DLEFT-MKC: Dynamic Late Fusion Multiple Kernel Clustering with Robust Tensor Learning via Min-Max Optimization
Authors: Yi Zhang
Siwei Wang
Jiyuan Liu
Shengju Yu
Zhibin Dong
Suyuan Liu
Xinwang Liu
En Zhu
Venue: ICLR 2025 Spotlight (2025)
Keywords: multiple kernel clustering; multi-view clustering; late fusion MVC
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: First to incorporate min-max paradigm into tensor-based MKC; first to dynamic reconstruct base partitions from late fusion MKC.

Abstract: Recent advancements in multiple kernel clustering (MKC) have highlighted the effectiveness of late fusion strategies, particularly in enhancing computational efficiency to near-linear complexity while achieving promising clustering performance. However, existing methods encounter three significant limitations: (1) reliance on fixed base partition matrices that do not adaptively optimize during the clustering process, thereby constraining their performance to the inherent representational capabilities of these matrices; (2) a focus on adjusting kernel weights to explore inter-view consistency and complementarity, which often neglects the intrinsic high-order correlations among views, thereby limiting the extraction of comprehensive multiple kernel information; (3) a lack of adaptive mechanisms to accommodate varying distributions within the data, which limits robustness and generalization. To address these challenges, this paper proposes a novel algorithm termed Dynamic Late Fusion Multiple Kernel Clustering with Robust {Tensor Learning via min-max optimization (DLEFT-MKC), which effectively overcomes the representational bottleneck of base partition matrices and facilitates the learning of meaningful high-order cross-view information. Specifically, it is the first to incorporate a min-max optimization paradigm into tensor-based MKC, enhancing algorithm robustness and generalization. Additionally, it dynamically reconstructs decision layers to enhance representation capabilities and subsequently stacks the reconstructed representations for tensor learning that promotes the capture of high-order associations and cluster structures across views, ultimately yielding consensus clustering partitions. To solve the resultant optimization problem, we innovatively design a strategy that combines reduced gradient descent with the alternating direction method of multipliers, ensuring convergence to local optima while maintaining high computational efficiency. Extensive experimental results across various benchmark datasets validate the superior effectiveness and efficiency of the proposed DLEFT-MKC.

Published as a conference paper at ICLR 2025
DLEFT-MKC: DYNAMIC LATE FUSION MULTIPLE KERNEL CLUSTERING WITH ROBUST TENSOR LEARN- ING VIA MIN-MAX OPTIMIZATION
Yi Zhang
Siwei Wang
College of Computer
Intelligent Game and Decision Lab Academy of Military Sciences Beijing 100091, China wangsiweil 3@nudt edu. cn
National University of Defense Technolgy
Changsha, Hunan 410073, China zhangy@nudt. edu. cn
Jiyuan Liu, Shengju Yu, Zhibin Dong, Suyuan Liu, Xinwang Liu" & En Zhu College Computer, National University Defense Technolgy Changsha, Hunan 410073, China
(xinwangliu, enzhu 1arahueemna @nudt. cn
ABSTRACT
Recent advancements in multiple kernel clustering (MKC) have highlighted the effectiveness of late fusion strategies, particularly in enhancing computational ef- ficiency near-linear complexity while achieving promising clustering perfor- mance. However, existing methods encounter three significant limitations: (1) reliance fixed base partition matrices that do not adaptively optimize during the clustering process, thereby constraining their performance to the inherent rep- resentational capabilities of these matrices; (2) focus adjusting kernel weights to explore inter-view consistency and complementarity. which often neglects the intrinsic high-order correlations among views, thereby limiting the extraction of comprehensive multiple kernel information; (3) lack of adaptive mechanisms accommodate varying distributions within the data, which limits robustness and generalization. To address these challenges, this paper proposes novel al- gorithm termed Dynamic LatE Fusion Multiple Kernel Clustering with Robust Tensor Learning via min-max optimization (DLEFT-MKC), which effectively overcomes the representational bottleneck of base partition matrices and facili- tates the learning of meaningful high-order cross-view information. Specifically, it is the first to incorporate a min-max optimization paradigm into tensor- based MKC, enhancing algorithm robustness and generalization. Additionally, it dy- namically reconstructs decision layers to enhance representation capabilities and subsequently stacks the reconstructed representations for tensor learning that pro- motes the capture of high-order associations and cluster structures across views, ultimately yielding consensus clustering partitions. To solve the resultant opti- mization problem, we innovatively design strategy that combines reduced gradi- ent descent with the alternating direction method multipliers, ensuring conver- gence local optima while maintaining high computational efficiency. Extensive experimental results across various benchmark datasets validate the superior ef- fectiveness and efficiency of the proposed DLEFT-MKC.
INTRODUCTION
Multiple Kernel Clustering (MKC) has emerged as a crucial technique in machine learning, aimed at analyzing complex linearly-inseprrable data by projecting data features into higher-dimensionaal or even dinnnte-iimeniionll spaces, Reproducing Kernel Hilbert Space (RKHS), thus transform- ing data into linearly separable entities (Filippone et al.) 2008 Marin et al. 2017 Blanco Valencial et al. 2017]. Given today's 's era of big data, where almost all data encompass multiple distinct rep- resentations views, MKC algorithms have naturally garnered considerable attention and study in the field. They integrate multi-source information within the kernel space, subsequently assign-
ing samples distinct clusters (Gönen & Alpaydın 201] Kumar & Daumé 2011 Chitta et al.)
corresponding author

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2012 Tang et al. 2022). Specifically, MKC primarily learns an optimally combined kernel by min- ing information from multiple views, subsequently serving the clustering tasks. This methodology particularly beneficial across various real-world applications, including image recognition, natural
language processing, anomaly detection, and bioinformatics (Peng et al. 2019 Zhou et al. 2020
Zhang et al.) 2022a Wang et al., 2022 Yu et al., 2023b a Tang et al. 2023 Yu et al. 2024).
Some representative MKC algorithms believe that the optimal kernel is a linear combination of base
kernels (Huang et al. 2012 Gönen Margolin 2014] [Bang et al. 2018). The neighborhood kernel learning methods seek non- lonelinearraaao combination of base kernels for better representability of the optimal kernel (Liu et al.; 2017, 2020). A matrix- regularization is introduced to consider the selection of kernels (Liu et al. 2016 Hu et al. 2211)) Recently, min-max framework has been introduced to seek optimism in pessimism (Liu) 2023b), and some variants are proposed (Liu et al. 2021c Liu 2023a). An overall process fusion manner is proposed to deepen the degree of fusion between views (Zhang et al. 2022b). Furthermore, tensor-based multi-view clustering algorithms also received lot of attention due to their capability to study the high-order correlations among views and achieve encouraging performance. [w et al.] (2019) reorganize the affinity matrices into tensor form and Iearn its intrinsic tensor based on low-rank tensor approximation. The weighted t-TNN is introduced to reflect the importance of different cigenvalues (Gao et al., 2020), A new graph learning paradigm is proposed to enable the affinity graph propagated from KKM to enjoy the valuable block diagonal and sparse property through an explicit theoretical connection between the clustering indicator matrix and affinity graph (Ren et al. 2021). Chen et al. (2022c) stacks multiple affinity representations in a low-rank constrained tensor to recover their comprehensiveness and higher-order correlations. Late fusion MKC methods propose to first cluster each individual view and then fuse these results into cohesive solution (Wang et al.) 2019b). The development of late fusion strategics has further transformed MKC techniques (Zhang et al., 2021] Liu et al.] 2021b); these approaches not only benefit exploring cluster structures during the fusion process but also significantly enhance computational efficiency, achieving near- complexity.
Despite the encouraging improvement in clustering performance, several critical challenges remain unaddressed in MKC. First, late fusion MKC relies on fixed initial base partitions that do not adap- tively optimize during the clustering process, presenting a bottleneck in performance due to their inherent representational limitations, suboptimal starting points can severely compromise final out- comes, Additionally, many existing MKC algorithms focus on adjusting kernel weights to explore inter-view consistency and complementarity, and often overlook the intrinsic high-order correlations among views, thereby limiting the extraction of comprehensive multiple kernel information. Fur- thermore, existing MKC methods frequently lack adaptive mechanisms that accommodate varying distributions within the data, thus limiting their robustness and generalization.
To tackle these issues, we propose a novel algorithm termed Dynamic LatE Fusion Multiple Kernel Clustering with Robust Tensor Learning via Min-Max Optimization (DLEFT-MKC), Specifically, we, for the first time, incorporate a min-max optimization paradigm into rensor-based MKC, rep- resenting a pioneering exploration aimed at enhancing both performance robustness and general- ization in clustering. Additionally, DLEFT-MKC dynamically reconstructs and calibrates the base partitions, effectively overcoming the representational bottleneck of initial base partitions. Further- more, stacking the dynamically adjusted partition matrices into tensors while applying t-TNN con- straints promotes the learning of meaningful higher-order correlations and cluster structures across views. We design an innovative and efficient strategy that combines the reduced gradient descent method (RGDM) with the alternating direction method of multipliers (ADMM) to solve the resultant max-min max-min-max max optimization problem, ensuring convergence to local optima while maintaining high computational efficiency. For evaluating the proposed algorithm, we conduct comprehensive exper- imental studies in terms of clustering performance, evolution and convergence, cluster partitions, parameter sensitivity, ablation study, and time complexity. Extensive experimental results across various benchmark datasets validate the effectiveness and efficiency of our proposed DLEFT-MKC.
The primary contributions of this paper are summarized as follows,
This study is the first to incorporate a min-max optimization paradigm into tensor-based MKC, which represents pioneering exploration of min-max optimization aimed at en- hancing both performance and robustness in clustering.
We propose groundbreaking approach for the dynamical reconstruction and calibration of base partition matrices from LFMVC, effectively overcoming their representational bot- tleneck and enhancing clustering performance.
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We stack the reconstructed representations into tensors and optimize dynamic partitions using tensor techniques, significantly enhancing our ability to learn high-order correlations and uncover latent structures across views.
To solve the resultant optimization problem, We design an innovative and efficient strategy to combine the RGDM with ADMM. Extensive experimental results across various bench- mark datasets validate both the effectiveness and efficiency of our proposed algorithm.
2 RELATED WORK
2.1 MULTIPLE KERNEL k-MEANS CLUSTERING (MKKM)
The k-means clustering algorithm aims to partition data points into k clusters by minimizing intra- cluster distances and maximizing inter-cluster distances. Its objective can be articulated follows:
min S.C 1=1 Σ k,, - c,ll², s.t. $1 1,
(1)
where æ; represents the i-th data sample.e,denotes the center of the j-th cluster,S R serves as the clustering assignment matrix. If the i-th sample is assigned to the j-th cluster, then S,, 1. and n and denote the number of samples and clusters, respectively.
Many approaches capture structural information by mapping features into Reproducing Kernel Hilbert Space (RKHS) address complex data that linearly inseparable. Notably, the dimen- sionality of mapped features can be extremely high or even infinite; therefore, kernel methods are typically employed to compute the kernel matrix, thereby avoiding explicit mapping. By defining F= ( the clustering assignment matrix S relaxed into the real domain, where L E R is diagonal matrix with each diagonal element representing the reciprocal of the sum of its correspond- ing column in matrix S. Consequently, the kernel K-means clustering algorithm can be articulated as follows:
min II (K(I ())) s.t. F FFF-I
FF")),
(2)
where K denotes the kernel matrix calculated using implicit mapping function ((-).
Following the framework of multiple kernel learning (Rakotomamonjy et al.) 2008), the kernel K- means method can be extended to multi-view scenarios, assuming that optimal consensus kernel matrix can be derived as a linear combination of predefined base kernel matrices. Therefore, the framework of multiple kernel K-means clustering can be formally articulated as follows:
min Tr (I (TT,IIFFF)) FFT)). s.t. F ==FII- E F.7
(3) where K, = >p== 72 K2 denotes combination of kernel matrices from different views, K, is the kernel matrix of p-th view, Y serve as the corresponding weight coefficient for each kernel view with A fr E RTT Σ/e 7 1, % 0, vp).mn denotes the number of views. According to the existing literature, the optimization problem of MKKM can typically be solved using coordinate descent optimization techniques that iteratively optimize specific variables while keeping others fixed.
2.2 LATE FUSION MULTI-VIEW CLUSTERING
Recently, the literature Wang et al. 2019b) has proposed a method known as Late Fusion Multi-view Clustering (LFMVC) address the computational complexity challenges associated with MKC. Unlike traditional multiple kernel K-means methods that represent distribution information from different views through weighted combination of kernel matrices (K,} Raxx LFMVC integrates information at the decision level by utilizing smaller base partition matrices {F,) Ra*k to capture data distributions for each kernel view. This strategy significantly reduces both time and memory overhead during the MKC process. Specifically, Late fusion MKC aims to learn a consensus clustering partition matrix F" E = by integrating individual base partition matrices pm Its objective function emphasizes maximizing alignment between weighted base partitions generated from different views and the consensus partition:
Tr F " Tip FDT,), s.t. F F I, I, T
V
(4)
F.Tp.y
where y denotes the weight coefficients of various kernel views, V = (7 R >p 0, Vp). and T, E kk is the p-th permutation matrix, for the better alignment among base partitions from various views.
7 - 7p
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We can observe from Eq.44) that LFMVC aims to optimize its objective function by maximizing all involved variables. To achieve this objective, a coordinate descent method has been developed for optimization purposes. As analyzed in previous studies (Wang et al. 20196), LFMVC's near-linear computational complexity and efficiency enable it to handle large-scale clustering tasks effectively.
2.3
PRELIMINARIES OF 3-ORDER TENSOR
2.3.1 TENSOR SINGULAR VALUE DECOMPOSITION (T-SVD)
For tensor A E IR X its t-SV can be factorized as A = U= V where E Re Xns and S E Ih X are orthogonal tensors, and E 18 X is f-diagonal tensor, whose each frontal slices is a diagonal matrix. According to the literature (Kilmer et al. [2013] Kilmer & Martin, 2011 the above t-SVD problem can be efficiently settled by matrix SVD in the Fourier domain, i.e. A = 5 V k = 1,2, :. 7gg-
2.3.2 T-SVD BASED TENSOR NUCLEAR NORM (T-TNN)
For a tensor A E RniXngxns, its t-TNN can be expressed as. E Emin(ou (Ax), where (Aa) denotes the i-th largest singular value of Au.
Σx Axll+
Note that according (Zhang et al.) 2014 Semerci et al. 2014), t-TNN is proven be valid and the tightest convex relaxation to 1 -nor of the tensor multi-rank.
3 PROPOSED
3.1 FORMULATION
We propose novel dynamic late-fusion multiple kernel clustering algorithm based on robust ten- sor learning through min-max optimization, effectively addressing the representational bottleneck of base partition matrices and facilitating the acquisition of meaningful high-order cross-view formation. Specifically, we first incorporate min-max optimization paradigm into tensor-base based MKC, which represents a pioneering exploration of min-max optimization designed to enhance both performance and robustness in clustering. Additionally, the proposed algorithm dynamically reconstructs and calibrates the base partition matrix, effectively overcoming constraints imposed by initial representational limitations. Furthermore, stacking the dynamically adjusted partition matri- ces into tensors while applying t-TNN constraints promotes the learning higher-order correlations and cluster structures across views.
To do so, we first introduce the dynamic partitions [Fp]pm to reconstruct the base partition ma- trices {F of late fusion strategy based MKC. Next, We maximize the alignment between the reconstructed and base partitions to ensure the quality reconstruction, and dynamically optimize this alignment during the subsequent iterations. Furthermore, explore and capture higher-order intrinsic correlations across views, we stack the dynamic reconstruction {Fp)p into a tensor and optimize it with t-TNN. Additionally, we impose an orthogonal constraint on to preserve its capacity to reveal the clustering structure. Thus we can obtain the following expression:
Tr(f,, F2)
p||F| Ø. s.l. F.
I, Vp.
F. F.
Next, we attempt to directly learn the consensus clustering partition by incorporating E5. and permutation matrices {Ty}7= with the paradigm of LFMVC. In addition, due to the different contri- butions of various views, we assign kernel weight coefficients y to each view in order to sufficiently mine and learn each kernel view with particular emphasis.
Finally, pioneeringly introduce the min-max paradigm into the resultant objective function, which minimizes the function w.r.t & and maximizes w.r.t. Fr F, and T... Therefore, the final objective function can be expressed as follows,
max min max Tr( Tr(F* F. T. F
E"" p=1 F2TTT)) +
IF F2) Fll
s.t. F. 1
This max-min-ve paradigm denotes that we maximize the alignment between the consensus clus- tering partition and base partitions while optimizing the kernel weight coefficients minimize the
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objective function, preventing premature convergence local optima. In this way, the proposed algorithm can robustly learn the optimal consensus clustering partition even under challenging con- ditions.
3.2 OPTIMIZATION
To solve the resultant max-min-nme max optimization problem of DLEFT-MKC in 6). we combine the optimization strategies the reduced gradient descent method (RGDM) and Alternating Direc- tion Method of Multipliers (ADMM), updating one specific variable while keeping others fixed. To facilitate the divisibility of F, we introduce an auxiliary tensor variable according to the principles of ADMM and obtain the augmented Lagrangian function follows,
L(A, Fp, T
F
(F FØT, + XF, F2) p pllAle A' 111A- (F 2
8.t. F, ,,
V. A F
where, Y E Raxxxx represents the Lagrange multiplier, with م>> 0 acting as the penalization factor. alternating optimization strategy allows for the decomposition of the problem in Eq.(7) into five distinct uub-problems. Each sub uub-problem independently optimizes its respective variables while keeping others fixed.
update mm By fixing the other variables, (Fp) mm can be updated as follows,
max
p=1 (I F,T, + XE, F2) ||A (f
# lle, s.t. F, F. Fp II
Y
(8)
Then by expanding the Frobenius norm and simplifying this problem, we can obtain the following problem w.r.t. each Fp:
max Tax (T. + Ny + μAp Y,)), s.t. F, P2
where A, and Y, represent the p-th slice A and Y, respectively. By setting M, 7Ø FTT F, + HA,, Y, the problem in (9. can be effectively addressed by applying the economic rank-k SVD of Mp. Assume that the matrix M, possesses rank-k truncated SVD representation given by M, Ug V." where U₂ E R Eg E BAxk V R Then, the problem in Eq. has a closed-form optimal solution given by.
F, = V.V...
(10)
update and F": By fixing the other variables, we derive a min-max optimization problem w.rt. and F as follows,
Σ TT XF Fp), s.t. A, F
min max F
(11)
To solve it, we begin by rewriting it as an optimal value function of the maximization problem as follows,
min ((r), G(>)= = max Tr(F"T
2 F2)
(12)
YEA
According to Theorem 4.1 in the literature (Bonnans & Shapiro tion ((>) in 12 is differentiable, and 0g(()) TT Tr
1998 the optimal value fun- F- F,T, XF, F.). where F-
arg max Tr F (Em "E,,,,) ;))
Therefore, reduced gradient descent strategy can be employed to address the optimization problem in Eq. (2). According to the literature (Liu 2023b), we firstly calculate the gradient of g(>) w.l.. 7, and subsequently optimize 7 along a descent direc- tion while satisfying the constraint A, with the optimal F".
To do so, we should guarantee the cquality constraint and positivity constraint of y. First, we designate To non-zero component of and 79(r) the reduced gradient of G(y). By following Rakotomamonjy et al. (2008); ; Liu (2023b), 79(7) can be expressed as follows,
(v9(T)), = 09(>) ag(r)
Vp u, [v9())). E.
(8g())
09(7))
(13)
a%
0%
p 1. 0% O1p
where u typically denotes the index of the largest component of 7,, as suggested by Rakotomamonjy et al. 2008), which is regarded providing improved numerical stability.
Next, to ensure that 7y remains positive at all times, we design the calculation strategy of the descent direction for updating 7 follows,
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if 7o = and 179(>)) 0,
Up
[v9()), (
if =p=u. otherwise.
(14)
After deriving the descent direction Y using Y + aV, where is chosen step size that could be determined using line search strategies such as Armijo's rule. Overall, the algorithm for solving the optimization problem in (2) is outlined in Algorithm[T the appendix.
from Eq. 14), we can then update the weights
update A: By fixing the other variables, the A sub- sub-problem constitutes t-TNN minimization problem and can be articulated as follows,
1 11A-OF+) + Y
min pl All
(15)
Let B F+ Y. the sub-problem 15 can addressed using the tensor tubal-shrinkage of B, as detailed in Theorem
Theorem 1 Zhou et al. 2019a Given A, B E R X I mn n2), we can have U V by t-SD. The global optimal solution mina |||A|| 1 I B & pro-
vided by the tensor tubal-shrinkage operator, i.c., (B) U iffu (B) VT, where
B fft 3). and P-(B) is tensor whose i-th frontal slice is P.(B;) (aag(Ei.e,....... with ; = sign(a; (B.) )m)aaaa (B,) B.) O).
update {Tp)p=> By fixing the other variables, (T,)²= sub-problems can be addressed as fol- lows,
IIxX Tp
Tr(F FTT,) s.L. T, T,
T- = I,
(16)
which can be readily solved in manner similar to that of the problem in Eq.
update Y and μı: The penalty factor y and the Lagrange multiplier are updated follows,
Y = Y ( A). 7 x μ4
(17)
where the literature typically sets > 1 to enhance convergence speed (Chen et al. 2020), and we
2 in this paper.
In conclusion, we present the algorithm process of DLEFT-MKC in Algorithm 2 in the appendix.
4 EXPERIMENT
4.1 EXPERIMENT SETTING
We utilize multiple benchmark datasets evalu- ate the clustering performance of DLEFT-MKC, including: Live BBCSpor ProteinFold Wil-
Dataset
#Samples
#clusters
Liver
345
6
Bhesport
544
2
ProtcinFold
694
12
27
low Plan PsortNeg Scene15 (Lazebnik al., 2006 CCV Flowerl02"] Reuters Tab. sum-
Plant
940
69
marizes the detailed information regarding the uti- lized datasets. These datasets exhibit considerable variation in sample sizes (345 to 18,758), kernel counts (2 to 69), and cluster numbers (2 to 102), thereby offering balanced experimental platform for evaluating the clustering performance dif- fering algorithms. For all datasets, the true number of clusters k is predetermined and provided
PsortNeg
1444
69
5
Scene
4485
3
15
CCV
6773
3
20
Flower102
8189
4
Reuters
18758
5
Table 1: Summary of datasets used.
ht h/tpaa///////ee.../// ies. adu/ datasst//
attp: / http:r//ml..ucd..e//atteee.
attp: btsp:///kk...r http::/ddk........dddd/eeeeeeeed pteee
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as input. We apply four commonly used criteria: clustering accuracy (ACC), normalized mutual information (NMI), purity (PUR), and rand index (RI). We evaluate the proposed DLEFT-MKC in terms of clustering performance, evolution and convergence, cluster partitions, parameter sensitiv- ity, ablation study, and time complexity. The complete experimental results, along with sufficient instructions for reproducibility, are provided in the appendix.
Along with DLEFT-MKC. we compare it against numerous MKC algorithms selected from recent literature. Specifically, Avg-KKM and SB-KKM serve as baselines that perform KKM on average kernel and single kernel without additional operations. We also include classical MKC algorithms such as MKKM (Huang et al., 2012), LMKKM (Gönen & Margolin| 2014), ONKC (Liu et al.) 2017), MKKM-MR (Liu al. 2016), and LKAM (Li et al., 2016). Furthermore, we incorpo- rate some recent methods including subspace-bu basedd, graph-based, and tensor- -based approaches, i.e., LFMVC (Wang et al., 2019b), NKSS (Zhou et al. 2019b), SPMKC (Ren & Sun 2020), HMKC (Liu et al. 2021a), SMKKM (Liul (023b), OPLFMVC (Liu et al. (02Ib). LSMKKM (Liu et al.
2021c). AIMC (Chen et al. 2022a), OMSC (Chen al., 2022b), HFLSMKKM (Liu) (203a)). LTBPL
GMC Wang et al. 2019a),
(Chen et al. 2022c), UGLTI. (Wu et al., 2019), WTNNM
Gao et al. 22200). KCGT Ren et al. 2021).
For all algorithms, we adhere to guidelines in the literature for parameter configuration. In addition, each experiment conducted 20 times using k-means reduce the adverse impact of randomness. The average results, along with standard deviations, are then reported.
4.2
EXPERIMENTAL RESULTS
4.2.1 CLUSTERING PERFORMANCE
Tab presents comparison of ACC among the aforementioned algorithms, where "-" signifies that the results are unavailable due to an out-of-memory error, with the top three results being high- lighted. Note that comparisons regarding NMI, PUR, and RI are included in the appendix. Then the following observations can be drawn:
(1) Recent advancements in clustering algorithms have demonstrated that tensor-based clustering methods yield significant performance improvements, particularly when compared to traditional MKC algorithms. For instance, the algorithms LTBPL, UGLTL, WTNNM, and KCGT, which are based on tensor learning, consistently outperformed conventional methods like SMKKM and HMKC across all ten datasets. Specifically, UGLTL and WTNNM achieved average ACC improve- ments of approximately 14.8% and 7.8%, respectively, over the hest-performing traditional MKC.
(2) Despite these advancements, these tensor-based approaches still face challenges related to com- putational efficiency and scalability. For example, LTBPL, UGLTL, and WTNNM are unable to handle the Reuters dataset effectively, In contrast, our proposed DLEFT-MKC demonstrates a sig-
Algarilime
Liver
BBCSpprt
ProtzinFald
Willow
Pbat
PannNNy
Suanel
CCV
Flawcrift2
Avg-KKM
54.2g
63.2
20.00
22.2
61.3 2.0.
:1.0
43-2
19.0
27.1 2...8
15.5
57.
T1.4
1.3
1.3
51.21:1
2..
20.1
1..
MKKM
53.0103
23
21
22.0
56.1 2.0
51.9g0
18.0200
22.4.
45
LMKKM
53.7
63.9
22
22.0
40.9
18.0
GNKC
1.
63
LG
22.
41.4
40.2
22.4
30.5
41.8
MKKM-MR
34.7
50.3 a.
21.2
40.2 n.
46.2
LKAM
0..0
78.9 n5
1.2
2..1
2.3
41.4
LEMVC
5..
2.0
3..
2..5
29.5
4.
1.0
25.1
38.4
45.7
NKSS
55.0
64.1113
26.4
25.5
39.2
48.2
40.1
20.
41.7
37.4
SPMKC
54.5g..
51.
17.8
26.3
51.4g.1
25.0
18.0
16.2
25.0.04
26.820
HMKC
0..
35.
1..5
64.2
19.1 10000
n..3
10.
47.7
16.8
SMKKM
23.9100
14
34.7 1.0
22.4
10.5
41.0200
1.0
22.2
42.0
45.6
OPLEMVC
54.6 0.
89.2
31.1 2.0
27.3
47.1
46.1 298
43.9 1.
23.7
30.4
43.9
LSMKKM
58.5 58.3.1.0
73.4
L.
21.
57.1
45.7
44.5
21.5
43.8.1
17.1
AIMC
48.2
OMSC
53.0
89.0 n.
2..0
56.5 0.D
D.D
2..
38.9
4..4
HELSMKKM
57.4
11.
24.2
43.
31.3
41.7
18.
35.8
37.5
GMC
51.0103
88 28.2
20.360
21.200
39.4
25.2
26.0
16.8
34.1200
LTRPI.
58-3100
96.5 06.5gg
22.1
28.8.00
48.2
29.1
53.6..0
9..1
37.1
68.0
92.2.00
94.4
65.8
WTNNM
6..
0.0
43.2 17
82.0
68.0 6..3
64.8....0
76.1 1:12
47.7 o..
61.7.00
KCGT
6..
7.
33.4
2.2
26.1
52.4
44.9
45.6
45.6
23.9 0.5
39.5 39.5
43.0
DLEFT MICC
86.4
90.2
66.5
84.9
94.1
96.0
96.2
81.
79.9
97.0
Table 2: Empirical comparison of the proposed DLEFT-MKC with dozens of recent MKC algo- rithms on ten benchmark datasets in terms of ACC, The best result is bolded and highlighted in red, the second-best and third-best ones represented in blue and orange, respectively.
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nificant performance enhancement by surpassing UGLTL and WTNNM in terms of ACC by 32.8%, 0.1%. 15.4%, 47.8%. 25.5%, 3.8%. 1.8%. 37.8 37.8%, 14.1% as well as 33.1%. 4%, 23.3%, 52.9%, 26.1%, 31.2%, 20.1%, 33.8%, 18.2% across all datasets.
(3) The comparative analysis of various algorithms reveals that while late fusion strategies have improved clustering performance, they are not without limitations; for instance, LFMVC and OPLFMVC showed reduced complexity but struggled with unstable performance due to their heavy reliance on initial base partitions too much. Through dynamic restructuring of partitions, DLEFT-MKC significantly enhances performance; it exceeds LFMVC and OPLEMVC by 43.1% and 44.4%, respectively when averaged over ten datasets.
In summary, these results validate the effectiveness of our proposed DLEFT-MKC in enhancing clus- tering performance across multiple datasets; significant improvements in ACC- CeC--aveaggng around 10% over existing tfate-of-the-all view clustering research domains. By leveraging tensor learning alongside a min-max optimization framework, our approach addresses existing challenges and sets search this domain. DLEFT-MKC's ability to maintain high accuracy while significantly reducing computational complexity demonstrates its robustness and efficiency, qualities that particularly essential for real-world applications requiring large-scale data processing.
cettriihmms-udddrseeoo its potential a leading solution in multi-
benchmark for future
4.2.2 EVOLUTION AND CONVERGENCE
We calculate the error value and clustering performance at each iteration to analyze the evolution of DLEFT-MKC, illustrated in Fig. [ As observed, the error curve initially oscillates, followed by a sharp decrease, and ultimately converges rapidly. The corresponding clustering performance im- proves significantly during the initial oscillation phase (learning process) before stabilizing, thereby effectively demonstrating both the necessity and efficacy of the learning process. This analysis highlights how DLEFT-MKC adapts over iterations, leading to enhanced clustering results.
Ebagee
Scmer5
3.25
311
Medin
connnn/erteeeee 80
The waw.aarraacc Saraloo
Namber afranaii
The Number of Faansiooa
The Number
The Nu ober Becationa
Figure I:
The evolution of error values and clustering performance during the clustering learning
process of our proposed DLEFT-MKC across iterations.
4.2.3 CLUSTER PARTITIONS ANALYSIS
We further analyze the learned cluster partition and illustrate the visual results in Figure 2 As ob- served, through the learning process of DLEFT-MKC, the cluster partition becomes increasingly clear and distinguishable, manifested as a more pronounced block diagonal structure. This obser- vation further reinforces the effectiveness of our proposed DLEFT-MKC in achieving well-defined clusters. These results demonstrate that DLEFT-MKC not only enhances clustering performance but also facilitates better interpretabilityy of the clustered data.
4.2.4 PARAMETERS SENSITIVITY ANALYSIS
In order to further investigate the impact of two parameters DLEFT-MKC, we conducted sep- arate experiment to analyze their sensitivity and effectiveness, as illustrated in Figure 3 As shown, two trade-off parameters introduced by DLEFT-MKC exert significant effects, demonstrating regu- larity and consistency across various datasets. This indicates that each term in Eq,(6) plays a crucial role, suggesting that DI DLFFT-MKC exhibits stability within small ranges of parameters while main-
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Published as a conference paper at ICLR 2025
Amplec
100
430
400
700
800
Faudes Ftantonn
Derms sbridon5
Deci's Complnnc
260
5000
500
500
1900
1000
1000
3000
0000
1900
2000
500
1200
500
The leftmost figure denotes the clustering partition learned by avg- wvg-KKM. The four right figures represent the clustering partitions of DLEFT-MKC during the learning process.
Figure 2:
taining good generalization ability. In addition, this analysis can guide the adjustment strategies for DLEFT-MKC across different datasets.
Pve
CDV
Power
800
ACTEV
C
ACCINN
40S0 8 3
5 5
Tower 82
AC220
Figure 3: The effect on clustering performance with varying parameter (1st line) and n2nd line) of the proposed DLEFT-MKC.
4.2.5 ABLATION STUDY
To investigate the factors contributing to the superior performance of the proposed DLEFT-MKC, we conducted series of ablation experiments focusing four key components: the active reconstruc- tion of the base partitions, tensor learning guidance, the alignment strategy utilizing permutation matrices, and the min min-max optimization paradigm. Specifically La denotes the absence of the min-max paradigm. As shown, our proposed DLFFT-MKC always achieves ei- ther superior or competitive performance, while Ls. although performing well but lacking stability, indicates the robustness afforded by the min-max paradigm. Additionally, L. also demonstrates significant performance drop, highlighting the importance of the permutation matrix. Furthermore,
Tab Ln denotes p X 0, 0, L3 denotes À = 0, La denotes the exclusion of the permutation, and L5 denotes
L1. L2, and L3 remain inferior to DLEFT-MKC, thereby underscoring the efficacy of our proposed dynamic restriction late fusion strategy utilizing tensor learning. These findings collectively vali- date the effectiveness of our proposed algorithm and underscore the importance of integrating these components for optimal clustering performance.

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Published as a conference paper at ICLR 2025
Alanriluus
Liva
BRCSpust
ProteisFold
Willow
PoonNeg
Sccnel5
Floe
C1
54.1 2..4
L.
2..
22.2 u.3
56.0 u.5
2..
4..8
1..
27:2
45.1 2.3
C2
62.3 0..0
84.8
6..7
+..2
98.7 6..7
95.8
4..8
7.
74.6+11
+
51.0
0.7
13.7
1.
31.9
44.4
54.6427
25.
60.6
51.7
C4
82.3 0..
86.8
2..
2.
+2.7
59.6
s..3
73.0
-0.4.15
87.1
60.6 2.000
2..1
26.1
2
91.0.00
9..
g6.1
1.
81.7
1.7
Proposed
86.4 0..
99.2
99.2
6..
84.9
94.1
96.0
96.2
81.5
79.9
9
Table 3: Ablation study of the proposed DLEFT-MKC, The best result are highlighted in bold.
4.2.6 RUNNING TIME COMPARISON
Finally, to evaluate the complexity of the algorithms, we report the time consumption in Fig. 4 and Tab. 9 in the appendix. The analysis of running times across various clustering algorithms reveals significant disparities computational efficiency. For instance. LFMVC takes 77.1s and 41.4s on Flower102 and Reuters; in contract, tensor-based clustering like WTNNM and KCGT exhibit markedly higher running times, with taking 34717s and 5976s on Flower 102 and KCGT reaching 1424s on Reuters. This stark difference highlights the computational demands associated with existing tensor learning approaches, which seriously undermines their application prospects in real-world scenarios. Notably, our proposed DLEFT-MKC addresses this important problem caused by tensor learning while delivering superior clustering accuracy in less than a minute on the same datasets. Overall, DLEFT-MKC not only demonstrates advanced clustering performance but also significantly reduces computational overhead, thereby validating its effectiveness for large-scale clustering tasks.
LIARO
0011
20OT
Logeitee Runnin
Figure 4: Time complexity comparison of all algorithms on benchmark datasets. For better clarity, we scaled the values and adopted logarithmic values in second.
4.2.7 ADDITIONAL EXPERIMENT
To further validate the robustness of DLEFT-MKC, we conducted additional experiments with noisy data. The visual comparisons of clustering results are presented in i in the appendix. As shown, DLEFT-MKC not only outperforms baseline algorithms terms of clustering performance also shows minimal fluctuation when subjected to noise interference, highlighting its superior robustness.
5
CONCLUSION
This paper introduces novel Multiple Kernel clustering framework known as Dynamic LatE- Fusion Multiple Kernel Clustering with Robust Tensor Learning (DLEFT-MKC) via min-max op- timization, which is simple yet effective and efficient. Specifically, For the first time, DLEFT-MKC integrates min-max optimization paradigm into tensor-based MKC, enhancing both performance and robustness; the framework dynamically reconstructs base partitions from LFMVC, effectively overcoming their representational bottleneck. Additionally, tensor learning is employed to cap- ture the high-order correlations and uncover latent structures across views. To solve the resultant optimization problem, we design an innovative and efficient strategy to combine the RGDM with ADMM. Experimental results demonstrate that proposed DLEFT-MKC significantly outper- forms other state-of-the-art MKC algorithms in terms of clustering performance and computation efficiency across benchmark datasets.
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Published as a conference paper at ICLR 2025
ACKNOWLEDGEMENTS
This work is supported by National Science and Technology Innovation 2030 Major Project un- der Grant No. 20227D0209103; National Natural Science Foundation of China under Grant No. 62441618, 62421002, 62325604, 62276271, 62476281; National Natural Science Foundation of China Joint Found under Grant No. U24A20323.
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A
APPENDIX
We include the main notations used in this manuscript Algorithm (Min-Maa Optimization For and )r ) Algorithm 2 (DLEF-MMKC) the complete experimental results terms of ACC (Tab. 5
NMI (Tab. 6. PUR (Tab. 7 and RI (Tab. running time comparison (Tab. 2 clustering evolution (Fig. 5). parameter analysis (Fig. 6 noise experiments (Fig. and theframework diagram (Fig.
in the appendix due to the space limitation of the paper.
Notation
Explanation
Notation
Explanation
n,
The number samples, clusters, and
K, gaxn
The p-th base kernel matrix
P.A
The tradc-off parameters
F E B
The consensus partition matrix
The i-th data sample
Ruxk
The p-th base partition matrix
a(
The 3-th largest singular value
Fp & Raxa
The p-th reconstructed partition matrix The p-th perturbation matrix The n-th order identity matrix The tensor stacking the matrices F The auxiliary tensor variable
|| l
The t-SV based tensor nuclear
T, RR Rexk
Y R
The kernel weight coefficient
I Rexu
(Y RM
E 7 1,
Vp
F Raxkxm
(7 Rm
2p 7p 1, 7, 2 0.
A Bankkm
Table Main notations used in this manuseript.
Algorithm Min-Max Optimization For y and F
Input: F", (Fp, pp k. A.
Ouput: Weight coefficients y and consensus clustering partition F*
1: while not converge do
2: calculate (())) via kernel k-means with K K
3: Calculate the reduced gradient [Vg())] via Eq. 13
4: Calculate the descent direction V in Eq. (4..
5: Update weight coefficients 7 < 7 + with the step size a.
6: if max by Yoial I 10 then
7: Converge.
8: end if
9: end while
Algorithm DLEFT-MKC
Input: Base partition matrices {F, {
p= the number of clusters k. trade-off parameters À and p.
Output: Consensus clustering partition F*.
Initialize F ( Fm), E, Fp, T, 1,
Vp, 1
A F 0. 0.1,
m F2F,T
2: Calculate F arg T. Tr F" 2pm
3: while not converge do
4: Update reconstructed partitions {Fp 1mm p= via Eq.
5: Update weight coefficients 7 and consensus parition F by solving Algorithm
6: Update permutation matrices pm by solving Eq. 16h.
7: Update auxiliary tensor A by solving Eq.
8: Update the Lagrange multiplier and the penalty factor a via Eq. 17
9: end while
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Reuters
45.5
47.2
1.5
1..
46.2
u.0
4..741.
37.7
26.8 1.000
46.8 46.811..3 10.33
45.5 1.7
43.9 1.0
47.1
4.00
0.0
37.5 4..8
43.0
97.0 4.0.0
Table 5: Empirical comparison of the proposed DI DLEFT-MKC with dozens of recent MKC algorithms on ten henchmarks datasets in terms of ACC. The best result bolded and highlighted in red, the second-best and third- bes-eesttt ones are represented in blue and orange, respectively.
Flower 102
27.1 27.1 0.8
33.0 1.0
1.5
40.2
41.4
38.4 1.2
41.7 2.088
25.6 0.4
47.7 1.33
42.5 42.5 1.88
30.4
1.0
4..0
38.0
35.8 0.8
34.1 0.00
65.812. 2.3
61.7 0.9
39.5
79.9
39.5
19.6 10.6
20.1 1:0.2
18.0 18.0 1.0.5
18.6
22.4
21.2
20.4
20.5
100.2
16.21002
1002
32.8 1.0.5 3..5
1.77
23.7 23.7
10.0
21.5
2.00
2.00
18.5 +0.3
16.8 16.8 10.4
1..7
47.7 10.0
20.5
81.5
CCV
19.6
20.0
Scene 13
43.2 1.8
39.3 3 0.2
0.11
1.
39.9
38.4
4.5
1.0
4..4
38.0
50.5 1.1
43.611. 1.0
43.9
44.5
44.5
41.7
41.7
26.9
0.7
94.4
7..1
45.5
96.2
40.4
PsortNeg
41.0 1..0
55.3 1..3
1.00
103
103
40.2
39.7 20.3
0.4
45.5 0.3
48.2
25.0
49.1 10.0
1..0
46.1 12.3
45.7
40.0
2.00
3..3
25.2 +0.0
92.2
64.0
44.9
96.0
29.1
61.3 0.0
51.2 1.1
56.1 0.6
0.6
41.4
5..3
u.u
59.5 U.6
1.1 51.4 1..4
64.2 0.1
49.5 1.5
57.1
47.9
56.5 0.0
39.4 39.444. 0.0
48.2 0.0
1.2
0..
5..4
Plant
39.2 39.2
47.3
43.6
94.1
Willow
2.3
1.0.3
1.3
1.2
22.
22.6104
22.910.4
1011
2.5
25.5
26.3
3..
22. 10.4
1..
24.8
2..5
28. 0.00
24.2
21.2
28.8 28.8
4.0
32.0
26.1 +0.4
84.9
ProteinFold
29.0 29.0
1.3
1..
1.77
36.3
31.8
11.
1.4
36.4 1.0.7
17.8
35.5
1..
12.
36.3
33.610.0
31.8
33.8 1.1
29.3
32.7 11.11
51.1
1.7
33.4
66.5
BBCS
1.4
71.4011
1.5
14
63.4 13.4114
63.2
U.5
22
112
51.3119
11.7
1:8.7
6..2
332
1.0
70.4
1..0
51.6113
20.2
10.
102
102
00.0
1.
99.210.1
Liver
0..2
1..
1.3
1.1
52.9
51.3100
6..0
u.u
55.9
54.510.
55.4
0.0
54.6
58.3100
52.8
5..
57.4 0.0
51.0
58.3
0.6
0.0
5..8
86.4
MKKM-MR
HF LLMKKMMM
DLEFT-MKC
Algorithms
Avg-KKN KKM
OPLEMVC
LSMKKM
SB-KKM
МKKM
LMKKM
ONKC
LKAM
LFMVC
NKSS
SPMKC
HМКС
SMKKM
AIMC
OMSC
GMC
LTBPL
UGLTL
WTNNM
KCGT
15

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Published as a conference paper at ICLR 2025
Reuters
27.4
25.5
20.3
22.3
20.7
1.0
27.4
16.8 16.8
10.
0.6
30.5
27.710.2
24.8 1.5
27.0
24.3
24.7 1.00
18.6 1..88
23 4.04
93.2
Table 6. Empirical comparison of the proposed DI DLEFT-MKC with dozens of recent MKC algorithms on ten benc menchmark datasets in terms of NMI. The best result is bolded and highlighted in red, the second-best and third- best ones represented in blue and orange, respectively.
Flower 102
46.0 0.5
48.7 1.4
42.7 1.0.
56.1 56.1 104
56.7
56.9
54.9 2.4
58.6 1.2
42.3 0.2
61.5 1.4
0.5
47.2
60.0
54.6
5..8
2..3
41.9 0.00
0..4
74.7 2..4
55.7 0.3
94.3
16.8 20.4
17.7 1:0.1
15.0 10..4
10.4
14.4
18.5
18.0 1..0
104
0..
20.1 10.33
16.9 16.9 10.2
12.1 10.1
27.6 27.6.102 1.2.2
10.3
18.1
17.8
10.0
19.1
15.110.2
15.6 15.6202 10.
58.9 58.9 1.055
38.9 1..11
20.8
77.4
CCV
16.8
4.7
37.9 1.1
0.1
38.6 1.1 0.1
38.8 0.1
37.7
37.3106
0.1
4..
39 39.4
39 3.13
45.9 0.1
40.6
41.3
41.4
41.6100
39.0
44.0
18.9
3.31
24.4
74.7
43.5+04
94.0
Scene
PsortNeg
10.7
1.00
32.2 1.2
10.2
2.77
21.6
21.8
1888 2.3
25.9 113
3.2
24.9 10.0
10.0
19.1 19.1 10.1
1.9
17.0
17.9
12.0
8.3
1.6
6.0
20.1
0.00
22.8 10.4
89.6
39.1
Plant
26.5+u0 0.0
16.9 1.1
19.5 1..5
1.5
10.5
20.4
u.u
u.8
12.6
32.9 1.4
1..
13.3
20.8
13.9
20.9 0.0 0.0
18.6
0.8 D.D
10.7 0.0
43.7 1.7
0.11
19.8
80.1
26.5
20.4
24.2
Willow
5.7 5.7.0.2 2.2
7.8 0:03
5.7 1.11
1.11
6.1 2044
6.3
8.3
2.93
5.4
7.1 7.1.0.1
11.9
10.4
8.5
6.310.1
6.6
7.7
5.010.2
2.7.0.3
7.2
1..7
13.3102
8.1
76.5
ProteinFold
41.1 1..1
111
1..
34.7
4..4
1.2
46.2
41.7
46.5 10..
1..
45.3 1..1
1111
4.1
4..
45.2
42.9
3.00
44.7 4.06
25.9 1.0.
43.7
73.2
1..0
80.2
40.3
BBCSport
1.1
1:22
1:22
1.2
43.6 1.2
44.0
1.1
4.1
65. 1.0
58.0 58.9 3.0
51.3 1.4
29. 29.9 131
1..4
1.0
7.0
65.0 14
69.6
0.D
35.4 1.2
78.3
10.
96.6
1.0
11.0
97.61022
1.10000
2.2
0.8 1.2 102
0.3
0.3
0.3
2.20000
1.2
1.0
1.0
0.0
0.9
1.500
0.4
0.9
1.9000
0.0
0.0
0.3
1..0
1.0101
45.2
Liver
2.5
MKKM-MR
LLMMKMMM
DLEFT-MKC
Algorithms
Avg-KKN KKM
OPLEMVC
LSMKKM
SB-KKM
МKKM
LMKKM
ONKC
LKAM
LFMVC
NKSS
SPMKC
HМКС
SMKKM
AIMC
OMSC
GMC
LTBPL
UGLTL
WTNNM
KCGT
16

---

Published as a conference paper at ICLR 2025
Reuters
53.0 0.4
53.9 1 0.0
52.9 12.910.5 1.5.
1..
5
0.0
53.2 U.4
46.9 4..
1.2
27.4 0.0
53.9 1.1.
53.3 1.00
51.7 1.2
5.2
52.8+0.0
49.8 0.0
46.9 0.8
97.6
50.5
Table 7: Empirical comparison of the proposed -LEET-MMC with dozens of recent MKC algorithms on ten benchmark datasets in terms of PUR. The best result bolded and highlighted in red, the second-best and third-hest ones are represented in blue and orange, respectively.
Flower102
32.3 32.3 0.6
38.4 38.4 1.0.
27.8 1.44
1..8
0..
44.6 1..6
1..8
48.7 1.44
54.5 1.8.
48.6
34.7 1
50.2
44.6
42.3
43.7 10.6
38.9 38.9 10.0
80.4
0.06
45.8
89.1
45.6
31.2
CCV
4:0.5
23.3 2..3
1.0.2
22.2 10.5
2..
24.6
23.7
232
28.2
23.6 23.6
10033
20.8
10.4
10.4
1.5
26.9 10.
24.7
28.6
27.
21.5
20.9
40.9
4..0
27.5
82.7
23.7
Scene15 15
1..
42.8 2.1
2.1
44.3 1.2
1.2
44.3
43.6
42.4-10
46.020.
0.5
44.1 102
42.
10.1
10.1
48.4 1.3
16.8
49 1..
48.620.0
44.8
0..0
27.9
42.4
95.4
80.9
49.1
96.2
53.1
49
PsortNeg
43.3 1.0
1.0
61.6 0.0
56.6 1.22
10.2
4..3
4..7
0.5
48.2 2.3 0.3
54.3 1.6
1.6
27.8 2.1
1011
53.0 10.0
42.2 2.01
49.5
43.5 0.0
32.8 2.66
27.1 1000
31.2 0..1
92.2 10.0 0.02
0..1
48.0
96.0
47.2
47.6
61.3100 0.9
53.2 0.5.
56.1 1.08
49.0 49.0 0.1
56.7 56.720..1 10.1
5..5
59.5
54.0 20.7
59. 59.0 0.1
64.2 1.0.
54.3 5..3
1.3
50.5 2.44
58.5
55.5
5..6
54.7 54.7 1.00
39.7 1.00
49.7 0.00
70.2
68.0 2.1
56.3
94.1 10.1
Plant
61.3
Willow
27. u.2
30.1 1..4
27.210.2
27.4
27.3404
27.
2.44404
29. u.4
27. 27.4
30 10.2
34.2
2.3
3..3
27.7
28.2+00
30.4
27.6
23.3
30.1 30.1 +.00
40.9
0..
0.3
84.9
ProteinFold
37 1.7
1.2
1.11
1.
1.3
11.9144
1.8
39 1.5
44.8 4..8
4.6
23.7 1.0. 10.77
42.9 11. 1.9
41.8 1.55
36 36.4 4.6
42.6 42.6
38.9 0.0
0.0
41.8 D.9
32.6 D.D
64.8
1.1
1.2
78.3
38.7
BBCSport
68.1 0.0.
0.1
0.88
0.7
0..1
68.0
0.5
7..7
72.6
56.1 1.5
91.113.7
0.9
2.1
79.21.0.5
80.5..0.
8.00
0.14
88.2
96.5 0..
99.1 1.2
0.0
78.2
99.2
Liver
58.0
58.0 58.0 0.0.
58.0 0.0
0.0
0.0
58.0
5..0
6.00
0.00
58.0
58.0 58.0 0.0.
58.0 0.0
0.0
1.00
58.0 1.00
58.3 58.3
58.0
0.00
58.040.0
58.0 58.0 0.0
58.3 18.33+0.0
58.0 0.00
0.00
5..0 0000
86.4
HF LSMKKMMM
DLEFT-MKC
MKKM-MR
Algorithms
Avg-KKN KKM
OPLEMVC
LSMKKM
SB-KKM
МKKM
LMKKM
ONKC
LKAM
LFMVC
NKSS
SPMKC
HМКС
SMKKM
AIMC
OMSC
GMC
LTBPL
UGLTL
WTNNM
KCGT
17

---

Published as a conference paper at ICLR 2025
Reuters
21.8 1.4
23.6
21.8 1.4
20.3
1.3
2.6
2..0
22.1
13.6
0.1 11.1
22.6 1.0.5
22.1 22.1 1.088
20.6 20.6
1.0.
21.6
20.0
17.8
13.6
19.3 19.3
95.4
Table 8: Empirical comparison of the proposed DLEFT-MKC with dozens of recent MKC algorithms on ten benchmark datasets in terms of RI. The best result is bolded and highlighted in red, the second -es-eettttttt and third-best ones represented in blue and orange, respectively.
Flower102
15.5
1.66
12.1 12.1 10.4
20.
20
27.2
25.
27.6
14.5104
34.2 1.1.1
28 18.8
1..
29.7
29.8
26.0
23.2
2.0
66.4
11.2
26.54.0.7
80.8
6.6 10.22
6.7 1.011
1.2.
5.6 5.0
7.7
7.2
6.11
U.2
6.2 6.2.
4.2
14.0 10.2
7.5 1.2
7.9 10.6
7.3
9.0
4.0
6.3
5.6 +0.2
35.8 3.1 1.1
26.1 20.0
9.72
71.8
CCV
6.6
Scene15 15
26.
21.4 21.4
0.1
22.6 1.11
22. 0.1
23.5
22.7
2..8
27.4
22.8 0.1
32.5 1.1
1.09
26.6 1.0
26.3
28.0
25.9
25.7
2.5
23.
92.3
17.1
20.7
92.7
21.2
PsortNeg
13.1
31.6 0.0
26.8 1.2.2
16.9
163
16.0
1.2
19.
0.1
21.810011
13. 10000
17.6
13.8100
15.1200
9.240.0
4.0
-0.5
2.7
82.8
10.1
18.8
90.5
1.2
24.6 1.2
1..4
9.8
19.0
9.1
21.7
8.5
19.1 19.1 1.00
31.0 1.22
16.9 0..8
11.3 1.8
19.7
13.5
20.8
12.0
-0.0
7.5
0.4
3..
17.5
86.2
Plant
13.9
Willow
3.1 #0.1
4.7 0.2
3.2 1.11
1.11
3.3
3.2
3.2
2.1
4.6
4.6 4.6.1..1
1.11
8.2 00.3
3.2 10.2
5.0 1..4
3..1
4.9 0..0
0.1
0.5
4.7 20.0
18.3 10.9
9.2
4.22
71.1
4.3
3.8
ProteinFold
14.4
15.
12.110.7
7.8
18.0
17.2115
1111
16.1
1..5
4.4
19.0
17.6119
2.34
19.9
19.0
1..
18.6 18.6+10
2.9
43.
2..4
16.9
58.4
15.7
BBCSport
1.9
2.22
1..2
1.5
3..
39.
1.2
3..0
44.3 4..3
79.3
1..8
81.1
61.6118
66. 20.0
7..7
22. 1.0
82.6
0..0
0.3
0.0
58.5+1.6
98.8
21.8
0.3 0.0
2.2 0.1
0.7 0.1
0.2 0.2
0.5
-0.0
3.6=0.0
0.4
1.1
0.1 0.1
0.9 0.0
0.0
0.3 0.0
0.5
2.400.0
-0.0
00.0
1.9
-..2
0.2 0.2
0.2
0.7
50.0
Liver
0.2
MKKM-MR
KLMMMKMM
DLEFT-MKC
Algorithms
Avg-KKNn KKM
OPLEMVC
LSMKKM
SB-KKM
МKKM
LMKKM
ONKC
LKAM
LFMVC
NKSS
SPMKC
HМКС
SMKKM
AIMC
OMSC
GMC
LTBPL
UGLTL
WTNNM
KCGT
18

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Published as a conference paper at ICLR 2025
Reuters
13.057
54.484
301.36
2159.1
398.62
5656.5
41.380
134155
22978
554.61
937.09
49.162
9531.5
19.736
138.50
5800.1
11424
46.6 46.609
Table 9: Running time comparison of the proposed DLEFT-MKC with dozens recent MKC algorithms on ten benchmark datasets, As seen, our proposed DLEFT-MKC alwaas achieves the leading computation efficiency, while other tensor- tensor-based -bssed MK algorithms. in contrast. requires an unacceptable amount of time consumption.
Flower
35.725
158. 48.49
73.470
414.45
99 19,434
1106.2
77.743
5642.9
21134
1871.2
393.37
85.706
2059.6
43.917
383.98
43898
486.11
867.96
34717
5976.3
58.356
CCV
13,428
34.872
25.943
2797.8
30.355
569.29
24.395
2473.3
4984.5
139.55
68.214
6.4595
989.30
4.2910
42.777
2621.3
196.31
317.14
13255
1443.1
12,042
268
Scene
4.2126
13.745
12.313
665.57
79.285
12,420
168.39
8.0331
732.98
1457.5
42.189
18.915
2.3429
262.69
2.2632
24.322
1054.8
37.545
6638.9
62.496
1298.4
599.97
6.8630
PsortNeg
0.7345
29. 550
10.838
35.544
23,433
72.125
3.4399
977.77
35.600
45.927
14.805
1.4863
8.7318
1.1370
10.648
50.044
22.933
1149.6
119.22
8567.7
559.06
25.598
Plant
0.3875
16.860
2.2695
12.656
8.8523
28.169
1.5419
320.96
63.280
20.778
13.746
1.3099
3.4047
0.7790
5.1713
84.367
17.564
518.88
34.669
2800.6
197.81
6.2585
Willow
0.3319
0.9097
0.4163
4.3772
1.1470
0.8178
1.6871
0.6584
11.178
16.217
2.7640
1.7630
0.3462
2.0128
0.1251
3.3521
85.690
1.1743
47.276
1.1987
12.762
9.3431
0.6476
ProteinFold
0.5382
6.9803
0.8964
387.04
2,2594
1.2601
1.8104
1.0644
23.620
17.952
15.854
2.8167
0.5692
2.4147
0.6204
4.2627
55.654
1.9474
73.129
2,9899
76.943
32.411
5.4057
BBCtt
0.1153
0.1621
0.1344
1.1007
0.5664
0.4537
0.6089
0.2971
2.3590
11.701
0.5222
1.0544
0.1658
0.9182
0.1153
1.9912
12.267
0.6732
7.3862
0.9727
9.1383
2.5097
0.1730
Liver
0.0358
0.1209
0.0454
12.791
0.2302
0.1530
0.2035
0.1386
0.8090
6.3990
0.6043
0.1669
0.0834
0.2166
0.0303
1.4543
12.877
0.4501
3,1920
0.2907
3.3709
2.0792
0.0875
HFL SMKKM
DLEFT-MKC
МKKM-MR
Algorithms
Avg-K KKM
SB- KK-K
МKKм
LMKKM
ONKC
LKAM
LFMVC
NKSS
SPMKC
HМКС
SMKKM
OPLEMVC
LSMKKM
AIMC
OMSC
GMC
LTBPL
UGLTL
WTNNM
KCGT
19

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Published as a conference paper at ICLR 2025
25
15
Figure 5: The evolution of error values and clustering performance during the clustering learning process of our proposed DI OLEFT-MKC across iterations.
The Numbber of tneations
The Number of ltte Inttttionn
20
The Numoor of Iibraton
The Nu ber of Ieeret
Reuters
Phant
AGC
16
Rauters
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0.08
0.00
0.04
0.02
100
R
1030000t100 Blsssnoo
The Number Ineritions
The Number of lte cntorns
-
10
The Numbe Itee diloor
---
The Number Pocatooo
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W Now
Flower 102
ASC
025
0.00
0.1
000
0.02
Numbee bwnalora
The Number of Itte Intrrtioons
10
The Number Ite Ienatioos
10
The Numoor attonn
ProsenFol
CDV
ProtbinFold
DCV
100
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0.00
0005
000
03 www. de Dig.oyanng
8
Por
The Numbhrr of Herations
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10
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0.25
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16
25
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The Number of Ioe aatoons
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20
Phonfheg
Plorhhog
15
025
S0d af P0R0RAA
015
01
Bin of 2000000
Cuuiii -
2
20

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Published as a conference paper at ICLR 2025
105
Hant
X
Figure 6: The effect on clustering performance with varying parameter À (line 1-2) and p (line 3-4) of the proposed DLEFT-MKC
10
1N/DDV
INIDOV
121
45
Flowerrtz
74
10"
9
8
9
IN/DOV
(SIDOV
(S4DOV
94
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74
129
34
10
8
70
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10
14
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12
12"
12
(670OV
Pcorloog
Peorling
12
10"1
9
21

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Published as a conference paper at ICLR 2025
Figure 7: Comparison of comprehensive clustering performance fluctuations between DLEFT-MKC algorithm and comparative algorithm under different degrees of
OLEFT OLEFTMM MOD
180M
LM000
OWKO
00 PMN
OP TAVE
OPLIAND
LSWAAA
Gaussian Noise/1)
Gaussian Noise/%
Piant
Realers
50
65
13
40
35
30
100
95
40
1901001
M-00
1900000
ORLAWNN
PMA
D.EFT WAO
LSSM
0699
300000
20
Gaussion Naise/N)
Gaussian Noise/%)
BBCSppott
Flovertat 12
Gaussian noise (0, 1, 2, 3, 5, 10, 20%)
80
25
20
22

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Published as a conference paper at ICLR 2025
T gr
TUAI SHiraag
Figure 8: The framework diagram of the proposed DLEFT-MKC algorithm.
E-praductt
Tensor learning
Clustering
Inokate
Consensus Partitions
Guide
Fusion
Reconstruction Paritions
Reconstruction
Reconstruction
LOss
Base Partitions
Data pre-process
Kernel Matrices
23


Reviews:
[{"review_id": "WdGtpGkXl3", "reviewer": "Reviewer_UZb7", "summary": "The paper presents a dynamic late fusion multiple kernel clustering with robust tensor learning via min-max optimization, named DLEFT-MKC. The proposed method aims to address the robustness and generalization problem via a min-max objective. It utilizes t-TNN constraints to explore the high-order relationships and cluster structures between views. The unstable performance caused by the base partition matrices is mitigated by the dynamically adjusted partition matrices. Additionally, late fusion is introduced to reduce its computational complexity. These modules help enhance the clustering performance. The authors also conduct extensive experiments to validate the effectiveness and efficiency of DLEFT-MKC in terms of performance and time cost.", "questions": "1.\tWhat does “robustness and generalization” refer to in this work? Does it mean less sensitivity to parameters?\n2.\tThe underlying reason explaining the min-max optimization’s positive effect on robustness should be better explained.\n3.\tEq. (4), $\\gamma_p$ is used to balance the effect of each view, while in Eq. (6), $\\gamma_p^2$ is adopted. We notice that the constraints on $\\gamma_p$ in Eq. (4) and Eq. (6) remain the same, but why Eq. (6) adopts $\\gamma_p^2$ instead?\n4.\tThough DLEFT-MKC is a late fusion approach, it is based on the third-order tensor and updating tensor requires t-SVD which is time-consuming. So why it has lower running on several datasets compared with other late fusion-based methods like OPLFMVC and LFMVC.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1.\tThe paper is well-written, and the motivations are clearly described;\n2.\tThe idea that incorporates a min-max optimization paradigm into tensor-based MKC to enhance both performance and robustness in clustering tasks is novel to a certain degree.\n3.\tThe paper conducts extensive experiments, and the results are adequately and logically discussed.", "weaknesses": "1.\tFor some justifications, more details should be provided.\n2.\tLeveraging t-SVD for tensor-based module optimization may recur high complexity, which is not consistent with the efficiency goal.\n3.\tThe work seems to combine Wang’s work about late fusion with tensor learning. Leveraging tensor learning for multi-view clustering result fusion is not a novel idea, which has been utilized in many previous works. The main contribution of the work lies in the min-max optimization, which has also adopted by some previous works. Therefore, the novelty of the work might not be so convincing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "we8mqRNZmQ", "reviewer": "Reviewer_xS8p", "summary": "The authors present a novel method for multiple kernel clustering called DLEFT-MKC, which addresses the problem of representational bottleneck of base partition matrices. This paper proposes a min-max optimization paradigm in tensor-based MKC to improve both performance and robustness. In addition, the authors use tensor learning to capture the high-order correlations between kernels. Finally, the effectiveness of DLEFT-MKC is demonstrated through extensive experiments on benchmark datasets.", "questions": "See the weakness.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1) This study is the first to apply a min-max optimization paradigm to tensor-based multi-kernel clustering, pioneering a new approach to improve performance and robustness in clustering.\n2）The overall structure of the paper is complete and well-organized. \n3) The formulas in the paper are written correctly and the derivations are correct.\n4) Another bright point in this paper is that it combines the RGDE with ADMM to optimize their problem, this combination is a well-established technique to solve convex optimization problems.", "weaknesses": "1) The complexity of the proposed method is not analyzed. \n2) The base partition matrices directly capture data information from multiple kernels. Why reconstruct the dynamic partition matrices? It would be better if the authors clearly introduced this design.\n3) Why are the standard deviation (std) values for your method so high on the Reuters and CCV datasets in Table 2?\n4) As far as I know, datasets like Caltech101-7, Caltech101-20, and Caltech101-all are the most commonly used for multiple kernel clustering, and it would be more convincing to compare the experimental results on these datasets. Please show a comparative experiment of the proposed method on at least one of them.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "oMYOzZ4Yeq", "reviewer": "Reviewer_Noi6", "summary": "This paper proposes a novel DLEFT-MKC, which effectively overcomes the representational bottleneck of base partition matrices and facilitates the learning of meaningful high-order cross-view information. Specifically, it is the first to incorporate a min-max optimization paradigm into tensor-based MKC, enhancing algorithm robustness and generalization. Extensive experimental results across various benchmark datasets validate the superior effectiveness and efficiency of the proposed algorithm.", "questions": "As mentioned above.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The paper is well-organized and written without technical errors.\n2. The motivation behind the paper is clear. The proposed method is innovative as it is the first to incorporate a min-max optimization paradigm into tensor-based MKC and the first to dynamically reconstruct base partition matrices from late fusion multi-view clustering.\n3. The experiments are comprehensive and solid, involving comparisons with over 20 competing algorithms. The proposed method consistently achieves the best performance, and the experiments have been evaluated from multiple perspectives. \n4. The experimental results demonstrate a substantial improvement in clustering performance, which is particularly exciting when coupled with a significant increase in computational efficiency.", "weaknesses": "1. The notation is somewhat difficult to follow. Some symbols are used without explanation. A table summarizing the notation would be beneficial.\n2. The case where $\\gamma_p = 0$ and $\\Delta \\leq 0$ is missing from the discussion in Eq.(14).\n3. There is insufficient discussion on the proposed algorithm. The objective expression appears complex, and the max-min-max form raises questions about how it ensures rapid convergence. While the experimental results show a notable improvement in computational efficiency compared to most other algorithms, there is a lack of analysis on computational complexity to explain this phenomenon.\n4. The cluster assignments in Fig.2 are not easily distinguishable, which is insufficient to validate the effectiveness of the learning process. The authors should provide additional visualizations to substantiate this contribution. For example, on the Reuters dataset, most algorithms achieve an average accuracy around 45%, whereas the proposed algorithm achieves an accuracy of 97%. Such a high accuracy rate raises concerns, and the authors need to provide an explanation.\n5. The author should provide the code and sufficient information for readers to reproduce the experimental results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "xFXOKlMafq", "reviewer": "Reviewer_JfTa", "summary": "This paper presents a novel algorithm, Dynamic Late Fusion Multiple Kernel Clustering with Robust Tensor Learning (DLEFT-MKC), which addresses key limitations in existing multiple kernel clustering methods. It dynamically optimizes base partition matrices, introduces min-max optimization for enhanced robustness and generalization, and reconstructs decision layers to capture high-order associations across views. These advancements improve computational efficiency and enable comprehensive extraction of multiple kernel information, leading to effective consensus clustering partitions.", "questions": "The authors claim that existing multiple kernel k-means methods often lack adaptive mechanisms for varying internal data distributions, stating that their proposed method addresses this issue. However, where is this adaptation reflected in the paper, and how does the method achieve it?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The experimental results are comprehensive, covering datasets of varying scales, comparative methods, and diverse visualization experiments.\n\n2. The overall writing is clear and easy to understand.", "weaknesses": "1. The motivation for this paper is unclear and lacks persuasiveness. For instance, the authors mention that one of the drawbacks of existing methods is their reliance on fixed initial base partitions, but they do not define what \"initial base partitions\" are. Additionally, while the authors highlight that current multiple kernel k-means methods often lack adaptive mechanisms for varying internal data distributions, the paper does not provide a solution to this issue.\n\n2. A framework diagram of the proposed algorithm should be included in the manuscript to aid understanding.\n\n3. The paper's novelty is insufficient, as it primarily integrates three related components from existing work, suggesting that it does not offer any new contributions.\n\n4. The paper lacks a theoretical analysis of computational complexity.\n\n5. Replacing Figure 4 with a table to present the comparison of runtime would provide a clearer and more intuitive representation.\n\n6. The reference list contains inconsistencies, with some conference names abbreviated and others in full, and some entries including DOI information while others do not. These details should be made consistent.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Spotlight)