, similarly, can obtain the following inequality,
Tr IP. 2 Σ (..11.11..) vr),s Čis (1+ log Cúj Ci.
(129)
Then, for the term Tr P,C.C'C ), we can get
Tr Σ P.T.C.C
1 Σ
Σ P.C.
C², + C
(130)
Ci.
ij
Accordingly, for the term Tr
C.P C,D G. G.G.D.D...CC.. ). we have
Tt Σ a P. C.D.E.E,GGGG.........
"
S
1
Σ
"
a c C.
C + +
(131)
is Ci.
Afterwards, for the matrix P. P, CD, CD, G.G. suppose C.j fer for any and j where faz is constant, then we have
Tr IIP.P..DDDEE..C.......... = ΣΣ (P..P.) C.2 (, C.S
ΣΣ P.P. (D,EE ),,, Čxsfrafas
$ ΣΣ (P. C. (D,E, C. Čos (Pr + faá).
(132)
Subsequently, based on the commutative property and symmetry, we can get
ΣΣ (P...).., Cu (D.E,.GG,,E,,,, Easlh. ΣΣ
C., (D,,G,,, 14 C.
= Σ P. (PP.,.D,,EGGG,EEDD.....
Ōufi C₁,
Σ C. C.,
(133)
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
and
C
ΣΣ (P, P.) C.2 G. G,, Cessas =
- EP C,,.P,CD,EGGG.EEDD.
(134)
C..
Consequently, combined with Eqs. (132), (133) and (134), we can obtain the following inequality.
C
Ir Σ P.P.P. CD,.D. E.D.
a Σ (,..
(135)
C
r=1
For Tr (P. ld, 1 ,,C,C,C C. after element unfolding, we have
Tr (P, La, 1 P.C,VC") ΣΣ (P. La, Pr), Cu(V,
Σ Ia 1. C,,(Wo),
(136)
1a I.),, C,.(V,)) Ci. ( + fi.))
For the first term, we can get
EP. Cr.(V.)isCrsfl Ep..Lu,,,, is C.d
(137)
Further, in conjunction with the fact that v, is diagonal matrix, we can derive the following equality,
ΣΣ (P. la i, C Crpsi
ΣE (P. 1a 1 Pr), C,)())) O.gh
= Σ Σ la II P.)
(V. C..f7.
P, = Σ P P. P, CV,
Cuft
(138)
C2
= Σ PP J. P,
1.1 Cr.
Combined with Eqs. (136), (137) and (138), we can get
C²
Tr (P.
S ΣΣ (P. I,P,V.V) ij C..
(139)
So, this point, according to Eqs. (128), (129), (130), (131), (135) and (139), we can have
C²
F(c.ū) Σ x Σ
P P. P.P.C.DDEEEGGGEEDD
1s Ca.
Ci.
2
Σ
a G.G,G,,.DD)
P
Čis
1 Ci.
in
C2 Ch
C.D G.D.E.GGGG.GG P
Oid
id
(140)
Ci
ld, 1.
(T.TT,)) Vr) Cis
1+
Ci.
C²
P. Id, 1 I.P..OV.)
1. Ci.j
C² + +
Σ P C
Cis
i
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Accordingly, F(C, C) an auxiliary function, and under the updating rule Eq. (11), the loss is monotonically decreasing. Therefore, get the following inequality.
J E C
D(h) J (D C C D
C(A)
D
F(!) a b(Th) E(h) F(J (M) (gh)) p
P(A) E(h)
(141)
During optimizing C,, after removing irrelevant items, the Lagrange function can be simplified as
(2,) Tr C, 2a E, P. + Q2,C 1., - 22,1,,
(142)
G,
For the matrix E, C. in conjunction with the non-negative property of C,, we have
Tr (XX.G.GGE.D..C). > Σ I(X.G.C,E.D.), (Cr)is 1 log (Cr)is
(143)
(Cr)
For the term Tr C, la, 1, according the eyclic property of trace operation, we have
1+ log (Cr)ss (Cr)ss
Tr (2, la 17) Tr (1a [2,C, 2 Σ (1a, 24)ss (Cr)is)
(144)
For the matrix G. CT we have
Tr GP, E, & E.G. (C.) (Cr)
(145)
2(Cr)ss
For Ir P.T.C. combined with the transpose property of trace operation, we have
+ +
TTC.PP.C.) Tr (C.P.C) & E(P.C)i
(146)
2(Cr)is
For the matrix C G, G, E. after element folding, we can get
D. CT suppose that for any i and j. (Cr)s, = wss wry((, (Cr)a.s where wa, is a constant,
Tr (C. G.G. D.D.C. C.D.
(147)
$ S ΣΣ C_
+
w
Combined with the symmetry, we can further have
ΣΣ (D.)., E. E.G.E,DDD). =
C. C E. (C.)i,
(148)
ECC.DEE (C,)
(C,)i,
For the second term in Eq. (147), based on the element merging. we can get
-E Cr)ijur (C,)²,
(149)
CC by (C,)
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead Combining Eqs. (147), (148) and (149) yields
(C,) Tr (C, G. C.D. EΣ [(..E.,....GG..
(150)
(Cr)
For the term Tr (,,C,, 1d, 17, Cr). we have
Tr (12 C, 1d, 1d, C,) - Tr c,0,,))) (1
(1,,1)),,, (C ΣΣ (Ld (L,, (Rt))))))) Cr)ssureuss
(151)
$ ΣE (La, (Q.)),
+ wtz)).
For the term ΣΣ (1s, 17),, (2r)))) (Cr)), (w in conjunction with the element merging property, we can obtain
E1(, (h. = E (, (Cr) = E (1.,11, (C
(Cr)
(152)
According to the facts that 1a, 12, is symmetric and that 22, is diagonal, we can get the following equality,
ΣΣ (1a, (1), (C.haal ΣΣ (1a, 12.).. (R))),
(153)
Further, conjunction with the element commutative law, we have
ΣΣ (1a, (C))) E (1a, 12),, (2))
(154)
Using element combination property yields
ΣΣ (1a, 11.1);1; 1,
ΣΣ
1a, 17, C,0.) (C.)uuuf
ΣΣ
(1d, 1. C,0,, (C+),,
(155)
ΣΣ (1d, 1., C,, (Cr)
1a (Cr)
Based on Eqs. (151), (152), (153), (154) and (155), therefore, we have
(Cr)
Tr (2, 1a, C,) < Σ (I.
(156)
(Cr)ss
Combining Eqs. (143), (144), (145), (146), (150) and (156), we can get
(C.) 20, Σ E( (C,)i. 1+ (Cr)ss
F(C,, (,C,,č)= = > G.G. 19 (Cr)..
(Cr),.)
+ (.P.C.DDE,,.CCEE....... (C,) + (Cr)s A ((,,)) (C + (C,)), (Cr)ia
(Cr)is
+(I.,,,,..., (r))s
(C,) 2 2 112r) (Cr)is
(Cr)
1 log
(C )
(157)
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
F(Cr, C₂) is an auxiliary function and with the updating rule Eq. (13), the loss value decreases monotonically. Consequently, obtain the following inequality,
E 1
C(++1)
C( D P(h)
E F aaah,,)4 ..th)) P( E(A) F()) (h) (6th)
(158)
E 1
C(++1)
C D (())
During optimizing D, owing to it being non-negative, get
Tr
Σ a
E.D"
= =
Σ a
Dad
iJ
(159)
Dij
2 Σ
a P. J,G,,,
Dis
1 + log
D Dij
i.
Suppose that for any and it holds for D,, ds,cDa,t under constant ds Then, utilizing element-wise expanding, we have
Tr [PP.CCC//BBCCC..DEE........
ΣΣ E (PPCIC.) D., (E. G,G. Ex), Dij= ΣΣ (P.CIC,) D., (. G Er),, 1s Diydaidis S
(160)
ΣE (P..IIC) Das (E, G, Diy de,).
For the first term, we have
ΣΣ (PP,CIC. D., GE, D.),
Σ (P.CIC. in
Diyck
(161)
D
E (IP.IIC.TP..C.DDE,.....
19 D
Further, combining the symmetry, we have
ΣΣ (P,CIC,) (PCCC,) D., (E. G, G, D.
ΣE (P.CIC. D.g (, G.G.G.G.. E,) jit
D Dh
(162)
D2
Σ (PP.ICC.TPP.CC.DDDE..G.C..
1g D
Combining Eqs. (160), (161) and (162) yields
D
(IP.C.C./TP.....///C..EE.E (/Pp://))))(/(.CCC///../...CCCC G.G.
(163)
i.j D
Therefore, we have
D D (164)
Tr Σ 2 C.P.P.C...T..CCC.DEE.......
Σ a P...II..
r=1
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
Based Eqs. (159) and (164), can get
D
F(D, = E 2 [P..IC./TP..CC..DDDC...G.
Dij
(165)
Dij Dij
2 [,, X,,,C,, E, Da.
1
So, F(C, C) is an auxiliary function. Under the updating rule Eq. (15), the loss is monotonically decreasing, As a result, we have the following inequality.
J E C(++1), cl++1), D(++1) P()) E(M) F()), a( alM,...N) (M)
(166)
I E C(++)), C(++1), D(A), P(N) E(h), F(h) a b(m)
During optimizing Pr, the Lagrange function can be simplified
c(Pr,T,) =r (a,, CD, G.G 2 C,D, E. G,G. C.
(167)
P in conjunction with the non-negativity of elements, we can have the Then, about the matrix G.G C.P following inequality,
Tr EXD,G,G,E,D,,CPP.)) 2 E.X,C, 1s (P.) in (Pr)..)
1 log Priss
(168)
For the term Tr C. after unfolding element by element, we have
Tr U.P.D.EE,GGG,E.D.. ΣΣ (P, (,,, )1. (P)
n.19t.
(Pr
E.C.CD...GGG.E.D.. 1, 2(Pr))..
(169)
ΣΣΦ, (., G. G. G.G.G.D...C ). (.,(P)...
42
2
where P, is acquired based on the assumption that (Py) i is ar, times of (P r), for any l and I. 4r,1 is constant.
Further, for C.P,, )r. (P. get
Lh 2 by element exchange and combining the symmetry, we can
z C,C
(D,
(P,((,
(170)
(Pr)
G,G.D.EE 1. P(Pr)
Therefore, for P.C.D.E we can obtain the following inequality,
(P,)
T.I.C.D.EEG.C,E.D,.CEEI. s ((P,CP,),)((C..pppppp. E[.,,CD,EEG.GEEDDD.CC i (P,)
(171)
For the term C.D.E.G,GG. E, P.T. combining Cauchy inequality, we can get
(Pr) + (Prils
Tr G.G.D,E, s Σ G.G.D.E,GG,GEDD,,,,,. ig (Pris
(172)
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
For the term C. likewise, have
TT - EE(5)) (c,c')),, (Pr) + (P-))
(173)
(Pr)s.
Subsequently, for term P. 1a, 1 P,C. utilizing cyclic property and element-to-element expanding, we can
obtain Tr TT PPC P.C) Tr (1a, [L..I.P.CC,,CP,,,
Σ (I. (P.)i.
1. 2(PT)
(174)
(Ld, (Pr).s (CT,C") (Pr)es L.
For the second item, combined the symmetry, we can derive the following equality,
(1d.1d)) (Pr) (cr,cT),, (Pr)ss R. ΣΣ (1d. (1d.1))),, (Press (CT, (Pr)., R..
(Pr).
(175)
(P (Pr),s
Therefore, we have
(Pr).
Tr Ia s EΣ [I..1fP.cr,.")
(176)
(Pr)is
Likewise, for P. 1a, I have
(1 1 1+log log (Pr)is
Tr (TT.CP,.11.11) = Tr (1. 2 Σ (14 I (,, (P.)).. in
(177)
(Pr)s.)
Hence, combining Eqs. (168), (171), (172), (173), (176) and (177), can have
(P,)
(Pr) + (Pr))
F(P,, P.) = P,C.DE,G.C,EDD,CC..
E(C.CT),,
Pi.
2
(Pr)is
P (P)) + (Pr)e
(Pr)
a Σ (CD.E.,,GGEED,..,..,,. ij
Σ 1 [1,.I.PP,C,,TT
(Pr)ss
1. (Pr)ss (178)
(Pr)ss
20 Σ [(X,C,,GE,DDD (P.). in
1 +
(Pr)as
2 -2 (1..111,C))
(P.),.g
1 log
(r))s
P (P))ss
Consequently, F(P,,P,) ).) is an auxiliary function, and under the updating rule Eq. (17), the loss function is monotonically decreasing. Accordingly, have the following inequality,
T E (h+1) ((++l) ((h-l) D (h+1), 1(h+1) (A) (h) ( ((h)) S (h)
(179)
I (l+1)) ((++1) ((h-1) ((++1) (()) E p(() a b(h)
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diagonal ma- Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
During optimizing E, since its optimal solution can be acquired via singular value decomposition, we can get that after each iteration, there always has
E B C(++)) cf D(++1), p(h-)) E(++1) F (000))0) I
(180)
E (1+1) C(++1), c(++1,, D(++1), p( E(h) F (h) 6(hh)
J
During optimizing Fy. its optimal solution is obtained by Eq. (76). Therefore, the objective value is decreasing when optimizing F,. Accordingly, we have the following inequality,
J E 11+ ((++1)
(( D ((++1), 1(++)) E(++1), F a, (h) , T B ( g(++1) C(A-1), (1+11) p(++1) g(h+1) F( ath) blh))
(181)
During optimizing a,, according to Eq. (82), its solution can be directly obtained. Therefore, we have C.++), ((++1), P((+)) E(++)) F(h-1), aafh+111, (BN) E(++1) F(h-1), 1) (M)) 6(T0) E 1 (h+))) C ( (l++)) p((+))
J E 1
C(++),
(182)
During optimizing by. its optimal solution is acquired by Eq. (86). Consequently, we have
J E (((+1) (1++1) c(f-1), D (l+11) p(t+1) ((++1), p(1+1), 11+11,, ((1+12)
(183)
J 1( (h++1) c(--1), (1+1), p(h+1) 1( p(++1), 1h+1), (9i))
Combining Eqs. (109), (141), (158), (166), (179), (180), (181), (182) and (183), we can obtain J E 1()+1), c(++1) c(h-1) D (1++1, p(++1) E(h+1), p(h+1), a I
(h+ 1), blh+11)
(184)
(b) C( c(h) ( ((h) E(h) F(h) (h) B(h))
which indicates that the objective value is monotonically decreasing with each iteration of the algorithm.
Combining the monotonic descent characteristic (i.e., Eq. (184) ) and the lower bound characteristic (i.e., Eq. (126)), can conclude that the proposed method is convergent.
G. Proof of Theorem 6
G.1. Computational Complexity
Proof. The computing cost of presented algorithm primarily comes from solving E,, C, C,,, D, P,, E, Fr, or and by.
During optimizing E,,, note that G.G.T is in Baxx, directly computing the term G. P. will require at least O(n²) cost, To decrease the computational cost, we observe that G. trix. Specially, its diagonal elements consist of "),((, (,
EE= (Gr) We can equiv- alently transform the term G, G.G. [P, J (X, O O,) [,,C/,, where O, Rd is equal to 1a, on this, computing G, G. C.P.I.
E (Gr).. ...
and O denotes the element-wise multiplication operation. Based C,C.C, D will take Old,n 2nd,k + 2nk²) cost. In the similar way, we can obtain that will need O(nk + 2nk22 + 2nkd,) cost. In addition, computing EF.
computing G. E,D will need Onk)²). Therefore, optimizing each E, totally requires O (mdd, + r2k²) cost.
During optimizing C, adopting the element- wise multiplication technique employed on E,, we can obtain that computing X,G. G, E,D., and C,, DE, G, E.D.D. will require Od d + d2k²) and (22, + d, d, k.2
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
respectively. Accordingly, computing | G, and Ex= Oddnn + dk2) and O(dk² + dkn) respectively where d is the dimension sum of all view data. Therefore, optimizing the matrix C totally requires O (dk² + dkn) computing cost.
_ df G, G, will require
During optimizing Cy,, computing the term X, G, requires Old,n+d,nkt computational cost. Computing [P, G, G, requires O(nk 2d,k² d + d2,²). Therefore, optimizing each C, totally requires O (d, + d,rh) cost.
G.G. E take Old,n + 2kd,,n + 2k²n) and O(nk + 4x2d, + 4k 2nk2) respectively. Consequently, computing P,,C,,, G.G. E, takes O(kdn+ +okn) and O(12+
During optimizing D. computing the terms E,CIC,",,,,.C E,T. and P.C.C,
Σ Σー ai.a, X,,,,, G, E, and Σx= af wk + vmk²) respectively. Therefore, optimizing D totally requires O (kdn + vk h vk + vnk").
During optimizing Pr. constructing E, and C, require O(d,n + d + d and Old,k? com- puting costs respectively. Constructing |P, G. C. requires Onkk 2d, d,nk). Therefore, optimizing each P, requires Ok, k2) cost.
During optimizing E, it involves conducting singular value decomposition on Er F, Constructing the term E'= b, F, and performing singular value decomposition on it require O(unk2) and O(nk²) respectively. Performing U multiplying V to generate the optimal solution takes Onkk2) Therefore, optimizing E totally requires O (wnk²) cost.
During optimizing F,, it involves conducting singular value decomposition on E E,. which requires O(k*) computing cost. Additionally, constructing the term E' E, requires O(nk²) cost. Therefore, optimizing each F, totally requires O (nk² + k3)) cost.
During optimizing a,, it involves calculating the term X,G, - C,,CIC, DE, Rd,xn and Raxx= respectively. Directly computing |X - DE, G Note that X, and G, are in DE, G will require at least O(m,,) com- putational cost. Especially, when the incomplete percentage is relatively small, i.e., ny is larger, the cost is almost close to (n²)). To reduce the cost, we observe that the value of X, [P, JIC.JDE, Gr f is the same as that of |.G.G,T [P,C G. F. In virtue of the element- multiplication technique adopted in analyzing E,,, we can obtain that computing the term |X [P, G. requires Od kn + 2d, + kd, cost. G.II. requires O(n + vkn + 2dk2 + dkn) cost. Therefore,
Accordingly, computing the term Σ optimizing all a, totally requires Ovknn + dk2 + dkn) computing cost.
During optimizing by. computing the term Tr E,,,F will require O(nk² + A-2) cost. Accordingly, computing Σ Tr (E T F,F,)) will need O(unk² + vh²). Therefore, optimizing all b, totally requires O (wnk² 2 + ok com- putational cost.
Based the above analysis, can obtain that optimizing all E,, C. C., D. E, F,, a, and b, will require O (ndk- + vnkk), , O (dk²² + dkm)), O (dnk + dk²). O (kdn + vh2n + k2d + +2d+uk3+ + unk"), Oldnk + dk-2,, O (unk²). O (onk2 2 + vk²). O(kkn + dk. + dkn) and O (unk² + =kk3) respectively. Therefore, the total computing cost is O(ndk: + onk" dk2 + vk=)). Considering that the view number v and the cluster number l are generally greatly smaller than the sample number n and that the dimension sum d is independent of n, we can obtain that the total computational complexity is On). That is, the computing cost is linear with respect to the sample size n.
G.2. Space complexity
Proof:
During optimizing E,, generating the term G.G. P,CCC, needs O(d,n + 2d, + 212 + 2nk) memory needs O(nk + 2k + 2kd, + 21:²d, + nkd,)) cost. For
cost. Similarly, generating G, G.G.E,E,,, the term EFT, it needs Onkk + k22 + nk?). Therefore, during optimizing the variable E,, it requires Okk2d, + nkdd, + nk?) memory cost.
E.D. needs tn +d,kk) memory cost. Generating the term During optimizing C. generating the term * [ DE, G. D. needs 2d +2 + d,,n) cost. Consequently, generating E G, G. and D C,,,, G, E.D.D. will need Oddn + Therefore, during optimizing C, it requires Odn + onk + wh² + dk) memory cost.
+ +dk) and Ovnk 2dk + 2nk + +dn) respectively.
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
During optimizing C,, generating the terms X,G, D and G, needs O(d,n + nk + k² + k, and (2 + 2Λ - nk + d,n)) memory costs respectively. Therefore, during optimizing C,, it requires Old,n + nk + k2 + d, memory cost.
During optimizing D, generating the term P,C,C,T,,,,C ory cost. Generating P. P,CCC G, E, needs O + 4h2 + 2nk). Accordingly, generating G, E, will need O(dn + 2dl + 2unk + 2nk²) and (2dk + 2 + 2unk) respectively. Therefore, during optimizing D, it requires O(dn - dk + unk + vk²) memory cost.
needs Oid,n 2d, 2nk + 21²) mem-
E C, G,G,G, E, and PPCC,
During optimizing p., generating the terms G, E, D. CT and C,C needs n(k +d k) and (d, needs Odd + 2h + nk + d,n).
memory costs respectively. Generating the term [P, DDE G, E, G, Therefore, during optimizing P,, it requires (d + nk 12 + d,kk memory cost.
1= E, will need Ovonk + ) Singular value decomposition and singular matrix multiplication will need O + k2). Therefore, during optimizing E, it requires O(unk v memory cost.
During optimizing E, generating F, needs O(nkk + 1²2 memory cost. Accordingly, constructing Σ
During optimizing Fr, generating E E, needs Onk + k2) cost. Performing singular value decomposition needs O(k2). Therefore, during optimizing F,, it requires O(nk + k²) memory cost.
During optimizing 0,, instead of calculating the term | DE, Gr we equivalently calculate the term G. - -P, G. which requires Od kn + kd, memory cost. Accordingly, generating En G. [P, JE, G. requires O(dn + vkn 2dk) cost. Therefore, optimizing all a, requires Oldn + vkn- dk) memory cost.
During optimizing br, constructing the term E E, needs Onk + k²) memory cost. Accordingly, constructing Em (Tr (T<(EEE,) (E), will need Ownn + wh??). Therefore, optimizing all b, requires O(unk + vk?² memory cost.
Consequently, we have that during optimizing all E,, C, C, D. E. E. a, and b,. it will need O nkd + vnk?), O(dn + unk + uk2 + ddk), O(dn + vnk + uk? + dk), Oldn O + m,²), Odd + vkn + dk) and O(unk: + vk²) memory costs respectively. Accordingly, the total memory cost is O(k2d + nkd + unk?). Owing to the data dimension sum d being independent of the sample size and the view number and the cluster number being far less than n, we have that the total space complexity is O()). That is, our memory cost is linear with respect to the sample size n.
vnk + vk²), O(dn -unk + vk + dk), O + vk?),
H. Experimental Background
H.1. Data Preprocessing
In experiments, each (observed) view data is normalized probability form. Specially, first map the data to have mean of 0 and a variance of 1 so as to prevent certain dimensions from encountering excessively large scales, and then employ the softmax function to transform negative elements into positive ones. Subsequently, we conduct normalization operation and transform each view data into a probability distribution. Afterwards, all comparison algorithms are evaluated on these preprocessed datasets.
H.2. Parameter Setting
C, Owing to the elements in X,, G,,, Pr. C. Cr, D, E,,, a and b being between O and 1, we can obtain that the terms
af | JPC, I and (P,C, C.) (i..., Tr(C P. C₂) are less than 1. Additionally, combined with the orthogonality of E and E,,, we have that the value of Tr (E Σr= E,,,F..) is generally much larger than that of a² |X, ..P.CC,JDD. Ip and that of Tr (CTP.C,) Therefore, we fine-tune À in wide range while fine-tuning 3 in relatively narrow range. Further, considering that Tr (E Et is upper-bounded by Vunk and that the sample size n is largely greater than the view number and the cluster number k; we search À and / in [10-2, 10-11, 10º, 10 102] and [10 10 100 100, 10 102]-
Venk respectively.
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
H.3. Implementation Details
Since P,C denotes the feature cluster, we normalize it column by column. During optimizing P, and C, we do column normalization them respectively. Similarly, C, denotes the feature cluster, and we also normalize it column by column. Additionally, E, denotes the sample cluster, and thus do row normalization on it. D denotes the association between feature cluster and sample cluster, and we do distribution normalization on it.
To evaluate the clustering outcomes from various perspectives, we adopt three commonly-utilzzed metrics, purity(PUR), accuracy(ACC)) F-score(FSC).
I. Symbol Description
Table 7 summarizes the symbols used in this manuscript and corresponding meaning well the size.
Table 7. Symbol Description
Notation
Description
Size
X,
Data matrix r-th view
d
P
Guidance matrix on r-th view Perspective-shared matrix Penspective-specifie matrix Association matrix between feature clusters and sample clusters Feature clusters on r-th view Index matrix r-th view Common feature clusters
d, x k
C
k x k
C
d, x k:
D
21 x k
E,
le
G
E
k
Fr
X 1 Space rotation matrix on r-th view
x k
a
View weight vector
b
Sample cluster weight vector Dish::
X 1 x & k My Yhy k x k
D,
D
D
4,
Lagrange multiplier matrix
V
Lagrange multiplier matrix
Ωr
Lagrange multiplier matrix
k x k
P,
Lagrange multiplier matrix
x k d, x Thr
H.
Temporary matrix
J.
Temporary matrix
k
L
Temporary matrix
k
M
Temporary matrix
k
A.
Temporary matrix
d, x k:
B,
Temporary matrix
X n
C,
Temporary matrix
k
K,
Temporary matrix
n X k
L,
Temporary matrix
d, x 2k
d,
Data dimension on r-th view
d
Data dimension sum
v
The number of views
n
The number of samples The number of clusters The number of samples observed on view r Matrix concatenation Element in the i-th row and j-th column of matrix.
k
Tr
[1]]]
(.)y
35

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
J. Hyper-parameter Sensitivity
In the paper, our model involves two hyper-parametersy and 3. To investigate the model sensitivity about hyper-parameters, we plot the performance changes with respect different hyper-pane parameert values, as suggested in Fig. where for the sake of simplicity, we in figure omit the coefficient 1/ 1/vnuk. One can observe that under given the performance relatively stable with respect to À while under given X, the performance fluctuates within tolerable range. Therefore, that the proposed model to certain extent is parameter-robust.
state
PUR
PUR
PUR
PUR
00.00
20.20
10F
10.00
104
10t
10
10"
101
100
129
10'
100
129
10
(a) FLOEVEN
(b) SYNTHREED
(c) DEOLOG
(d) YALTHREE
PUR
PUR
12.00
36.00
21.00
10:00
11.00
19.55
21.00
10.00
11.05
15:40
167
16
tor
10a
10²
10"
tort
0
16
10'
10"
10
fyy
(e) BGFEA
(f) AWTEN
(g) HDIGTWO
(h) YOUFOURV
Figure Performance Changes under Different flyper-parmmeet Values
K. Experimental Convergence
We have proven the convergence theoretically. Here, to confirm the convergence, we draw the objective value changes, as presented in Fig. 4. One can observe that along with iterations, our loss value is gradually decreasing, which demonstrates that our objective function is experimentally convergent.
L. More Ablations
L.1. The Ablation of Common Sample Clusters
In the manuscript, utilize common sample cluster matrix to formulate full spectral embedding. This strategy not only facilitates view information communication during the procedure of learning but avoids the fusion between view-specific sample clusters. To validate the effectiveness of this common sample cluster strategy, we organize relevant ablation. The comparison results are reported in Table 8 where OCSC and WCSC represent the clustering results without/with common sample clusters respectively. It can be observed that WCSC outperforms OCSC, illustrating that our common sample cluster strategy functional.
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
0.60
25.00
1.00
0.08
0.06
The OfjRciiv
The Value Objectvv
Objectev
The Va Value Objectvvv
0.02
0.00
20.00
0.00
+000
0.20
15,00
0.30
-000
0.40
10.00
0.20
0..0
5.00
0.10
+000
10
The Number of Iterations
The Number of Iterations
The Number of Iterations
The Number of Iterations
(a) FLOEVEN
(b) SYNTHREED
(c) DEOLOG
(d) YALTHREE
0.05
0.06
6.00
-2.50
0.04
0.04
5.00
The Value Ofjedde
The Value Ojjecetu
The Value of Objective
Value Ojjective
3.50
0.03
0.02
4.00
4.00
0.02
0.00
3.00
4.50
0..0
55.00
2.00
0.00
-0.06
6.09
10
15
20
25
30
10
20
30
40
50
10
15
20
25
30
20
30
40
The Number of Iterations
The Number Iterations
The Number of Iterations
The Number Iterations
(c) BGFEA
(f) AWTEN
(g) HDIGTWO
(h) YOUFOURV
Figure Loss Changes along with Iterations
Table Ablation for Common Sample Clusters
DATASET
FLOEVEN
SYNTHREED
0.2
0.5
0.8
0.2
0,5
0.8
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
OCSC
10.23
10.14
4.93
10,33
9.57
5.01
10.14
9.52
4.38
40.05
38.97
33.98
40.73
38.56
33.69
36.73
36.40
30.79
WCSC
11.17
10.80
5.89
11.25
10.53
5.88
11.54
10.74
5.83
42.50
42.12
35.21
42.83
42.83
35.24
41.04
41.04
34.59
DEOLOG
YALTHREE
OCSC
29.06
20.82
16.62
30.29
21.87
17.79
28.75
22.85
18.51
22.19
20.98
6.75
19.31
19.42
5.59
18.59
18.97
5.19
WCSC
BGFEA
AWTEN
OCSC
32
20
21.88
20.45
20.21
WCSC
22.08
22.08
20.08
22.64
22.40
20.16
22.48
22.28
20.20
20.74
12.24
11.01
20.17
12.01
10.97
20.18
12.06
10.94
HDIGTWO
YOUFOURV
OCSC
9.57
9.65
9.79
8.79
10.73
9.01
10.45
9.97
9.36
13.53
8.39
8.86
14.26
9.36
8.61
13.22
9.12
8.41
WCSC
11.73
11.51
10.03
11.75
11.60
10.11
11.79
11.70
10.22
15.99
10.91
10.48
15.95
10.85
10.47
15.95
10.80
10.47
L.2. The Ablation of Space Rotation
We adopt space rotation strategy to reorder sample clusters on each view so as to decrease the risk of misregistration. To validate its effectiveness, we conduct ablation experiments, and the results are summarized Table OSR and WSR represent the clustering results without/with our space rotation strategy respectively. As seen, after rotating, the clustering performance improved. This gives evidence that our rotation strategy beneficial for performance enhancement.
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Table 9. Ablation for Space Rotation
DATASET
FLOEVEN
SYNTHREED
0.2
0.5
0.8
0.2
0.5
0.8
A/B
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
OSR
10.57
10.13
5.18
10.35
10.13
5.13
9.98
9.39
5.07
38.17 17
38.17
33.68
37.67
36.33
33.55
38.50
37.00
33.65
WSR
DEOLOG
YALTHREE
OSR
30.62
21.51
30.34
21.89
18.16
30.23
22.01
18.07
22.26
21.05
20.49
20.72
WSR
31.84
22.32
18.45
31.84
23.74
18.41
31,28
24.25
19.56
24.94
23.55
7.89
21.73
20.61
6.30
21.55
20.70
6.70
BGFEA
AWTEN
OSR WSR
21.46
21.26
19.68
21.74
21.66
19.88
21,70
21.42
19,80
19.05
11.09
10.02
19.05
11.03
9.97
19.05
11.19
10.16
22.08
22.08
20.08
22.64
22.40
20.16
22.48
22.28
20.20
20.74
12.24
11.01
20.17
12.01
10.97
20.18
12.06
10.94
HDIGTWO
YOUFOURV
OSR WSR
10.27
10.19
9.22
10.34
10.17
9.42
10.60
10.27
9.42
14.27
9.01
8.75
14.21
9.27
9.27
14.25
9.24
9.27
11.73
11.51
10.03
11.75
11.60
10.11
11.79
11.70
10.22
15.95
10.47
10.47
L.3. The Ablation of View Weight
Orthogonal to previous techniques regarding views equally, in this manuscript we associate a variable for each view to automatically balance the view contributions. To verify its effectiveness, we ablate the view weighting. The comparison results are summarized in Table 10 where OVW and WVW represent the results without/with view weight respectively, Apparently, WVW does better than OVW, illustrating that the view weighting is effective.
Table 10. Ablation for View Weight
DATASET
FLOEVEN
SYNTHREED
0.2
0.5
0.8
0.2
0.5
0.8
A/B
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
OVW
10.06
9.61
4.78
9.92
9.41
4.69
10.43
9.44
4.75
40.95
40.95
33.82
41.52
41.14
33.89
39.21
39.97
33.19
WVW
11.17
10.80
5,89
11.25
10.53
5.88
11.54
10.74
5.83
42.50
42.12
35.21
42.83
42.83
35.24
41.04
41.04
34.59
DEOLOS
YALTHRFF
OVW
30.12
20.84
17.32
30.39
21.74
17.62
31.81
22.99
17.92
22.06
20.61
7.92
19.89
19.13
5.25
19.67
18.74
6.00
BGFEA
OVW
20.34
20.34
20.27
20.54
20.58
20.34
WVW
22.08
22.08
20.08
22.64
22.40
20.16
22.48
22.28
20.20
20.74
12.24
11.01
20.17
12.01
10.97
20.18
12.06
10.94
HDIGTWO
YOUFOURV
OVW
10.78
10.54
9.13
10.86
10.77
9.22
10.88
10.78
9.24
13.65
9.56
9.16
14.17
9.50
9.14
14.03
9.45
9.15
WWW
11.73
11.51
10.03
11.75
11.60
10.11
11.79
11.70
10.22
15.99
10.91
10.48
15.95
10.85
10.47
15.95
10.80
10.47
L.4. The Ablation of Sample Cluster Weight
We also associate weights for the sample clusters on all views to automatically balance them. The ablation results are presented in Table 11 where OSCW and WSCW represent the clustering results without/with the sample cluster weight respectively, As seen, the sample cluster weights indeed facilitate the clustering performance improvement.
M. Comparison Results under Other Missing Ratios
In order to further exhibit the strengths, we conduct experiments under more missing ratios. Specially, we organize the comparison under missing ratio being 0.3, 0.4, 0.6 and 0.7 respectively. Table 12, 13, 14 and 15 summarize relevant clustering results. According to these tables, one can observe that under these missing ratios, our proposed model still can provide competitive clustering results.
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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Table 11. Ablation for Sample Cluster Weight
DATASET
FLOEVEN
SYNTHREED
0.2
0.5
0.8
0.2
0.5
0.8
A/B
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
OSCW
10.68
10.25
5.52
10.64
9.94
5.52
10.85
10.17
5.49
40.92
40.57
33.68
40.97
41.13
33.63
39.72
39.79
33.16
WSCW
DEOLOG
YALTHREE
OSCW
30.44
21.01
30.71
22.89
17.74
30.44
22.34
18.03
22.27
20.55
20.25
5.53
WSCW
31.84
22.32
18.45
31.84
23.74
18.41
31,28
24.25
19.56
24.94
23.55
7.89
21.73
20.61
6.30
21.55
20.70
6.70
BGFEA
AWTEN
OSCW
20.98
20.94
18.88
21.38
21.72
19.43
21.18
20.98
19.01
19.52
11.42
10.39
19.52
11.25
10.37
19.52
11.25
10.35
WSCW
22.08
22.08
20.08
22.64
22.40
20.16
22.48
22.28
20.20
20.74
12.24
11.01
20.17
12.01
10.97
20.18
12.06
10.94
HDIGTWO
YOUFOURV
OSCW
11.03
10.93
9
11.13
10.98
9.51
11.19
11.11
9.5
15.07
9.96
9.55
14.94
9.86
9.53
14.91
9.89
9.54
WSCW
11.51
10.03
11.75
10.11
11.70
Table Clustering Comparison under Missing Ratio Being 0.3
DATASET
FLOEVEN
SYNTHREED
DEOLOG
YALTHREF
METRIC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
LRTL
10.58
10.67
8.18
38.36 18.36
39.36
32.75
27.42
20.55
16.42
20.09
20.76
7.11
TCTMC
11.31
10.21
8.26
38.74
37.72
31.43
31.74
18.43
15.48
20.63
17.65
8.31
AGCIM
9.41
8.46
8.36
33.67
33.67
28.33
31.82
21.26
17.67
21.88
18.45
7.33
LSIMV
9.06
10.13
5.32
36.43
38.83
33.27
31.74
17.88
15.88
18.47
15.73
7.23
GIMC
8.13
8.86
5.83
36.53
34.84
32.96
31.27
18.27
17.38
18.58
16.32
6.23
IMVCI
9.92
5.81
40.71
41.71
33.22
20.96
18.35
21.97
18.39
6.70
PIMVC
10.83
9.29
6,32
40.53
39.86
31.37
30.77
23.21
17.99
21.39
18.00
7.61
HCCGL.
7.14
6.83
5.72
38 .0
34.67
34.97
31.58
20.46
18.11
22.45
21.45
8.29
USETL
10.20
9.56
7.35
39.33
39.33
29.95
29.73
19.05
18.12
19.94
20.67
8.49
LBIMV
10.66
10.46
5,01
41.07
41.74
27.90
20.98
17.59
21.18
18.97
UIMC
10.88
10.44
6,24
36.17
39.56
33.27
33.02
19.43
15.82
20.82
18 18.61
5.36
OURS
11.63
10.90
5,96
42.40
34.92
32.12
22.91
22.91
BGFFA
AWTEN
HDIGTWO
YOUFOURV
LRTL
17.57
20.16
19.02
16.79
10.18
7.88
10.25
10.63
9.07
TCIMC
18.37
20.83
17.72
16.84
11.33
10.84
10.73
10.24
8.63
AGCIM
19.43
20.16
15.89
18.33
10.37
10.57
11.67
10.77
8.72
LSIMV
17.27
19.36
19.62
20.77
11.12
10.14
11.57
9.12
8.23
GIMC
19.79
19.13
18.66
19.78
9.32
9.89
10.94
8.11
7.73
10.84
10.27
8.42
IMVCI
18.78
19.79
15.74
20.53
8.47
10.79
11.27
11.12
8.74
PIMVC
20.64
20.51
17.23
20.61
9.87
9.35
10.49
8.11
15.24
10.36
8.75
HCCGL.
22.57
19.28
18.95
20.11
9.87
8.23
10.57
10.97
8.27
USETL
20.36
19.11
16.84
20.18
7.84
9.11
10.32
8.12
8.34
10.53
9.32
8.92
LBIMV
19.12
21.94
18.38
20.33
10.41
8.71
8.36
11.72
8.85
15.88
9.98
9.27
UIMC
19.87
18.27
18.46
18.69
11.37
9.36
9.26
10.43
9.17
OURS
22.16
21.88
20.06
20.85
12.14
11.02
11.81
11.57
10.07
16.02
10.82
10.48
N. Stability and Reliability
In order to demonstrate the stability and reliability of devised model, we record the standard deviations and present them in Fig. where D-1 ~ D-8 are the alternative name of datasets in Table correspondingly. According to this figure, we can observe that even under various missing ratios' scenarios, the value of standard deviation is fairly small relative to the mean value. This provides evidence that our model is stable and the produced results are reliable.
39

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Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Table 13. Clustering Comparison under Missing Ratio Being 0.4
DATASET
FLOEVEN
SYNTHREED
DEOLOG
YALTHREE
METRIC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
11.07
9.42
7.57
39.56
41.78
33.92
19.49
17.40
19.24
22.82
5.71
TCIMC
10.43
9.73
7.87
37.74
36.68
30.63
31.86
18.47
16,43
21.43
18.32
5.87
AGCIM
9.04
6.73
33.83
29.18
31.36
22.30
17.86
22.03
18.03
LSIMV
7.97
9.23
4.33
36.23
37.79
33.73
31.76
17.26
14,46
17.69
15.62
8.23
GIMC
7.84
7.73
6.74
36.74
32.57
33.26
31.26
18.32
16.47
19.47
16.57
6.73
IMVCI
10.27
9.62
5.65
41.68
39.68
34.72
30.56
21.56
18.38
19.52
20.64
7.31
PIMVC
10.71
9.38
5.12
42.17
37.17
32.87
30.88
21.77
18.22
17.79
14.79
6.11
HCCGL
7.06
5.36
40.18
37.33
32.41
30.53
20.97
17.31
19.61
20.12
7.37
USETL
9.57
38
37.48
29.01
19.51
20.85
LBIMV
10.96
9.04
5,47
40.51
40.53
30.24
30.18
21,18
18.51
21.33
19.32
5,52
UIMC
11.18
6.06
35.67
40.67
33.16
20.26
16.53
20.64
6.67
OURS
11.25
10.47
5.86
41.83
41.67
34.85
32.12
22.63
18.47
22.21
21.33
6.23
BGFEA
AWTEN
HDIGTWO
YOUFOURV
LRTL
19.76
19.76
19.33
17.74
10.11
8.67
12.53
9.72
9.21
TCIMC
19.21
21.47
17.65
15.73
10.76
9.79
11.53
9.83
8.73
AGCIM
20.48
20.40
15.78
18.00 28.00
11.42
10.32
10.86
11.01
8.46
LSIMV
19.34
18.57
18.88
19.28
11.09
9.72
11.89
7.65
7.89
GIMC
17.87
18.86
20.23
20.97
10.25
10.35
11.73
8.93
8.17
11.72
9.32
8.74
IMVCI
22.03
19.68
15.33
20.56
9.20
10.05
11.83
10.98
9.03
PIMVC
22.98
16.72
20.37
9.08
9.23
15.88
10.47
HCCCLL
20.52
18.36
18.57
20.17
10.21
8.14
11.53
10.26
9.74
USETL
20.61
18.12
17.21
20.13
7.39
8.97
9.43
9.21
7.87
11.32
9.74
8.77
LBIMV
19.16
21.43
17.74
20.09
9.45
8.83
8.70
10.44
8.90
15.42
10.32
9.24
UIMC
22.37
19.21
18.2
20.14
11.63
9.47
9.96
10.78
8.63
OURS
22.28
22.16
20.11
20.58
12.06
10.98
11.94
11.61
10.09
16.00
10.87
10.48
0.8
0.8
0.6
D-1
D-2
D-3
D-4
D-5
D-6
D-7
D-8
D-2
D-3
D-4
D-5
D-6
D-7
D-8
(a) Learned View Weights
(b) Learned Sample Cluster Weights
Figure 5. Visualization of View Weights and Sample Cluster Weights
O. Weight Visualization
We associate views and sample clusters with learnable weight vectors respectively adaptively measure their contributions, To validate that it indeed learns different weights for views and sample clusters respectively, we visualize these weights, as illustrated in Fig.5 5. D-1 D-8 are the alternative name of datasets in Table 1. It is easy to see that each dataset, the view weights learned are different and the sample cluster weights learned are also different, which indicates that our optimization procedure functional. Additionally, conjunction with Table 10 and 11, we have that the weight strategy brings performance increasement. Therefore, we can state that the devised weight strategy works well with incomplete multi-view clustering.
40

---

Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Lincar Overhead
Table 14. Clustering Comparison under Missing Ratio Being 0.6
DATASET
FLOEVEN
SYNTHREED
DEOLOG
YALTHREE
METRIC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
LRTI
11.05
10.47
7.34
37.83
40.83
30.68
31.42
20.70
16.26
18.06
19.82
5.98
TCIMC
10.56
10.38
35.43
30.55
31.27
20.78
16.47
20.36
18.36
8.10
AGCIM
8.97
8.24
6.95
33.67
33.67
29.41
32.91
23.07
17.55
19.98
17.03
6.22
LSIMV
8.24
8.58
7.82
35.86
35.86
34.55
32.58
17.56
15.37
17.49
15.74
6.37
GIMC
8.37
8.47
7.32
34.62
32,97
35.37
31,94
17.67
16.58
18.85
17.46
6.73
IMVCT
11.16
9.68
5.84
41.03
37.18
30.47
31.49
23.49
17.87
20.21
19.33
7.23
PIMVC
11.13
9.28
36.14
35.18
31.47
30.18
21.07
16.70
14.27
8.01
HCCGL
7.28
6.99
5.36
38.17
38.17
31,66
30.04
21.76
17.91
18.88
21.52
6.82
USETL
9.59
8,92
7.81
37.61
37.61
28.64
27.42
21.28
17.33
20.71
19.06
6.72
LBIMV
9.81
10.93
5.57
40.50
41.58
29.46
30.77
22.77
17.57
20.12
19.09
6.25
UIMC
10.81
9.29
5.93
34.67
38.33
33.09
29.94
19.89
16.53
20.64
18.42
6.86
OURS
11.54
10.74
5,98
42.50
41.50
35.08
31,28
24.02
18.53
21.33
20.36
5.87
BGFEA
AWTEN
HDIGTWO
YOUFOURV
LRTL
19.37
21.97
21
17.89
10.23
8.33
10.98
10.27
9.12
TCIMC
18.67
20.42
18.92
16.83
10.31
10.32
10.63
10.42
8.45
AGCIM
20.32
20.20
16.48
19.84
9.53
10.11
11.36
9.87
LSIMV
19.35
17.92
17.97
16.47
10.64
10.33
11.32
8,58
8.27
GIMC
19.13
17.85
18.16
19.32
9.26
9.38
10.63
8.47
7.78
10.74
9.52
8.53
IMVCI
22.68
22.68
16.83
20.58
8.92
9.69
10.97
12.43
PIMVC
20.60
20.54
16.79
20.51
10,28
9.67
10.24
8.73
8.66
15.48
9.32
9.62
HCCGL
20.56
18.32
18.77
20.23
10.83
8.44
9.43
11.24
USETL
19.06
18.01
17.33
20.21
8.59
10.23
8.37
8.89
9.11
9.27
LBIMV
19.76
15.35
7.51
9.37
9.64
14.32
9.22
UTMC
21.37
18:27
18.45
10.16
8.96
10.27
OURS
22.53
22.24
20.16
20.79
12.07
10.96
11.80
11.69
10.15
15.93
10.81
10.47
Table 15. Clustering Comparison under Missing Ratio Being 0.7
DATASET
FLOEVEN
SYNTHREED
DEOLOG
YALTHREE
METRIC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
PUR
ACC
FSC
LRTL
11.03
10.42
7.40
35,43
40.43
32.43
31.06
21.25
17.24
19.12
20.48
6.14
TCIMC
11.07
10.43
7.12
37.38
36.62
30.32
30.88
18.56
15.76
20.17
18.12
6.75
AGCIM
8.90
8.31
6.96
34.17
34.00
28.80
30.11
23.26
18.12
20.73
17.12
6.73
LSIMV
8.12
9.23
7.32
34.67
34.72
32.36
31.17
17.47
15.35
17.74
15.63
7.22
GIMC
7.73
8.23
31.87
33.43
32.87
18.32
19.74
15.78
7.21
IMVCI
9.70
9.95
5.66
39.58
38.74
31.59
30.24
23.24
17.38
20.16
20.79
8.62
PIMVC
11.27
8.36
7.48
34.33
32.50
31.39
30.19
21.08
18.13
16.52
13.97
7.11
HCCGL
7.13
6.78
5.76
36.00
36.00
32.97
29.57
22.17
18.38
20.67
20.27
7.14
USETL
9.41
8.85
7.33
36.51
36.51
28. 13
30.41
20.38
18.0 88
19.88
20.06
7.89
LBIMV
9.12
10.06
5.81
39.17
36.17
27.11
30.57
20.30
17.68
21.33
18.52
5.35
UIMC
10.74
10.37
5.98
42.67
30.92
31.18
22.47
17.20
6.32
OURS
11.81
11.01
5,90
41.83
41.50
35.00
31.28
24.02
18.68
20.91
19.82
5.83
BGFFA
AWTEN
HDIGTWO
YOUFOURV
LRTL
18.68
20.84
18.25
17.72
10.81
8.29
10.26
11.03
9.23
TCIMC
18.73
19.83
18.27
15.68
9.57
9.78
10.26
10.37
7.84
AGCTM
20.32
20.28
14.45
14.11
9.84
11.08
9.87
10.41
8.50
LSIMV
19.16
17.88
18.26
17.86
11.29
9.47
10.47
8.37
7.94
GIMC
19.72
19.53
18.13
18.37
8.97
9.63
11.33
8.86
11.28
9.32
8.43
IMVCI
18.73
20.73
17.47
20.05
8.11
10.22
11.25
11.32
8.28
PIMVC
21.74
20.74
16.38
20.36
9.25
9.39
9.51
8.47
16.36
10.89
8.94
HCCGL
21.35
18.36
18.27
20.21
10.14
7.87
9.15
9.32
USETL
19.24
19.09
15.83
20.14
8.74
9.46
8.64
8.47
8.47
9.49
8.32
9.56
LBIMY
20.84
20.47
18.36
20.07
8.34
8.61
8.11
10.49
9.03
15.23
9.74
8.72
UIMC
20.72
17.75
18.33
18.47
11.32
9.36
9.67
9.06
OURS
22.60
22.40
20.19
20.32
11.92
10.95
11.91
11.80
10.17
15.98
10.88
10.48

---

Bifurcate then Alienate: Incomplete Multi-view Clustering via Coupled Distribution Learning with Linear Overhead
45
45
PUR ACC
PUR ACC
40
40
35
FSC
35
FSC
30
30
25
25
68
20
20
15
15
10
10
10
5
5
D-1
D-2
D-3
D-5
D-6
D-7
D-8
D-1
D-2
D-3
D-4
D-5
D-6
D-7
D-8
(a) Standard Deviation under Missing Ratio Being 0.3
(b) Standard Deviation under Missing Ratio Being 0.4
45
PUR
PUR ACC
40
40
35
FSC
30
30
9
0
98
04
D-1
D-2
D-3
D-5
D-6
D-8
D-2
D-3
D-4
D-5
D-6
D-7
(c) Standard Deviation under Missing Ratio Being 0.5
(d) Standard Deviation under Missing Ratio Being 0.6
45
PUR
PUR
40
ACC FSC
ACC
35
FSC
30
30
25
25
68
20
15
60
10
A
D-1
D-2
D-3
D-4
D-5
D-6
D-7
D-8
D-1
D-2
D-3
D-4
D-5
D-6
D-7
D-8
(c) Standard Deviation under Missing Ratio Being 0.7
Standard Deviation under Missing Ratio Being 0.8
Figure 6.
Errorbar of Standard Deviation and Mean Value
42


Reviews:
[{"review_id": "234UZ5vAPM", "reviewer": "Reviewer_7KrE", "summary": "This paper introduces a new incomplete multi-view clustering (IMC) algorithm named BACDL. It simultaneously explores both perspective-shared and perspective-specific determinants through coupled distribution learning, with linear overhead. The approach bifurcates feature clusters and enhances discrimination via alienation and mutual exclusion learning. Extensive experiments validate the effectiveness of BACDL across multiple large-scale datasets and benchmarks, with results outperforming several state-of-the-art methods in terms of accuracy and efficiency.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Sincerely thank Reviewer 7KrE for the very constructive comments.\n\n**Q1:** Providing iteration level convergence and approximation guarantees would enhance the theoretical rigor.\n\n**A1:** Many thanks! This is a challenging task for authors during the rebuttal period. At this moment, authors have no inspiring ideas about it. So sorry for this. We will strive to explore this topic in future work.\n\n**Q2:** Ambiguities in theoretical details. Like the convergence level, global optimal.\n\n**A2:** Thanks! We will carefully proofread the manuscript to state them more precisely. Specially, the convergence  refers to the objective function value. The global optimal refers to the approximate solution to the proxy problem.\n\n**Q3:** Lack of justification for method choices. Like why NMF.\n\n**A3:** NMF, subspace, kernel and neural network are four common means to tackle IMC problems. Kernel and neural network can perform nonlinear mapping well, while usually suffering from intensive complexity due to the large-sized feature matrix and complex network structure. Moreover, the selection of kernel type and network architecture also heavily relies on empirical knowledge. In virtue of the superior high-dimensional data processing capability, subspace effectively builds affinity via utilizing multiple potential spaces. However, due to the full-sized self-expression characteristics, it generally encounters cubic computing overhead. Unlike them, NMF mines shared low-dimensional structures by decomposing the view data matrix to discover latent clusters. In the decomposed basis vectors, each element can be seen as a contribution to the features, which makes the analysis results more intuitive and helps to understand the reasons for clustering. So, in the paper, we adopt the NMF technique.\n\n**Q4:** Performance gains and statistical validation.\n\n**A4:** In experiments, we run 50 times, and record the average value of clustering results. We will add these statistical information in the next version, and further analyze its features.\n\n**Q5:** Lack thorough review on NMF-based approaches. \n\n**A5:** Thanks! We will thoroughly review the mentioned works in the next version.\n\n**Q6:** Writing and structural Issues. Like section transitions, example clarifications.\n\n**A6:** Thanks! We will carefully polish the manuscript. For the matrix $\\mathbf{G}_r$, it consists of 0 and 1, and there is only one 1 in each column.\n\n**Q7:** Complexity analysis. Considering the number of iterations required for convergence.\n\n**A7:** Good suggestion! The number of required iterations indeed affects the time complexity. Deriving the iterations required for convergence is a promising research direction. We will make efforts to explore this topic in future work.\n\n**Q8:** Reproducibility issues.\n\n**A8:** We will release the source code in the final version.\n\n**Q9:** What \"global optimal\" refers to.\n\n**A9:** It refers to that of the proxy problem.\n\n**Q10:** Why feature clusters' perspective? Why not sample clusters' perspective?\n\n**A10:** Kindly note the feature cluster matrix aims at mapping original data to a potential space, and learns the marginal distribution of original data. Bifurcating it will be conductive to extracting representative features from original data. It can be seen as a feature extractor. The sample cluster matrix mainly plays a role in grouping formed representations to generate clusters. A high-quality cluster partition relies on more discriminative representations. So, we adopt the feature clusters' perspective. Please see the following comparisons. BSC: Bifurcate sample clusters; BFC: Bifurcate feature clusters.\n\n|FLOEVEN||||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n||0.2|||0.5|||0.8|||\n||PUR|ACC|FSC|PUR|ACC|FSC|PUR|ACC|FSC|\n|BSC|10.42|10.11|5.12|9.63|10.12|5.21|10.52|10.13|5.01|\n|BFC|**11.17**|**10.80**|**5.89**|**11.25**|**10.53**|**5.88**|**11.54**|**10.74**|**5.83**|\n|**SYNTHREED**||||||||||\n|BSC|41.21|40.51|35.02|**42.88**|41.01|34.21|40.21|39.94|32.43|\n|BFC|**42.50**|**42.12**|**35.21**|42.83|**42.83**|**35.24**|**41.04**|**41.04**|**34.59**|\n|**DEOLOG**||||||||||\n|BSC|31.21|21.73|18.12|30.76|21.42|17.14|30.87|**24.62**|19.21|\n|BFC|**31.84**|**22.32**|**18.45**|**31.84**|**23.74**|**18.41**|**31.28**|24.25|**19.56**|\n|**YALTHREE**||||||||||\n|BSC|23.21|21.67|7.62|20.52|19.43|6.11|20.62|19.62|6.53|\n|BFC|**24.94**|**23.55**|**7.89**|**21.73**|**20.61**|**6.30**|**21.55**|**20.70**|**6.70**|\n|**BGFEA**||||||||||\n|BSC|21.32|21.42|19.72|21.32|21.22|19.67|21.42|21.22|19.87|\n|BFC|**22.08**|**22.08**|**20.08**|**22.64**|**22.40**|**20.16**|**22.48**|**22.28**|**20.20**|\n|**AWTEN**||||||||||\n|BSC|19.32|**12.26**|10.64|19.86|11.94|10.22|20.01|11.46|10.31|\n|BFC|**20.74**|12.24|**11.01**|**20.17**|**12.01**|**10.97**|**20.18**|**12.06**|**10.94**|\n\nEvidently, BFC is more desirable than BSC in most cases, revealing that feature clusters' perspective is more preferable.", "claims_and_evidence": "The claims in the submission are not fully supported by clear evidence. Specifically, Theorem 5 lacks clarity regarding whether convergence refers to the objective function value or iterations. Additionally, the term \"global optimal\" in Theorems 1 and 2 is used imprecisely, without specifying if it refers to the global minimum of the original problem or the proxy problem. This paper also fails to clarify whether the function is strictly convex, which is crucial for ensuring a global minimum. These issues weaken the theoretical rigor of the claims.", "methods_and_evaluation_criteria": "This paper validates the proposed method using multiple large-scale datasets, which is appropriate for assessing its effectiveness and efficiency. The experimental design is well-structured, and a variety of widely used evaluation metrics are employed, providing sufficient evidence to support the claims. The experiments are comprehensive and thoroughly validate the proposed approach.", "theoretical_claims": "I have checked the correctness of the proofs for the theoretical claims in the paper, particularly for Theorem 1 and Theorem 2. However, there is an issue with the claim of \"global optimal\" in these theorems. This paper does not clearly specify whether \"global optimal\" refers to the global minimum of the original problem or the approximate solution to the proxy problem. Additionally, the use of the term \"global\" is imprecise, as it does not account for cases where multiple global solutions may exist, especially if the function is not strictly convex. This lack of rigor weakens the theoretical clarity of the claims.", "experimental_designs_or_analyses": "The experiments are very thorough. However, in the analysis of the results, the paper should provide a more quantitative evaluation of the performance gains. Additionally, hypothesis testing should be included to confirm the statistical significance of the observed performance improvements.", "relation_to_broader_scientific_literature": "The key contributions of the paper are well-positioned in relation to the broader scientific literature. The paper reviews several state-of-the-art methods, and compared to these existing approaches, the main contribution lies in simultaneously considering both perspective-shared and perspective-specific determinants. Additionally, the paper introduces a new algorithm designed to solve complex optimization problems. One of the notable aspects of the proposed algorithm is its linear overload, which distinguishes it from other methods in the literature. This contribution offers a novel approach to tackling the problem, improving efficiency and effectiveness in ways not previously explored.", "essential_references_not_discussed": "A thorough review of the existing works that are based on NMF should be included, such as:\n\n[1] Wen, J., Zhang, Z., Zhang, Z., Zhu, L., Fei, L., Zhang, B., & Xu, Y. (2021, May). Unified tensor framework for incomplete multi-view clustering and missing-view inferring. In Proceedings of the AAAI conference on artificial intelligence (Vol. 35, No. 11, pp. 10273-10281).\n\n[2] Wen, J., Xu, G., Tang, Z., Wang, W., Fei, L., & Xu, Y. (2023). Graph regularized and feature aware matrix factorization for robust incomplete multi-view clustering. IEEE Transactions on Circuits and Systems for Video Technology, 34(5), 3728-3741.", "other_strengths_and_weaknesses": "Strengths:\n\n1.Well-written and organized. This paper is clear, logically structured, and easy to follow.\n\n2.Novel Approach. Introduces BACDL, addressing the limitations of current methods by capturing both perspective-shared and perspective-specific determinants, improving clustering performance.\n\n3.Comprehensive Literature Review. Relevant works are reviewed in depth, providing a strong foundation for the proposed method.\n\n4.Thorough Theoretical and Experimental Analysis. This paper includes detailed theoretical proofs, optimization steps, and a well-structured experimental setup. Performance is validated using large datasets and state-of-the-art baselines.\n\n5.Efficiency: The algorithm exhibits linear overhead in terms of time and space, demonstrating scalability to large-scale datasets.\n\nWeaknesses:\n\n1.Convergence and Approximation Guarantees. Theorem 1 proves the validity of the proxy problem, demonstrating that the proposed proxy problem is a valid approximation that can represent the original problem for optimization. However, Theorem 2 proves that solving this proxy problem guarantees a decrease in the objective function value, but it does not provide an analysis of convergence at the iteration level. In other words, while the objective value decreases, there is no clear proof that the algorithm will converge to a fixed point in a finite number of iterations. For approximation methods, in addition to convergence in terms of the objective value, it is important to prove an approximation guarantee, which would clarify the relationship between the solution to the approximate problem (i.e., the proxy problem) and the solution to the original problem. Many approximation methods provide such guarantees, and incorporating this would enhance the theoretical rigor of this work.\n\n2.Ambiguities in Theoretical Details. (1) In Theorem 5, it is unclear which level of convergence is being discussed. The theorem mentions the convergence of the algorithm, but it is not specified whether this refers to convergence in the objective function value or convergence in terms of iterations. It is crucial to clarify this to prevent potential misinterpretation, as readers might assume iteration convergence is guaranteed when only objective value convergence is proven. (2) Similarly, the use of the term “global optimal” in Theorem 1 and Theorem 2 needs to be more precise. This paper uses the term \"global solution\" without clearly specifying whether it refers to the global minimum of the original problem or the approximate solution to the proxy problem. It would be more precise to refer to the \"global minimum\" of the proxy problem, especially in the context of convexity. Additionally, since a semi-definite Hessian matrix ensures that the function is convex, it only guarantees a global minimum if the function is strictly convex (i.e., the Hessian matrix is positive definite). If the function is not strictly convex, multiple global solutions might exist. The paper should clarify these points for better theoretical rigor.\n\n3.Lack of Justification for Method Choices. Although non-negative matrix factorization (NMF) is used in the proposed method, this paper does not provide a clear justification for why NMF is preferred over other potential techniques. There are various other matrix factorization methods available, and it would be useful to explain why NMF is particularly suited for this incomplete multi-view clustering (IMC) problem. Additionally, the choice of perspective-shared and perspective-specific determinants is made from the feature clusters' perspective, but the paper does not provide any reasoning as to why this approach is chosen rather than considering the sample clusters' perspective. Moreover, while distribution learning and mutual exclusion learning are critical components of the algorithm, their roles and necessity are not clearly explained.\n\n4.Performance Gains and Statistical Validation. Although this paper demonstrates significant performance gains of the proposed BACDL algorithm over several state-of-the-art methods, these gains are not sufficiently quantified statistically. For example, the performance comparison could benefit from hypothesis testing to determine whether the differences observed in performance metrics (e.g., clustering accuracy, purity) are statistically significant. This would provide stronger evidence that the observed improvements are not merely due to random variation. \n\n5.Lack Thorough Review on NMF-based Approaches. Although the related work overview is comprehensive, the proposed method is based on Non-negative Matrix Factorization (NMF). Therefore, a thorough review of the existing works that are based on NMF should be included, such as:\n[1] Wen, J., Zhang, Z., Zhang, Z., Zhu, L., Fei, L., Zhang, B., & Xu, Y. (2021, May). Unified tensor framework for incomplete multi-view clustering and missing-view inferring. In Proceedings of the AAAI conference on artificial intelligence (Vol. 35, No. 11, pp. 10273-10281).\n[2] Wen, J., Xu, G., Tang, Z., Wang, W., Fei, L., & Xu, Y. (2023). Graph regularized and feature aware matrix factorization for robust incomplete multi-view clustering. IEEE Transactions on Circuits and Systems for Video Technology, 34(5), 3728-3741.\n\n6.Writing and Structural Issues. (1) In the introduction, this paper mentions several limitations of the proposed method but fails to address the consequences of these limitations. (2) Section Transitions: There is a lack of smooth transitions between some sections, which affects the readability of the paper. For example, the introduction of Definition 1 (lines 180-188) feels abrupt and lacks context—it's unclear why this definition is introduced at this point. Similarly, Theorem 3 is presented without explaining its relevance or application, leaving readers to wonder what it contributes to the overall algorithm. (3) Example Clarifications: In some sections, such as the one introducing the index matrix G_r, it would be helpful to provide a concrete example to better illustrate its structure and functionality. \n\n7.Complexity Analysis. The complexity analysis section provides a general overview of the computational costs, but it fails to account for the number of iterations required for convergence. Given that the algorithm involves several steps of optimization, the number of iterations could significantly affect the time complexity, especially for large-scale datasets. This paper does not clarify whether more iterations are needed when handling larger datasets or whether the algorithm's performance scales well with the number of iterations.\n\n8.Reproducibility Issues. While the paper provides extensive experimental results, it does not offer any code or datasets, which makes it difficult for other researchers to verify the results or build upon this work. Providing access to the implementation would greatly enhance the transparency and reproducibility of the research.", "other_comments_or_suggestions": "1.In Algorithm 1, it should be (g^h-g^{h+1}) rather than (g^{h+1}-g^h).\n\n2.Figure 2: Include units for runtime to provide clearer context.\n\n3.Ensure that all definitions (e.g., Definition 1) are properly referenced, specifying the exact source from which they are drawn. Additionally, this paper should cite the relevant literature for the Majorization-Minimization (MM) framework used in the optimization step, as this method is central to the proposed approach.", "questions_for_authors": "1.How do the authors ensure that the algorithm converges in a finite number of iterations? Can the authors provide proof of convergence in terms of the number of iterations?\n\n2.Can the authors provide an approximation guarantee for the solution to the approximate problem used in the optimization? How does this compare to the exact solution?\n\n3.This paper uses the term \"global optimal\" in the theoretical proofs. Is this referring to a global minimum of the objective function, or is it specific to the approximation method?\n\n4.Why are perspective-shared and perspective-specific determinants modeled from the feature clusters' perspective rather than from the sample clusters' perspective?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Banc3XN2PW", "reviewer": "Reviewer_xp7C", "summary": "A BACDL algorithm with dual-determinant learning is specially designed for incomplete multi-view clustering (IMC) in this paper.  It bifurcates feature clusters and further alienates them through mutual exclusion learning to strengthen the discrimination. It alleviates the dimension inconsistency, and bridges all views by unifying the association between feature clusters and sample clusters. The full clustering embedding is formulated by weighted space rotation and remapping. Theoretical analysis demonstrates its linear overhead and convergence. Experimental results under multiple missing ratios validate its effectiveness.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1:** View guidance may impair the efficiency goal.\n\n**A1:** Thanks. The computing cost of view guidance is linear, and thus hardly affects the efficiency goal. It requires constructing $\\mathbf{X} _r\\mathbf{G} _r\\mathbf{G} _r^{\\top}\\mathbf{E} _r\\mathbf{D} _{\\gamma}^{\\top}\\mathbf{C}^{\\top}$, $\\mathbf{C} _r\\mathbf{C}^{\\top}$ and $[\\mathbf{P} _r\\mathbf{C}|\\mathbf{C} _r]\\mathbf{D}\\mathbf{E} _r^{\\top}\\mathbf{G} _r\\mathbf{G} _r^{\\top}\\mathbf{E} _r\\mathbf{D} _{\\gamma}^{\\top}\\mathbf{C}^{\\top}$, which takes $\\mathcal{O}(d_rn+d_rnk+d_rk^2)$, $\\mathcal{O}(d_rk^2)$, $\\mathcal{O}( nk + 2d_rk^2 + d_rnk)$. So, it totally takes $\\mathcal{O}(d_rnk+d_rk^2)$. This is $\\mathcal{O}(n)$ and consistent with the efficiency goal.\n\n**Q2:** Why introducing common orthogonal sample clusters?\n\n**A2:** They gather all view sample clusters to formulate full embedding. Due to the missing instances, the sample clusters on each view are incomplete. The orthogonality plays a role in enhancing the separability of learned common sample clusters to better group samples.\n\n**Q3:** Does the missing ratio affect the complexity of BACDL?\n\n**A3:** It does not affect the complexity. The index matrix containing missing ratio owns the properties, i.e., there is only one element that is 1 while the other elements are 0 in each column. We have $\\mathbf{G} _r\\mathbf{G} _r^{\\top}$ is diagonal with elements either 0 or 1. So, $\\mathbf{X} _r\\mathbf{G} _r\\mathbf{G} _r^{\\top}$ equals to $\\mathbf{X} _r\\odot\\mathbf{O} _r$ where $\\mathbf{O} _r=\\mathbf{1} _{d _r}\\left[\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{1,j}, \\cdots,\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{n,j}\\right]$. This takes $\\mathcal{O}(d_rn)$ cost and is irrelevant to the missing ratio.\n\n**Q4:** Why guaranteeing C column-normalized?\n\n**A4:** Each column of feature clusters denotes a probability distribution on all feature dimensions. Each column sum needs to add up to 1. After bifurcating, each part also should conform to this point. On the basis of column-normalized $\\mathbf{P}_r$, we derive that the premise for $\\mathbf{P} _r\\mathbf{C}$ being column-normalized is that $\\mathbf{C}$ only needs to be column-normalized.\n\n**Q5:** How does the element-wise operation reduce the computational overhead?\n\n**A5:** This mainly benefits from the equivalent element transformation. For $\\mathbf{E}_r$, directly calculating $\\left\\|\\mathbf{E} _r^{\\top}\\mathbf{G} _r\\right\\| _F^2$ will take $\\mathcal{O}(knn_r)$. As the missing ratio decreases, $n_r$ is gradually increasing. The overhead is almost close to  $\\mathcal{O}(n^2)$.\n\nNote that $\\left\\|\\mathbf{E} _r^{\\top}\\mathbf{G} _r\\right\\| _F^2$ is equal to $\\left\\|\\mathbf{E} _r^{\\top}\\mathbf{G} _r\\mathbf{G} _r^{\\top}\\right\\| _F^2$. Then, by the element-wise operation, $\\left\\|\\mathbf{E} _r^{\\top}\\mathbf{G} _r\\mathbf{G} _r^{\\top}\\right\\| _F^2$ is $\\left\\|\\mathbf{E} _r^{\\top}\\odot\\mathbf{Q} _r\\right\\| _F^2$, \nwhich takes $\\mathcal{O}(nk)$. $\\mathbf{Q}_r$ is $\\mathbf{1} _{k}\\left[\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{1,j}, \\cdots, \\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{n,j}\\right]$.\n\n**Q6:** Why learning the sample clusters on each view respectively (SEV)? Why not directly learning common ones? \n\n**A6:** The former may facilitate capturing the view characteristics. Learning common sample clusters (CSC) is a feasible scheme, and yet could weaken the view data diversity.\n\n|FLOEVEN||||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n||0.2|||0.5|||0.8|||\n||PUR|ACC|FSC|PUR|ACC|FSC|PUR|ACC|FSC|\n|CSC|10.47|10.64|4.78|10.52|9.32|**5.92**|10.25|9.36|4.57|\n|SEV|**11.17**|**10.80**|**5.89**|**11.25**|**10.53**|5.88|**11.54**|**10.74**|**5.83**|\n|**SYNTHREED**||||||||||\n|CSC|40.72|39.89|34.73|40.37|39.87|33.83|38.87|37.84|31.59|\n|SEV|**42.50**|**42.12**|**35.21**|**42.83**|**42.83**|**35.24**|**41.04**|**41.04**|**34.59**|\n|**DEOLOG**||||||||||\n|CSC|30.68|20.57|17.58|**31.93**|19.82|17.63|28.93|22.47|18.97|\n|SEV|**31.84**|**22.32**|**18.45**|31.84|**23.74**|**18.41**|**31.28**|**24.25**|**19.56**|\n\n**Q7:** When stacking sample clusters (STA), the performance?\n\n**A7:** Please see the following table. MSC: mapping sample clusters.\n\n|FLOEVEN||||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n||0.2|||0.5|||0.8|||\n||PUR|ACC|FSC|PUR|ACC|FSC|PUR|ACC|FSC|\n|STA|10.23|10.14|4.93|10.33|9.57|5.01|10.14|9.52|4.38|\n|MSC|**11.17**|**10.80**|**5.89**|**11.25**|**10.53**|**5.88**|**11.54**|**10.74**|**5.83**|\n|**SYNTHREED**||||||||||\n|STA|40.05|38.97|33.98|40.73|38.56|33.69|36.73|36.40|30.79|\n|MSC|**42.50**|**42.12**|**35.21**|**42.83**|**42.83**|**35.24**|**41.04**|**41.04**|**34.59**|\n|**DEOLOG**||||||||||\n|STA|29.06|20.82|16.62|30.29|21.87|17.79|28.75|22.85|18.51|\n|MSC|**31.84**|**22.32**|**18.45**|**31.84**|**23.74**|**18.41**|**31.28**|**24.25**|**19.56**|\n\nDirectly stacking sample clusters could lead to insufficient interaction among views. Moreover, this does not adaptively balance their contributions. Misregistration also will weaken the quality of sample clusters.", "claims_and_evidence": "This paper is organized in a clear manner and provides some insights for incomplete multi-view clustering. I understand the motivation of dual-determinant scheme. The feature cluster bifurcating highlights the novelty.", "methods_and_evaluation_criteria": "This paper compares the results with multiple methods in three metrics.", "theoretical_claims": "It is ok in convergence and complexity.", "experimental_designs_or_analyses": "The experimental designs are reasonable.", "relation_to_broader_scientific_literature": "The feature cluster bifurcating and alienating could provide further reference for incomplete multi-view clustering.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "The strengths:\n\n1. The motivation is clearly illustrated and the overall organization is logically-structured. \n2. The idea that bifurcates feature clusters and alienates via mutual exclusion learning is novel to a certain degree.\n3. The authors conduct extensive experiments and also make moderate discussions about the results.\n\nThe weaknesses:\n\n1. Introducing view guidance may induce additional computing complexity, which could impair the efficiency goal. \n2. The reasons why introducing common sample clusters and why setting them to be orthogonal are not illustrated in depth.", "other_comments_or_suggestions": "Refer to the above comments", "questions_for_authors": "1.  Does the missing ratio affect the complexity of BACDL? If yes, how? As illustrated in the updating rule, the index matrix is associated with the data matrix. \n2. Why it is necessary to guarantee the matrix C to be column-normalized?\n3. How does the element-wise multiplication operation decrease the computational overhead?\n4.  Why learning the sample clusters on each view respectively? Why not directly learning common sample clusters? Is this beneficial for the performance improvement?\n5. Rather than formulating the full clustering embedding by mapping the sample clusters of each view, when stacking them, there is no common plane, how does the model perform?\n\n.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HjiOhcrvOz", "reviewer": "Reviewer_KBMg", "summary": "This work aims to alleviate the issue of single-determinant paradigm in incomplete multi-view clustering. It introduces distribution learning and associates each type of determinants to bifurcated feature clusters. Through mutual exclusion learning and view guidance learning, it eliminates the dimension inconsistency and enlarges the determinant distinction. All views are interconnected together based on the distribution principle. After rotating and remapping, incomplete sample clusters are formed into full clustering embedding. A nine-step updating rule with overall linear overhead and theoretical convergence efficiently minimizes the objective function.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1:** More descriptions on the sample cluster distribution in Eq.(3).\n\n**A1:** Thanks. Each row of the sample cluster $\\mathbf{E}_r$ represents a probability distribution. So, for each row of $\\mathbf{E}_r$, its sum needs to be 1. For handling the incompleteness, we introduce the index matrix $\\mathbf{G}_r$ which consists of 0 and 1. We can formulate the observed sample clusters as $\\mathbf{G} _r^{\\top}\\mathbf{E} _r$. Hence, we have  the incomplete sample clusters need to satisfy $\\mathbf{G} _r^{\\top}\\mathbf{E} _r\\mathbf{1} _k =\\mathbf{1} _{n _r}$.\n\n**Q2:** More illustrations on the space rotation influence.\n\n**A2:** It mainly serves as reordering sample clusters to make them as consistent as possible. Note that the common sample clusters are subject to the similar constraints as space operation, which can help tackle the negative elements that rotation operation brings. Please see the following performance comparisons. DM: direct mapping; RO: rotation operation.\n\n|FLOEVEN||||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n||0.2|||0.5|||0.8|||\n||PUR|ACC|FSC|PUR|ACC|FSC|PUR|ACC|FSC|\n|DM|10.57|10.13|5.18|10.35|10.13|5.13|9.98|9.39|5.07|\n|RO|**11.17**|**10.80**|**5.89**|**11.25**|**10.53**|**5.88**|**11.54**|**10.74**|**5.83**|\n|**SYNTHREED**||||||||||\n|DM|38.17|38.17|33.68|37.67|36.33|33.55|38.50|37.00|33.65|\n|RO|**42.50**|**42.12**|**35.21**|**42.83**|**42.83**|**35.24**|**41.04**|**41.04**|**34.59**|\n|**DEOLOG**||||||||||\n|DM|30.62|21.51|17.96|30.34|21.89|18.16|30.23|22.01|18.07|\n|RO|**31.84**|**22.32**|**18.45**|**31.84**|**23.74**|**18.41**|**31.28**|**24.25**|**19.56**|\n\nOne can observe that after space rotation, the performance receives improvement.\n\n**Q3:** The initialization lacks clarity.\n\n**A3:** Thanks! We create a random matrix with element from 0 to 1, and then do column-normalization. We use it to initialize $\\mathbf{P}_r$, $\\mathbf{C}_r$ and $\\mathbf{C}$. For $\\mathbf{F}_r$ and $\\mathbf{E}$, we use random orthogonal matrix to initialize them. We use the row-normalized random matrix with element from 0 to 1 to initialize $\\mathbf{E}_r$, and the random matrix with element from 0 to 1 to initialize $\\mathbf{D}$. For $a_r$ and $b_r$, we initialize them with $1/v$ and $1/\\sqrt{v}$.\n\n**Q4:** Analysis of each term's contribution.\n\n**A4:** The first term is deemed as a error reconstruction, and extracts feature clusters and sample clusters as well as their association. It also adaptively balances the important of each view. The second term separates feature clusters from sample clusters to highlight their discrimination by point-to-point alienation. The third term mitigates the misregistration and constructs complete sample clusters.   \n \n**Q5:** Why do the common sample clusters not adhere to the similar constraints as the view sample clusters?\n\n**A5:** We introduce orthogonal transformation to reformulate the view sample clusters, which inevitably brings negative elements. Note that the essence of view sample clusters is a probability distribution, and accordingly all elements in them are non-negative. After transformation to produce common sample clusters, there will have negative elements. So, we make the common sample clusters adhere to similar constraints as the orthogonal transformation. \n\n**Q6:** How is $n_r$ merged during updating the view coefficient? Affect the running speed?\n\n**A6:**  Updating the view coefficient needs to compute $\\left\\|\\mathbf{X} _r\\mathbf{G}_r\\right\\| _F^2$. Note that for each column of $\\mathbf{G}_r$, there is only one 1 and other elements are 0. We have $\\left\\|\\mathbf{X} _r\\mathbf{G} _r\\right\\| _F^2$ is equal to $\\left\\|\\mathbf{X} _r\\mathbf{G} _r\\mathbf{G} _r^{\\top}\\right\\| _F^2$. The diagonal elements of $\\mathbf{G} _r\\mathbf{G} _r^{\\top}$ are 1 or 0 and other elements are 0. Consequently, we have $\\mathbf{X} _r\\mathbf{G} _r\\mathbf{G} _r^{\\top}$ aims to select some columns of $\\mathbf{X}_r$. So, we have that $\\left\\|\\mathbf{X} _r\\mathbf{G} _r\\right\\| _F^2$ is equal to $\\left\\|\\mathbf{X} _r\\odot\\mathbf{O} _r\\right\\| _F^2$. $\\mathbf{O}_r$ is $\\mathbf{1} _{d _r}\\cdot[\\sum _{j=1}^{n_v}(\\mathbf{G} _r) _{1,j},\\cdots,\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{n,j} ]$. Before merging, the computing overhead is $\\mathcal{O}(n n_r)$. After merging, it is $\\mathcal{O}(n)$ and not affected by $n_r$.\n\n**Q7:** When accumulating sample clusters vertically, how does the algorithm behave? Reasons?\n\n**A7:** For the performance comparison, please refer to A7 in Reviewer xp7C. We attribute this phenomenon to three points. Accumulating sample clusters on all views vertically could result in  inadequate communication between sample clusters across different views, which is not conductive to formulating rich representations. This also could not automatically measure the importance of different sample clusters.  Meanwhile, the misregistration will disturb the cluster structure, and consequently degrades the formulated representations.", "claims_and_evidence": "Utilizing dual perspective determinants to encode cluster representations is under-studied in incomplete multi-view clustering. I agree with that. Table 4 and 5 illustrate its effectiveness.", "methods_and_evaluation_criteria": "The authors conduct comparing experiments under several missing ratios to illustrate the effectiveness in tackling IMC issues", "theoretical_claims": "Yes. They are correct.", "experimental_designs_or_analyses": "Yes. I checked experimental designs. They are valid.", "relation_to_broader_scientific_literature": "The key contributions in this paper are the dual perspective determinant cluster coding and the solution with overall linear overhead, which may serve as a groundwork for subsequent studies.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n1)  The algorithm analysis is comprehensive, covering complexity, convergence, sensitivity, etc.  \n2)  Experiments are solid, involving diverse missing ratios and data scales. Ablation is also thorough. \n3)  The flow chart of the proposed model is intuitive, and the function of each part is clear. \n\nWeaknesses:  \n1)   During formulating Eq.(3),  the descriptions about the sample cluster distribution needs further enhancement.  \n2)   The space rotation operation utilized could alter the value of learned sample clusters. This influence requires more illustrations.  \n3)   The initialization lacks clarity, for instance, the cluster association.  \n4)   A systematic analysis of each term's contribution to the objective loss would significantly strengthen the motivation.", "other_comments_or_suggestions": "See the weakness above.", "questions_for_authors": "1. As mentioned, the final clustering results are derived by performing spectral clustering on the common sample clusters. The sample clusters from each view contribute to the formation of the common sample clusters.  A question arises: why do the common sample clusters not adhere to the similar constraints as the view sample clusters?\n \n2. The number of samples observed on each view $n_r$ is embedded into the original data matrix, how is it merged during updating the view coefficient? If not merged, is the computing overhead impacted by it? In practice, whether does it affect the running speed?\n\n3. When generating the clustering embedding by accumulating sample clusters on all views vertically, how does the algorithm performance behave? possible reasons?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9tlGQ9Pnx7", "reviewer": "Reviewer_hkPK", "summary": "In the paper, the authors propose a dual-determinant incomplete multi-view clustering algorithm BACDL. They partition feature clusters through a bifurcation scheme, and alienate bifurcations to differentiate determinants. With coupled distribution learning, it alleviates the dimension inconsistency by introducing view guidance.  Then, they bridge all views based on the principle between marginal distribution and conditional distribution, and reorder all incomplete sample clusters in potential space to construct the clustering embedding.  Finally, compared experiments with different missing ratios are organized to reveal the merits of BACDL.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "**Q1:**  More details regarding Remark 1.   \n\n**A1:** Thanks. Due to $\\mathbf{X} _r\\in\\mathbb{R}^{d_r\\times n}$ and $\\mathbf{G} _r\\in\\mathbb{R}^{n\\times n_r}$, computing $q=\\|\\mathbf{X} _r\\mathbf{G} _r-\\widehat{\\mathbf{A}} _r\\mathbf{E} _r^{\\top}\\mathbf{G} _r\\| _F^2$ will take  at least $\\mathcal{O}(n n_r)$ overhead. Note that $\\mathbf{G}_r$ consists of 0 and 1, and there is only one element that is 1 in each column. $\\mathbf{G} _r\\mathbf{G} _r^{\\top}$ is diagonal with 0 and 1. $\\mathbf{X} _r\\mathbf{G} _r\\mathbf{G} _r^{\\top}$ and $\\mathbf{E} _r^{\\top}\\mathbf{G} _r\\mathbf{G} _r^{\\top}$ equal to $\\mathbf{X} _r\\odot\\mathbf{O} _r$ and $\\mathbf{E} _r^{\\top}\\odot\\mathbf{Q} _r$. $\\mathbf{O} _r$ is $\\mathbf{1} _{d _r}\\cdot[\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{1,j},\\cdots,\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{n,j}]$ and $\\mathbf{Q} _r$ is $\\mathbf{1} _{k}\\cdot[\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{1,j},\\cdots,\\sum _{j=1}^{n _v}(\\mathbf{G} _r) _{n,j}]$. So, we have $q$ equals to $\\|\\mathbf{X} _r\\odot\\mathbf{O} _r-\\widehat{\\mathbf{A}} _r\\mathbf{E} _r^{\\top}\\odot\\mathbf{Q} _r\\| _F^2$. Computing the latter needs $\\mathcal{O}(d_rn)$, which is $\\mathcal{O}(n)$.\n\n**Q2:** Interpretation regarding TCIMC and LSIMV. \n\n**A2:** TCIMC uses the tensor Schatten p-norm to explore complementary and spatial structure. Despite enhancing view interaction, it induces intensive time overhead due to the tensor operation, and is unsuitable for large-scale tasks. LSIMV constructs sparse and structured representations. It adopts a norm based sparse constraint to generate low-dimensional features, and uses local embedding to do aggregation. This needs a relatively larger memory cost due to the almost full-sized graph. \n\n**Q3:** Does the orthogonal rotation affect the cluster label quality?\n\n**A3:** This rotation reorganizes the sample clusters to relieve misregistration. Although altering the distribution, kindly note that the final common sample clusters are also required to be orthogonal, which keeps pace with the rotation. See the following comparisons. NRN: not involve rotation; ART: contains rotation.\n\n|FLOEVEN||||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n||0.2|||0.5|||0.8|||\n||PUR|ACC|FSC|PUR|ACC|FSC|PUR|ACC|FSC|\n|NRN|10.57|10.13|5.18|10.35|10.13|5.13|9.98|9.39|5.07|\n|ART|**11.17**|**10.80**|**5.89**|**11.25**|**10.53**|**5.88**|**11.54**|**10.74**|**5.83**|\n|**SYNTHREED**||||||||||\n|NRN|38.17|38.17|33.68|37.67|36.33|33.55|38.50|37.00|33.65|\n|ART|**42.50**|**42.12**|**35.21**|**42.83**|**42.83**|**35.24**|**41.04**|**41.04**|**34.59**|\n\n**Q4:** How to set the numbers of feature clusters and sample clusters?\n\n**A4:** The model based on eq (1) mainly shows that this paradigm can factorize multi-view data as feature clusters and sample clusters as well as association to mine latent patterns. For generality, the numbers of feature clusters and sample clusters can be any value (smaller than the feature dimension and sample size). In practice, we set their values just as $k$. The benefits are multifold. Dataset will be divided into $k$ groups, whether from the view of samples or features. The misregistration will be relieved owing to the small sample cluster number. Also, this makes the association small-sized, saving resources.\n\n**Q5:** Does assigning view guidance to samples (VGS) facilitate the results? \n\n**A5:** This a solution to avoid dimension difference. The following experiments reveal the performance. DAM: devised assigning mechanism.\n\n|FLOEVEN||||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|VGS|10.83|10.32|5.47|10.77|9.73|5.58|10.63|10.07|5.37|\n|DAM|**11.17**|**10.80**|**5.89**|**11.25**|**10.53**|**5.88**|**11.54**|**10.74**|**5.83**|\n|**SYNTHREED**||||||||||\n|VGS|41.53|40.93|33.97|41.74|42.17|34.82|39.92|39.83|33.64|\n|DAM|**42.50**|**42.12**|**35.21**|**42.83**|**42.83**|**35.24**|**41.04**|**41.04**|**34.59**|\n\nThe reasons for superior results of VGS are that this operation filters out noisy and data redundancy. For inferior results, possible reasons are that the operation causes information loss in diversity.\n\n**Q6:** Performance when using the association in eq (1).\n\n**A6:** This will build an association on each view for feature clusters and sample clusters separately. SAS: separated association scheme in eq (1); DUA: devised unified association. \n\n|FLOEVEN||||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|SAS|10.95|10.50|5.35|10.54|10.24|5.76|11.43|10.60|**6.01**|\n|DUA|**11.17**|**10.80**|**5.89**|**11.25**|**10.53**|**5.88**|**11.54**|**10.74**|5.83|\n|**SYNTHREED**||||||||||\n|SAS|40.51|40.57|**35.95**|41.48|41.37|34.46|40.76|**41.76**|33.18|\n|DUA|**42.50**|**42.12**|35.21|**42.83**|**42.83**|**35.24**|**41.04**|41.04|**34.59**|\n\nUnder eq (1), the view communication could be inadequate due to the separability. Also, this could fail to propagate feature cluster information across views into sample clusters since the association is established individually.", "claims_and_evidence": "The provided evidence supports the claims.", "methods_and_evaluation_criteria": "The method is applicable to large-scale clustering scenarios.", "theoretical_claims": "Yes. No issues were found.", "experimental_designs_or_analyses": "Yes. The designs confirm the effectiveness of proposed BACDL.", "relation_to_broader_scientific_literature": "The ability to be applicable to large-scale scenarios highlights its practicality.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Strengths, \n\n-  The designed dual-determinant learning paradigm and coupled distribution learning paradigm for IMC issues are interesting, and owns reference value for further study.\n\n-  The solution is technically sound, making linear overhead and ensuring convergence. \n\n-  The introduction of sample cluster transformation for misregistration is novel. Experiments are evaluated from multiple perspectives.\n\n\n Weaknesses\n\n The discussions regarding Remark 1 appears overly concise. It is recommended to provide more details.  \n\n-  Regarding TCIMC and LSIMV (without feature cluster bifurcating), they receive preferable results on DEOLOG.  From table 2, there is a remarkable 2.19 PUR interval. These is no interpretation for this phenomenon. \n\n-  The application of orthogonal rotation in the potential representation space to reorganize the sample clusters could deteriorate the distribution attribute. This implies that the cluster label quality is highly dependent on the rotation.", "other_comments_or_suggestions": "None", "questions_for_authors": "-   About the number of feature clusters and the number of sample clusters, how to set their values? Based on eq (1), one can have that these numbers can be arbitrary and only the association needs to satisfy the dimensionality constraints.  \n\n-  The view guidance plays a role in avoiding the dimension difference, and is related to the common representation matrix to extract perspective-shared features. If assigning it to the samples (This also can learn shared features since all sample dimensions are consistent after projecting.  Moreover, the dual-determinant representations are concurrently learned in the same dimension space), will that facilitate increases in the results?  \n\n-  Instead of the unified association, if using the framework in eq (1) to construct the sample clusters and then remapping them, it seems that the loss 1 is not necessary. In this case, is the performance still competitive? Could you please provide some empirical evidence for this baseline?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: All reviewers recognized the quality of the work.  The work has the following strengths:
* Novel idea
* Effective and efficient algorithm
* Theoretical supports
* Comprehensive numerical evaluation

According to the feedback of the reviewers on the authors' rebuttals, all questions and minor issues have been addressed.
Decision: Accept (poster)