From Spectrum-free towards Baseline-view-free: Double-track Proximity Driven Multi-view Clustering
Authors: Shengju Yu
Zhibin Dong
Siwei Wang
Suyuan Liu
KE LIANG
Xinwang Liu
Yue Liu
Yi Zhang
Venue: ICML 2025 poster (2025)
Keywords: Multi-view Clustering
Anchor
Subspace Learning
Primary area: general_machine_learning->clustering

Abstract: Current multi-view clustering (MVC) techniques  generally focus only on the  relationship between anchors and samples, while overlooking that between anchors. Moreover, due to the lack of data labels, the cluster order is inconsistent across views and accordingly anchors encounter misalignment,  which will confuse the graph structure and disorganize cluster representation. Even worse, it typically brings variance during forming spectral embedding, degenerating the stability of clustering results. In response to these concerns, in the paper we propose a MVC approach named DTP-SF-BVF. Concretely, we explicitly exploit the geometric properties between anchors via  self-expression learning skill, and utilize topology learning strategy to feed captured anchor-anchor features into anchor-sample graph  so as to explore the manifold structure hidden within samples  more adequately.  To reduce the misalignment risk, we introduce a permutation mechanism for each view to jointly rearrange anchors according to respective view  characteristics. Besides not involving selecting the baseline view, it also can coordinate with anchors in the unified framework and thereby facilitate the learning of anchors. Further, rather than forming spectrum and then performing embedding  partitioning, based on the criterion that samples and clusters should be hard assignment, we manage to construct the cluster labels directly from original samples using the binary strategy,  not only preserving the data diversity but avoiding variance. Experiments on multiple publicly available datasets confirm the effectiveness of  proposed DTP-SF-BVF method.

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven
Multi-view Clustering
Shengju Yu
Zhibin Dong' |
Siwei Wang?
Suyuan Liu'
Ke Liang'
Xinwang Liu'
Yue Liu
Yi Zhang'
Abstract
1. Introduction
Current multi-view clustering (MVC) techniques generally focus only on the relationship between anchors and samples, while overlooking that be- tween anchors. Moreover, due to the lack of data labels, the cluster order is inconsistent across views and accordingly anchors encounter mis- alignment, which will confuse the graph struc- ture and disorganize cluster representation. Even worse, it typically brings variance during form- ing spectral embedding, degenerating the stability of clustering results. In response to these con- cerns, in the paper we propose a MVC approach named DTP-SF-BVE Concretely, we explicitly exploit the geometric properties between anchors via self-expression learning skill, and utilize topol- ogy learning strategy to feed captured anchor- anchor features into anchor-sampp sample graph as to explore the manifold structure hidden within samples more adequately. To reduce the misalign- ment risk, we introduce a permutation mechanism for each view to jointly rearrange anchors accord- ing respective view characteristics. Besides not involving selecting the baseline view, it also can coordinate with anchors in the unified frame- work and thereby facilitate the learning of anchors. Further, rather than forming spectrum and then performing embedding partitioning, based on the criterion that samples and clusters should be hard assignment, we manage to construct the cluster labels directly from original samples using the binary strategy, not only preserving the data di- versity but avoiding variance. Experiments on multiple publicly available datasets confirm the method.
In recent years, multi-view clustering (MVC) is becom- ing research hotspot because of its ability effectively mine potential patterss hidden in heterogeneous data, and is widespreadly deployed in various fields such as drug design and finance analysis (Xu et al., 2024; Yang et al., 2023a; Wang et al., 2023; Yu et al., 2024b; Wen et al., 2024a; Liang et al., 2023; Li et al., 2025; Ma et al., 2024b). As a powerful tool in MVC, anchor technique commonly utilized to filter noise points and decrease the computing overhead (Li et al., 2023; Yu et al., 2025b; Zhang et al., 2025). It first selects small number of significant samples to represent overall samples, and then replaces the sample-samp proximity relationship by building up the anchor-sample relationship (Ma et al., 2024a; Yang et al., 2022; Yu et al., 2024c). Fol- lowing this line, a series of prominent works have been successively proposed. For instance, Kang et al. (2020b) regard the centroids generated by k-means view as anchors and merge multiple graphs by splicing their left singular vectors, Xia et al. (2023) first project original samples to perform de-correlation and then select anchors in projection space according to the sample variance. Wang et al. (2022a) design a hierarchical k-means model output anchors and construct sparse similarity using the learned bipartite graph. Unlike them, Huang et al. (2023) leverage three diversity levels in neighbors to construct anchors and generate graph directly in the early-stage fusion.
respective
Although generating pleasing clustering results from various aspects, current methods usually focus only on the anchor- sample proximity relationship, and fail take into account the anchor-anchor characteristics. This is not reasonable since between anchors, there generally exist informative ge- ometric features. Overlooking them will not be conductive to constructing discriminative anchors and extracting the intrinsic similarity among samples. Additionally, due to the fact that clustering tasks do not involve any data labels, anchors could be misaligned across views, leading to the graph structure becoming chaotic. Wang et al. (2022b) pro- vide alignment scheme from the perspectives of feature and structure matching, nevertheless, it requires to select the baseline view. Also, the anchor generation, the anchor trans- formation, and the graph construction are separated from
effectiveness of proposed
'Scoool of Computer, National University of Defense Tech- nololgy, Hunan, China. 'Intelligent Game and Decision Lab, Bei- jing. China. yu-shengju@ foxmail.com. Correspondence Si- wei Wang <wangsiwei @ @wwdt.edu.cn... Xinwang Liu <xinwan- gliu@ nudt.edu.cn>>
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
Multi-view Data
Anchor Generation
Permutation
Anchor-Sample
Cluster Indicator Learning
Topology
Self-expression
Proximity
Learning
Learning
HHOOBD
0
view
Clustering Results
0
view
Sample:
Anchor:
Orthogonal Representation:
Multipiicatin:
Figure 1. The devised DF--SF-BVF multi-view clustering framework. It explicitly extracts the geometric characteristics anchor-anchi via self-expression learning, and delivers them into the topology learning of anchor-sample so as to exploit the manifold structure among samples more sufficiently. It introduces learnable permutation model for each view to alleviate the anchor misalignment. Instead of constructing spectrum and then conducting partitioning, it directly learns the cluster indicators via binary learning to avoid introducing variance. These three sub-parts are all jointly optimized within an unified learning model so as to move towards mutual reinforcement.
each other. These limitations hinder the interaction of view information across different levels and accordingly weaken the distinctiveness of anchors. Furthermore, the clustering procedure adopted by current methods is to first form spec- trum and then conduct embedding partitioning on it, which causes the generated clustering results containing non-zero variance, degrading the stability and interpretability.
but provides consensus structure for anchors on different views to force them rearranging towards correct-aligning direction. Subsequently, we give six-step updating scheme with linear complexity to optimize the resultant objective loss. Experiments on multiple benchmark datasets demon- strate that DTP-SF-BVE is effective in grouping multi-view data and owns competitive strengths against multiple clas- sical MVC approaches. For more clarity, we summary the contributions of this work as below,
With these concerns in mind, we design a MVC method termed DTP-SF-BVF in this paper, and its framework is pre- sented in Fig. 1. To be specific, we introduce self expresxpeesion learning mechanism explore the geometric characteris- ties between anchors, and integrate them into the topology learning of anchor-sample graph so to characterize the manifold structure inside samples more sufficiently. Then, we associate each view with a permutation model, which is learnable and works jointly with the anchor generation, to re- arrange anchors their original dimension space according to view-specific features. Owing the joint-optimization mechanism in the unified framework, consequently, it does not involve the selection of baseline view. Further, to climi- nate variance, based on the criterion that one sample should belong to only one cluster, we avoid the formation of spec- trum and choose to directly generate cluster indicators from original samples. When the sample belongs to its cluster, we manage to optimize its indicator as and otherwise 0. In addition to well preserving the data diversity, this paradigm also can skip the spectral partitioning stage and thereby al- leviate the computing burden. The cluster indicator matrix is shared for all views, which bridges all anchors, permuta- tions and views. Not only does it play an important role in gathering multi-view information at the cluster-label level,
1. We explicitly take into account the geometric features between anchors, and successfully integrate them into the anchor-sample graph through topology learning exploit the manifold characteristics hidden within samples more fully for better clustering.
2. We devise joint-alignment mechanism that not only eliminates the need for selecting the baseline view but also coordinates well with the generation of anchors.
3. We avoid the formation of spectrum by directly learning cluster indicators using binary strategy, which effectively clears the variance in clustering results, accordingly high- lighting the stability.
4. We provide a six-step optimization scheme with linear complexity for the loss function. Experiments validate the effectiveness of our proposed method from multiple aspects.
2. Related Work
Based on the fact that each view data typically owns self- unique features and consequently can compensate for the limitations of other views, multi-view clustering aims at
2

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
integrating information from diverse views obtain more comprehensive and accurate data representation, thereby achieving superior clustering effect than single-view coun- terparts (Liang et al., 2024a; Yu et al., 2023a; Wang et al., 2024; Wan et al., 2024; Wen et al., 2023; Zhang et al.; Tang & Liu, 2022a; Yu et al., 2024e; Yang et al., 2024). Anchor technology is recently introduced into multi-view clustering to increase the computing efficiency (Shi et al.. 2021; Yu et al., 2024d). It is intended to replace the full graph with small-sized anchor graph by utilizing some discriminative landmarks. Specially, given a dataset (Xp p E R4=x") ")== where dp. and u denote the dimension of data, the number of samples and the number of views respectively, anchor based multi-view clustering can be formulated as
et al., 2020a; Zeng et al., 2024; Chen et al., 2023a). In next section, we will elaborate in detail on the principles of our devised DTP-SF-BVF approach to alleviate these issues.
3. Methodology
To explore the geometric properties between anchors, in- spired by subspace reconstruction (Zhang et al., 2020b; Xia et al., 2022), we introduce self-expression learning for anchors. To be specific, we utilize the paradigm A Ap to explicitly extract the global structure between anchors. Especially, due to S, E Rm where m is the number of anchors, solving Sp will take O(m²) com- puting overhead, which is almost ignorable against (mm2) that solving Zp takes since m is far less than n. Then, to integrate the characteristics of anchor-anchor into anchor- sample so to exploit the manifold features inside samples, we adopt the idea of point-point guidance to adjust the an- chor graph. Specially, we utilize the element [Spla. to guide [z and [Zplj.t. i,j = 1, , m,, 1 = 1, 1,...m. n, which can be formulated as aims at restricting similar features to maintain the consis- tency. At this point, MVC objective can be devised as
x - Ap Ip +n 4p F 7 12p ZIpp (L)
where 1- = [ 1 2 2, = 1, 0, A, E Rd,xm. Zp Rmxnj²= pp=1 77 and y denote the anchor ma- trix, anchor graph and regularization hyper-parameters, re- spectively. The fusion graph Z E R aims at gathering the information from different views at the graph level. The non non-negative constraints and column sum constraints guar- antee the learned graph to satisfy the similarity requirements. After obtaining Z. the cluster labels can be received by first constructing spectrum on the fusion graph Z and then con- ducting spectral partitioning operation on the embedding.
mm - [[z [Zp],.. [Sp]i and
min (Z, Sphep
Σ
X - Ap + A1A - Spl
p=1
+ 3
Σ 1.j=1
[2pla - 12pl.=112 [Splas
Noticed that the final clustering results are heavily depen- dent on the quality of Zp while Z,p is related to anchor matrix Ay. consequently, many works focus on the generation way of anchors. For example, Chen al. (2023b) utilize tensor learning investigate the low-rankness within views and employ dynamic anchor learning strategy explore that between views. Yan et al. (2022) integrate anchor learning and feature learning together, and learn to generate anchors separately. Given the fact that similar samples typically lie in the same cluster and have homologous characteristics, Li et al. (2022a) devise an alternative sampling scheme, which independent of initialization, to generate anchors. Liu et al. (2024) narrow the distributions of anchors by leverag- ing the correlation information between views to enhance their distinction. These methods successfully construct rep- resentative anchors from different perspectives. neverthe- less, they generally pay only attention to the relationship between anchors and samples when constructing anchor graph, while overlooking the influence of geometric char- acteristics inside anchors. This could bring about the loss of some informative features. Anchors on different views also could be misaligned due to the unsupervised property of data, leading to the confusion of graph structure (Wang et al., 2022b). Besides, the clustering results outputted by current approaches usually contain variance when partition- ing the spectrum, which exacerbates the instability (Zhang
Subsequently, to eliminate the anchor misalignment issue, one straightforward idea is to compute the space similar- ity between anchor sets and then match anchors according to their distance. However, multi-view data generally has various dimensions, and accordingly anchors on different views also have various dimensions. is typically difficult to directly compute the distance between anchor sets with di- verse dimensions. Although one can project all anchors into common space to make them have the same dimension, it can not guarantee the distance similarity after projecting to be consistent with that before projecting. Additionally, deter- mining the appropriate projection dimension needs heuristic searching. The projecting operation also could lead to heavy information loss. Consequently, these strategies are not that sensible. To get rid of this dilemma, considering that the nature of anchor misalignment is that the order of anchors on different views is not identical, we can alleviate the mis- alignment issue by rearranging anchors. In particular, we associate each view with a learnable permutation matrix T, E mm to flexibly transform anchors according to the characteristics of respective view, i.c., ||,, A, The subsequent issue is how to make anchors rearrange to- wards the correct corresponding direction. Next, we solve this and the variance issue concurrently.
3

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
Due to variance arising from the construction of spectrum, avoid forming spectrum, and choose to directly learn the cluster indicators. Especially, we factorize the anchor graph as a basic coefficient matrix and a consensus matrix, and utilize binary learning to optimize the consensus ma- trix. This not only makes the consensus matrix successfully represent the cluster indicators, but also provides a common structure for anchors on all views, inducing them rearrang- ing towards the common structure. Further, since views typically own different levels of importance, we introduce weighting variable for each view to automatically measure its contributions. Therefore, our DTP-SF-BVE is devised as
Update Ty: The optimization w.r.t Tp in Eq. (3) can be
written as
min a | p TT + AA A,T
s.t. 1
10.1)
(6)
Expanding the objective by trace operation, Eq. (6) can be further equivalently transformed as
min Tr (T TT GpTp (AH, M, 2AS;) - 22 Tp
s.t. T 1 = 1, T, E (0. 1mmxm
(7)
where G, E Rmxm A Ap. Hp Rmxm S, Mp E R R = B and J, E Rmxm BJ Given the characteristics of feasible region, we can obtain the optimal Tp via traversal searching on the one-hot vectors {e:)
2 IX,,
A,
A C
AIITTT - Ap Sp + TTr LL,
s.t. a.t 1 1 1, 2 0, B, Bp = 1, T = 1, T,
Update B,: The optimization w.I.t B, in Eq. (3) can be written as
T₁ E m0.1)mxm Σ C = 1, 1,2, ,
min I1 (B, (9La + a²0,) B,CC Ca By
Ce (0,1) S = 1. 2 0, > [Splat 0,
s.t. B Bp Ik.
(8)
(3)
where O2 E RRmxm- = T. A A,T,, Then, we split the feasible region into [Bp] (,, = and BB
where Q = (A, E 14 B, E Rmnk Sp E Rmxm Tp E Ruxm, a Rexi C E p = 1......) .0. The second term aims capturing the char- acteristics between anchors. The third term is the ma- trix form of point-point guidance, and aims at delivering the characteristics of anchor-anchor into anchor-sample of the first term, where Lg E mM = Dp Sp D,- = (" , = 1,... m... This spectrum-free model directly outputs discrete clustering results via the consensus cluster indicator matrix C. a plays a role in ad- justing the importance between views. More illustrations for the objective function please refer to Section E in Appendix.
0, i # j. Further, combined with the fact that CCT is diagonal matrix, Eq. (8) can be equivalently transformed as
min [Bpl.] [Bl..
T
Σ CAA (3Ls + +a a2Or)) [Bplat
i=1
CX Cx
2a
Ap TT, j:: [Bp]is
s.t.
[Bp] Bpp.. mxm B Bplad -1=00 0,
[ T. "",,,,....,,,BBu
[Bp]a [Bp = 1)x1
(9)
It is a quadratically constrained quadratic programming and can be solved by current software.
4. Solver
Update Sp: The optimization w.r. Sp in Eq. (3) can be written as
We adopt the alternating optimization idea to minimize the loss function Eq. (3).
min Tr S,S, +2 S,
(-00z-
8 M, Sp
Update Ap: The optimization w.r.t A, in Eq. (3) can be written
21
(10)
s.t. S 1 1 S, 2 0,
S [ppk3 = 0.
min a 1X, A,T,B,CI - ApT A,
(4)
Noticed that the constraints can be equivalently transformed
By utilizing the derivative equal to zero, we can obtain
as V = = 1,0 S [Splas: e, [Spla. = 0,j
1,2, 2, ... ..... and therefore Eq. (10) is further converted as
Ap = a ET + AF,F,)
(5)
i in +2 (-0p
3 -. Mp
[Sp
where Ep E RmXn_ = T,B,C, F, E mm = TT SpSpp
-x
j::
(11)

---

From Spectrum-free towards Baseline-view-free: Double-track Proximity Driven Multi-view Clustering
Algorithm 1 The proposed DTP-SF-BVF
5. Complexity Analysis
Input: Multi-view data ppp hyper-parameters X, B. Output: Discrete cluster indicator matrix C.
Space complexity The space complexity of DTP-SE- BVF is mainly from optimization variables Ap. Tp. Bp v. According the fact that Ap R4 Tp E Rm=xm Bp Rmm* k Sp E m= C R and a E Buxi, we have that storing them will require O(dpm), O(m²), Omm)), Omm²), (nnk) and ( memory overhead, respectively. Thus, storing all optimization variables will take O + m²v u mkk + nk) memory overhead where represents the data dimension sum of all views and independent of the sample number n. Further, since the number of anchors m is generally greater than or equal to the number of clusters k. we have m²v 2 mkv. Besides, considering that m is generally much smaller than n and is also independent of n, we have that the space complexity of proposed DTP-SF-BVE is (nzk), which is linearly related to the number of samples n.
Initialize: (Ap: Tp: Bp, p C, az
Sp. C and a. p =
1: repeat
2: Update Ap via Eq. (5)
3: Update T2 via Eq. (7)
4: Update Bp via Eq. (9)
5: Update Sp via Eq. (11)
6: Update C via Eq. (14)
7: Update op via Eq. (16)
8: until convergent
It is a quadratic programming and can be easily solved.
Update C: The optimization w.r.t C in Eq. (3) can be written as
min Tr (C WC - ZC) C
Time complexity The time complexity of DTP-SF-BVF is mainly from the updating of all optimization variables. When updating Ap. constructing Ep and F, will take O(m kk + mkn) and O(m²) respectively. Constructing the item o E E. + FF, and solving its inverse will take O and O(m)) respectively. Thus, updating A, will take O(m k+ mm-n m²n + +d,um + dnmm). When updating Tp. constructing Gp, Hp. Mp and Jn will take Oldpm²) O(m²), O(mkn + m²n) and Oddmmn + mnk + m²k), respectively. Traversal searching hot vectors will take (mm!). Thus, updating Tр will take Odymm² (l6mm2+pppp +d + m² +mnn++ m²n + m2). When updat- ing B,, constructing Qp and the item CX, A. will take (dd,m²) and
(12)
s.t. Σe C. = 1,j= 1,2, .......... n, C {0, 13
where W E RiXE = E a B A Z E B = R= 2 2 " A,T,BBBB............. The constraints indicate that there is only one non-zero element in each column of C, and thus we can solve C column by column. Eq. (12) can be further transformed as
p ATA,T,,,,,,,, +
one-
min C WC.. C.. C
k
(13)
s.t. Σ 1m
C.. C.y=1,C, E {0, 13x
km² + k2 m)) respec- tively. Performing quadratically constrained quadratic pro- gramming will take O(m kk). Thus, updating B, will take Od,m²+
The item C. WC means that it takes a certain diagonal element of W, and C.. takes a certain element of Z Therefore, we can determine the corresponding index of minimum by 1" = arg min, Wt.- Zy.1, 1= 1. k. Then, the value of C.j can be obtained by
+ knd, + k2m + m²k)). When updating Sp. due to the construction of Qp and M, having been completed, it only involves the performing of quadratic programming, which will take ((m²). When updating C, constructing W and Z will take O(d,m² + dymnk + d2 + kmm2 + k²m) and O Ondym + nmm2 + rmmk), respectively. Since the value of C can be determined by comparing the diagonal element of W and the row of Z. updating C will take (d + dzkk² + +km2+ + k2m + ndym + nm3 + + nmk)). When updat- ing a, constructing 6, will take Odd,m² + dy + dakn). The value of a can be determined by Cauchy inequality, and thus updating a will take Odym + dpmnk + dpkn). Based on these, we have that updating all Ap. Ty. Bp. Sp. C and a will take 0 m² + knd + km + dk²).. Besides, considering that m is usually greater than or equal to k, dp is independent of n, n is largely greater than m. we can obtain that updating all variables will take Omm2n + dnm + mllo + m ku). which is also linearly related to the number of samples n.
11, i=1" 0, i 1,2, k
C₁
(14)
Update a: The optimization w.r.t a in Eq. (3) is
min Σ
s.t. s. A,T,,-,,,, 1, 0.
(15)
dam +om" +lo++
Since the item a, the optimal a can be determined via Cauchy inequality. Thus, we have
1 = |X, B, is a constant for b,
ap
(16)
p= bp
Algorithm summarizes the pipeline of DTP-SF-BVF.
5

---

From Spectrum-free towards Baseline-view-rre: Double-track Proximity Driven Multi-view Clustering
6. Experiments
Table Clustering result comparison (tean+ssd))
6.1. Experimental Setting
Desuar
DERMATO
CALTEZ
FWMM
ACC
We evaluate the algorithm performance on the fol- DERMATO, CALTE7, Cora, REU7200, Reuters, CIFI0Tra4, FasMNI4V.
Datasets
PSSCC
1348
305T 951
43.2222.80 2.3
33399
23.57
lowing 7 datasets:
MSCAA
53 82)
41.91:2151
34.33211077
MYSC
413802230
MLBSCC
57 37.28:20000
31.0
XCLES
46.1882211
Baselines We choose the following 17 classical MVC meth- ods as the baselines to demonstrate the effectiveness of pro-
PSONOP
2.38125571
21.00::255) 57)
21/02:1100 Ab)
4151c2283
MFSC
25:5020000
13
20/94 280
42.5:290211
ROMYS
18.58
6. 3.3
$2.02:1:55
2800021 301
1882:22550
27.22
posed DTP-SF-BVE VF: FMR (Li et al., 2019), PMSC (Kang
DOMVVC
08:88:00000
52.8033071
et al., 2020a), AMGL (Nie et al., 2016), MSCIAS (Wang et al., 2019), MVSC (Gao et al., 2015), MLRSSC (Brbić & Kopriva, 2018). MPAC (Kang et al., 2019). MCLES (Chen et al., 2020), FMCNOF (Yang et al., 2021). PFSC (Lv et al., 2021), SFMC (Li et al., 2022a), MSGL(Kang al., 2022), FPMVS (Wang et al., 2022c), UOMVSC (Tang et al.,
OHINTE
8243650 66
6354:5000
33.94
47
00:6::000
25.8
53.20
BASTAG
1418
53342250
22.00 59)
4225225371
25:44:222 251
Ous
03.8020000
16:23=00.00
54.30:00000
XX0%
TMR
2692:112
AMKIL
15
MOKIAS
153605190 190
42 1200
MYSE
536022911
MINE
48:31150001
630020000
KCLES
2023). PGSC(Wu et al., 2023), OrthNTF(Li et al., 2024b), HHung et al., 2023).
PNENOT
51100:4831
5.20/
44520-2221
45.10250000
1220025000
2.00
3.56210085
23 23.19:20700)
FOMVS
51:7003311
4150022000
3.80:30000
53:11:53300
30.10
Parameter Setup
We search the hyper-parameters À and 100, 1,00, 10 102, 1 and [2 12- 2,,-2,22 22, 24] re-
POSC
33550530
4.54:5000
22 2216:20042
OHNNTE
52.58
42:0020000
7.4:250000
25 (357:2000
11.9655000
58 DO1
in [10-1, spectively. For all competitors, we download their source code and tune the parameters according to their provided guidelines. Three popular metrics are employed to measure the clustering results. For fairness, we run 20 times and calculate the mean and variance of clustering results.
FASTNI
$1.8
4500022440
2.5020000
23:39:=135)
59.21
PASS
10.22 105
AMKIL
18.27 131
2ESICA000 D00
MSKIAS
12 1220007111
11411120
35
MEBSSC
MPAC
NCLES
36100:111
22 22-00120020
P9SC
621912730
SIMC
4214 801
633502000
3430
34 3404:21000
MILL
34.9920.33
37 21:77221271
6.2. Clustering Results and Analysis
DOMV3C
75177.20
61351200311
55.1220000
15:2322000
PODC
3314:5420
2623:1140 140
24.20
35.57:20035
The clustering results are reported in Table I, and from this table we can conclude that,
OHINTP
78.3k20
633312000
37.32:2000
34.77.01100
21:8::1000
16:4 41677-10011
Oup
75.22150001
41.2222000
38.30
4434:20000
2064
51391
1. Overall Effectiveness. Our DTP-SF-BVF consistently beats these competitors in terms of all three metries on DERMATO, CALTE7, Reuters and FasMNI4V. Particularly, it makes 6.91% improvement in Fscore than the second- best approach on DERMATO. In other cases, such on Cora, it is still able to provide comparable outcomes. These indicate that our DTP-SF-BVF is effective in partitioning multi view data and can achieve competitive clustering outcomes.
outperforms on all of these seven datasets and three met- ries, which reveals that our double-track proximity strategy can help extract representations more sufficiently.
4. Reliable Stability. The results outputted by our DTP- SF-BVF are all not with variance. This mainly benefits from avoiding the generation of spectrum. Not only does the spectrum-free property enhance the result stability, but allows the labels to be directly derived from original data, well maintaining the diversity. Despite zero-variance for methods like MPAC, SFMC and OrthNTF, the low-rank con- straint could damage potential graph structure, accordingly weakening their performance.
2. Anchor Suitability. In contrast with PMSC, AMGL, MCLES, FMCNOF, OrthNTF, FMR, PGSC, etc, which tackle MVC problems using tensor, kernel, latent space, co-training or matrix factorization means, our DFP-SF- BVF using anchor tool can produce better results than them in most cases. For instance, on Cora, it surpasses them
in terms of NMI with 38.59%, 42.37%. 27.00%, 42.63%, 43. 43.20% 23.07%, 42.55% respectively. These suggest that our adopted anchor means is recommendable.
5. Broader Applicability. Some methods like PMSC, PFSC, MFLVC, AMGL, UOMVSC, MCLES, PGSC, etc, can not work with large-sized CIF10Tra4 and FasMNI4V due to the intensive complexities or self- self-limitations, while our pro- posed DTP-SF-BVF operates normally with its lower com- plexities and meanwhile can produce superior clustering outcomes. So, DTP-SF-BVF enjoys broader applicability.
3. Ample Proximity. Different from FPMVS, FASTMI, SFMC, etc, which concentrate only on the sample proximity relationship. DTP-SF-BVF also successfully takes anchor-anchor characteristics into the measuring of overall similarity and accordingly brings performance en- hancement. Taking FASTMI as an example, DTP-SF-BVF
Due to the space limit, more conclusions are presented in the Section H of Appendix.
6

---

From Spectrum-free towards Baseline-view-fre Double-track Proximity Driven Multi-view Clustering
I FMR I PMSC IN AMGL in MSCIAS IN MVSC IN MURSSC MPAC MCLES IN FMCNOF IN PFSC SFMC IN MSGL IN FPMVS
3AMVC UOMVSC IN PGSC AEVC II OrtINTE FMVACC FASTII Ours
20
Logz Running Time
15
10
DERMATO
CALTE7
Cora
REU7200
Reuters
CIF10Tra4
FasMNI4V
Figure 2. The running time comparison seven public benchmark datasets.
To further exhibit the advantages of proposed DTP--SF-VF, we also compare it with several popular deep learning com- petitors, ADAGAE (Li et al., 2022b), DEMVC (Xu et al.,
Table Comparison with alignment competitors
DERMATO
CALTE?
Beacn
CIPIDlIn
ACC089
2021a), MFLVC (Xu et al., 2022) and DSMVC (Tang &
22-49-1000 1B00
12:30-311D)
66)
55441-220 2.25
Liu, 2022b). Comparison results are summarized in Table 2.
AEVE OPe
61821-379:
15-17-44000
4661-430
53 12000000
26.3 LUUNO
10.80 10.30)
54.36 10.000
23.44 23-56 11)
36300010 (0)
$43M-0555
51.364 (0.30
SNG29
UMIACT
36410-250)
11.0112.000
SINLLL 54
12,80 28.90012.06
12
57.800-0831)
JANYCC
41 (1440-4444)
26 (31155777)
31ATR 260
11102.101 T1:
51.01:1155 1.53)
Table 2. Comparison with deep learning based competitors
AEVC
8-44 E0)
13:20.00 800
91.55 (43
ACX(S)
15:55:7:01
43.9k-2.00
11.00:00
JANVVC
38380:300
20.80-1150 150
41.33+
4366c-1-5
DOMIN
3058440
34 10-
AEVC
17900:310 101
46 (110-1177
22:19:0000
36, 36199-0
17.15k00 37)
45.99-0055
MITNC
(7)
15 41 17)
OZ
$150:-0000
76:33/31300
2635520000)
44.58/22.10)
20164020000 D0)
31.371000000
NOWN
78386-8995
75 -0 47)
XMME5)
ARMGAA
18:171-1139 341
DOSIE
MITANC
16.15 1611511834)
3508110000
12:001181100
818L-055)
52551099)
4 (36210-1)
6.3. Running Time Comparison
45.20110000
4520010460
1000
To illustrate the efficiency of DTP-SF-BVF, we count the running time of each algorithm, and report the comparison results in Fig. 2. From this figure, we can draw that,
ABAGAB
53 34:2211731
3130:2040
25 25.9908011 14)
15.t::1 231
20.9:11.39
2269621
48.99
MECVC
DSMIVC
73 2333611 891
$1.00 1.29
3019 12>
25
33.00-20.22 32)
21 21.00c 1.5
SSndie 200
Ouss
28.2044000
41110:50000
28:304-0100 001
44 (41200000
20044:0000)
51.30
1. MVSC, PFSC, PGSC and MCLES consume significantly more time than others. This is mainly caused by the sub- space strategy they employed, which typically requires con- structing large-sized similarity and needs at least cubic com- putational overhead.
6. Competitive Ability. Against these competitors adopt- ing deep learning technique to tackle MVC problems, we can produce preferable results in most cases, such as on DERMATO and Cora as well as REU7200. This illustrates that even comparing deep learning methods, our devised DTP-SF-BVF still provides competitive clustering results.
2. MPAC, PMSC, FMR, MLRSSC, etc, take more time than us, which is mainly because MPAC and PMSC gather multi-view representations at the partition level, and FMR and MLRSSC utilize the kernel dependence measure to do data reconstruction.
Then, we also conduct comparison with some alignment methods, FMVACC(Wang et al., 2022b), 3AMVC (Ma et al., 2024a) and AEVC (Liu et al., 2024) to illustrate the strength of our proposed baseline-view-fiee alignment strategy. Comparison results are presented in Table 3.
3. FPMVS, FMVACC, MSGL and SFMC operate slower than us. Possible reasons are that the connection compo- nent constraints and feature matching constraints conducted anchor graph induce large proportion of additional calculating expenditure.
Flexible Alignment. Against FMVACC, 3AMVC and AEVC that require firstly selecting baseline view and then performs alignment, our results are more desirable in most cases. For example on CALTE7, we receive 27.32%, 6.84%, 37.11% improvement than FMVACC respectively. This is primarily because our joint-alignment property, besides not involving baseline view, also can coordinate with the generation of anchors, more flexibly transforming anchors.
4. FMCNOF and FASTMI enjoy slightly faster running speed, the reasons of which could be that FMCNOF de- couples dense optimization matrices by sparse factorization skills and FASTMI generates base clusterings via fast parti-
7

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
tioning on the view-sharing graph.
Table 6. The effectiveness of baseline-view-free strategy
5. AMGL, MSCIAS, UOMVSC and OrthNTF are generally faster than PMSC, MVSC, PGSC, MPAC, PFSC, etc, possi- bly because the former ones alleviate the computing burden of spectral partitioning and graph mergence via low-rank approximation or non-negative factorization.
Metric
Abla
DERMATO
CALTE7
Cor
REU7200
Reatets
CIFI0TI44
FusMNIVV
ACC
BVF
UA
85.47
50.73
50,66
76.59
52.44
31.65
26.22
16.67
54.26
45.29
26.83
25.91
57.36
53.68
$2.53
39.55
35.41
24.77
15:30
56.47
BVF
$9.97
43.70
3187
15.64
59.31
Phcore
BVT
79.47
57.92
78.12
30.69
41.12
28.55
21.18
48.4
17.90
20.64
47.41
51.37
6. All algorithms can normally work on DERMATO and CALTE7, while with the increase of sample size, PFSC, FMR, MCLES, PMSC, MPAC, MSGL, etc, are gradually ineffective, which is mainly due to the limitations of their innate computing requirement or memory cost.
To further show the merits of baseline-view- strategy, we organize the experiments involving baseline view (BV), The comparison results are reported in Table 7. Evidently, our strategy indeed brings performance improvement. More in-depth analysis please refer to Section J in Appendix.
6.4. A Alltion Study
Table 7. Comparison between baseline-view-Fee strategy and BV
Table The effectiveness of double-track proximity strategy
Dinaset
DERMATO
CALTE?
Cora
REU7200
Reuters
CIP10Tra4
FESMNNV
Matric
Able.
DERMATO
CALTE7
Coca
REU7200
Rester
CIFIOTis
FuMMNI4V
ACC(1))
ACC
71.51
49.05
30.35
16.75
54.26
47.05
26.09
57.36
52.15
BV Ours
72:37
62.43
45.78
22.87
22.98 26.83
$1.22
85,47
80.66
52.44
26.22
54.26
57.36
$3.07
6.00
23.19
15:48
S6.13
NMI
DTP
89.97
45.25
43.70
31.87
1564
59.21
NMD(%)
Foomm
ETP
$7.92
73.79
78.12
41.12
50.42
28.55
43.04
44.84
20.64
17.70
46.77
$1.37
BV Ours
71.97
89.97
41.24
45.25
34.76
43.70
6.35
31.87
15.73
15:64
59.21
$2.73
(sccune(3))
To validate the effectiveness of double-track proximity (DTP), we organize relevant ablation (Abla.) experiments and present the comparison results in Table 4 where SLA denotes the clustering results of considering only anchor- sample relation. (We omit the variance items since they are all zero.) As seen, our DTP is coherently better than SLA, which well illustrates that the devised DTP strategy can help achieve superior results.
BV Chrss
87.92
70.38
78.12
33.47
41.12
25.46
28.55
44.54
39.84
17.96 20.64
51.37
46.31
More ablation studies are placed into Section I of Appendix.
7. Limitations
DTP-SF-BVE contains srper-parameters À and 3, which requires additional efforts for fine-tuning. Consequently, designing non non-parametric version can further boost its practicality. Besides, we adopt the square weighting scheme with linear constraints to measure the contributions between views. Some other ingenious view schemes could be deeply investigated in the future so as to further increase the results.
Table 5. The effectiveness of spectrum-frres free strategy
Matx
AX
DERMATOO
CALTET
Cluss
CaP10Tta
ACC
57
$2.46 0080
3635 (UM0
$4.52.0030
57.3610000 (0.0
XMI
1F
6080
44.N6 -080
11.81-4.80
-4.0
54.31c 90
(7/00:0000
76129:000)
4L13::000)
28.59:00000
46.0:30.80
2164:0000
91.77:0000
0.0
8. Conclusion
T46.66
199038
In this work, we introduce double -racke proximity, which concurrently considers anchor-sample and anchor-anchor characteristics, to more fully extract multi-view representa- tions for better clustering. To reduce the mismatching risk, we adopt joint-alignment mechanism that does not involve the selection of baseline view and also can coordinate with the anchor generation. Furthermore, we avoid forming spec- trum and directly generate cluster indicators via binary learning strategy, which not only effectively eliminates the variance but well preserves original diversity. For the re- sulting optimization problem, we provide solution with linear complexities. Experiments on multiple public bench- mark datasets verify the effectiveness of DTP-SF-BVE In future work, we will extend our DTP-SF-BVE method to non-parametric scenarios to further enhance its practicality.
Table 5 summarizes the ablation results about the spectrum- free (SF) strategy, where CS denotes the clustering results containing spectrum as previous methods do. Evidently, in addition to owning the ability to generate preferable and stable clustering results, our SF also enjoys less time con- suming. This gives evidence that the designed SF is more suitable for MVC problems.
In the paper we adopt a baseline-view-ffee (BVF) strategy to decrease the anchor misalignment risk. To demonstrate its effectiveness, we report the ablation results in Table 6, where UA denotes the clustering results without involving alignment. According to this table, it is easy to discover that BVF makes more favorable clustering results than UA which well suggests that our BVF is functional.
8

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
Impact Statement
Huang, D., Wang, C.-D., and Lai, J.-H. Fast multi-view clustering via ensembles: Towards scalability, superiority, and simplicity. IEEE Transactions on Knowledge and Data Engineering, 25(11))11388111 11402, 2023.
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 feel must be specifically highlighted here.
Kang, Z., Guo, Z., Huang, S., Wang. S., Chen, W., Su, Y., and Xu, Z. Multiple partitions aligned clustering. In Proceedings of the 28th International Joint Conference on Artificial Intelligence, pp. 2701-2707, 2019.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant No. 62406329, 62476280, 62441618, 62325604, 62276271.
Kang, Z.,, Zhao, X., Peng, C., Zhu, H.,, Zhou, J. T., Peng, X., Chen, W., and Xu, Z. Partition level multiview subspace clustering, Neural Networks, 122:279-288, 2020a.
References
Kang. Z., Zhou, W., Zhao, Z., Shao, J., Han, M., and Xu, Large-scale multi-view subspace clustering in linear time. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pp. 4412 4419, 2020b.
Brbié, M. and Kopriva, I. Multi-view low-rank sparse sub- space clustering. Puttern Recognition, 73:247-258, 2018.
Chen, J., Mao, H., Peng, D., Zhang, C., and Peng. X. Mul- tiview clustering by consensus spectral rotation fusion. IEEE Transactions Image Processing, 2023a.
Kang, Z., Lin, Z., Zhu, X., and Xu, W. Structured graph learning for scalable subspace clustering: From single view to multiview. IEEE Transactions on Cybernetics, 52 (9)88976-8986, 2022.
Chen, M.-S., Huang, L., Wang. C.-D., and Huang, D. Multi- view clustering in latent embedding space. In Proceed- ings of the AAAI Conference Artificial Intelligence, volume 34, pp. 3513-3520, 2020.
Lao, J., Huang. D., Wang. C.-D., and Lai, J.-H. Towards scalable multi-view clustering via joint learning of many bipartite graphs. IEEE Transactions on Big Data, 10(01): 77-91, 2024.
Chen, M.-S., Wang, C.-D., Huang, D., Lai, J.-H., and Yu, P. S. Efficient orthogonal multi-view subspace clustering. In Proceedings of the 28th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 127-135, 2022.
Li, J., Wang, Q. Yang, M., Gao, Q., and Gao, X. Effi- cient anchor graph factorization for multi-view clustering. IEEE Transactions on Multimedia, 2023.
Chen, Y., Zhao, X., Zhang, Z., Liu, Y., Su, J., and Zhou, Y. Tensor learning meets dynamic anchor learning: From complete incomplete multiview clustering. IEEE Transactions on Neural Networks and Learning Systems, 2023b.
Li, J., Gao, Q., Wang, Q. Deng, C., and Xie, D. Label learn- ing method based tensor projection. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pp. 1599-1609, 2024a.
Li, J., Gao, Q., Wang, Q. Yang, M., and Xia, W. Orthog- onal non-negative tensor factorization based multi-view clustering. Advances in Neural Informatin Processing Systems, 36, 2024b.
Cui, C., Ren, Y., Pu, J., Li, Pu, X., Wu, T., Shi, Y., and He, L. A novel approach for effective multi-view cluster- ing with information-theorriicc perspective. Advances in Neural Information Processing Systems, 36, 2024.
Fu, Y., Li, Y., Huang, Q. Cui, J., and Wen, J. Anchor graph network for incomplete multiview clustering. IEEE Transactions Neural Networks and Learning Systems, 2024.
Li, R., Zhang, C., Hu, Q., Zhu, P., and Wang, Z. Flexible multi-view representation learning for subspace cluster- ing. In Proceedings of the 28th International Joint Con- ference on Artificial Intelligence, pp. 2916-2922, 2019.
Gao, Nie, F., Li, X., and Huang, H. Multi-view subspace clustering. In Proceedings of the IEEE International Conference on Computer Vision, pp. 4238-4246, 2015.
Li, X., Zhang, H., Wang, R., and Nie, F. Multiview cluster- ing: A scalable and parameter-free bipartite graph fusion method. IEEE Transactions Pattern Analysis and Machine Intelligence, 44(1):330-344 2022a.
Huang, D., Wang, C.-D., Lai, J.-H., and Kwoh, C.-K. Toward multidiversified ensemble clustering of high- dimensional data: From subspaces to metrics and beyond. IEEE Transactions on Cybernetics, 2022.
Li, X., Zhang, H., and Zhang, R. Adaptive graph auto- encoder for general data clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(12): 9725-9733 2022b.
9

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
Li, X., Pan, Y. P., Sun, Y., Sun, Q S., Tsang, W., and Ren, Z. Fast unpaired multi-view clustering. In Proceedings of the 33rd International Joint Conference on Artificial Intelligence, 2024c.
Nie, F,, Li, J., and Li, X. Parameter-free auto-weighted multiple graph learning: framework for multiview clus- tering and semi-supervised classification. In Proceedings of the Twenty- International Joint Conference on Artificial Intelligence, pp. 1881-1887, 2016.
Li, Y., Hu, X., Yu, S., Ding, W., Pedrycz, W., Kiat, Y. C., and Liu, Z. A vertical federated multiview fuzzy clustering method for incomplete data. IEEE Transactions on Fuzzy Systems, 33(5):1510-1555 1524, 2025.
Nie, F., Xie, F., Yu, W., and Li, X. Parameter-insensittee min cut clustering with flexible size constrains. IEEE Trans- actions on Pattern Analysis and Machine Intelligence, 2024.
Liang, W. Liu, X., Liu, Y., Ma, C., Zhao, Y., Liu, Z., and Zhu, E. Consistency of multiple kernel clustering- In International Conference Machine Learning, pp. 106500206677 PMLR, 2023.
Qiang, Q., Zhang, B., Wang, F., and Nie, Fast multi-view discrete clustering with anchor graphs. In Proceedings of the AAAI Conference on Artificial Intelligence, vol- ume 35, pp. 2360-9367, 2021.
Liang, W., Tang, C., Liu, X., Liu, Y., Liu, J., Zhu, E., and He, K. On the consistency and large-scale extension of multiple kernel clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024a.
Qin, Y. Wu, H., Zhang, X., and Feng, G. Semi-supervised structured subspace learning for multi-view clustering. IEEE Transactions Image Processing, 31:1-14, 2022.
Liang, W., Zhu, E., Yu, S., Xu, H., Zhu, X., and Liu, X. Scalable multiple kernel clustering: learning clustering structure from expectation. In Proceedings of the 41st In- ternational Conference on Machine Learning, pp. 29700- 29719, 2024b.
Shi, S., Nie, F,, Wang, R., and Li, X. Fast multi-view clustering via prototype graph. IEEE Transactions on Knowledge and Data Engineering, 35(1):443-455 2021.
Liu, S., Liang, K., Dong, Z., Wang, S., Yang, X., Zhou, S., Zhu, E., and Liu, X. Learn from view correlation: An anchor enhancement strategy for multi-view clustering. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pullern Recognition, pp. 26151-26161, 2024.
Sun, Y., Li, X., Sun, Q., Zhang, M.-L., and Ren, Z. Im- proved weighted tensor schatten p-norm for fast multi- view graph clustering. In ACM Multimedia 2024, 2024.
Tang, C., Li. Z., Wang, J., Liu, X., Zhang, W., and Zhu, E. Unified one-step multi-viww spectral clustering. IEEE Transactions on Knowledge and Data Engineering, 35 (06):6449-6460, 2023.
Long, Z., Wang, Q., Ren, Y., Liu, Y., and Zhu, C. S2mvte: simple yet efficient scalable multi-view tensor clustering. In Proceedings of the IEEE/CVF Conference on Com- puter Vision and Pattern Recognition, pp. 26213-26222 2024.
Tang, H. and Liu, Y. Deep safe incomplete multi-view clustering: Theorem and algorithm. In International Con- ference on Machine Learning, pp. 21090-21110. PMLR, 2022a.
Lu, Y., Lin, Y., Yang, M., Peng, D., Hu, P., and Peng. X. De- coupled contrastive multi-view clustering with high-order random walks. In Proceedings of the AAAI Conference Artificial Intelligence, volume 38, pp. 14193-14201, 2024.
Tang, H. and Liu, Y. Deep safe multi-view clustering: Re- ducing the risk of clustering performance degradation caused by view increase. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 202-211, 2022b.
Lv, J., Kang, 7., Wang, B., Ji, I.., and Xu, Z. Multi-view subspace clustering via partition fusion. Information Sciences, 560:410-444 44343,,333,.. 2021.
Wan, X., Liu, J., Liu, X., Wen, Y., Yu, H., Wang. S., Yu, S., Wan, T., Wang, J., and Zhu, E. Decouple then classify: a dynamic multi-view labeling strategy with shared and specific information. In Proceedings of the 41st Inter- national Conference on Machine Learning, pp. 49941- 49956, 2024.
Ma, H., Wang, S., Yu, S., Liu, S., Huang, J.-J., Wu, H., Liu, X., and Zhu, E. Automatic and aligned anchor learning strategy for multi-view clustering. In Proceedings of the 32nd ACM International Conference Multimedia, pp- 5045-5054, 2024a.
Ma, H., Wang, S., Zhang, J., Yu, S., Liu, S., Liu, X., and He, K. Symmetric multi-view subspace clustering with automatic neighbor discovery. JEEE Transactions on Circuits and Systems for Video Technology, 34(9):8766- 8778, 2024b.
Wang. J., Ma, Z., Nie, F., and Li, X. Fast self- clustering with anchor graph. JEEE Transactions on Neu- ral Networks and Learning Systems, 33(9):4199-44 4212, 2022a.
10

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
Wang, Q. Tao, Z., Xia, W., Gao, Q. Cao, X., and Jiao, L.. Adversarial multiview clustering networks with adap- tive fusion. IEEE Transactions on Neural Networks and Learning Systems, 34(10):7635-76) -647, 2023.
Xu, Ren, Y., Li, G., Pan, L.., Zhu, C., and Xu, Z. Deep em- bedded multi-view clustering with collaborative training. Information Sciences, 573:279-290, 2021a.
Xu, J., Ren, Y., Tang, H., Pu, x., Zhu, X., Zeng, M., and He, L. Multi-vae: Learning disentangled view-common and view- weww-peculaar visual representations for multi-view clustering. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 9234 9243, 2021b.
Wang, Q. Tao, 7., Gao, Q. and Jiao, L. Multi-view sub- space clustering via structured multi-pathway network. IEEE Transactions on Neural Networks and Learning Systems, 35(5):7244-7225 2024.
Wang, S., Liu, X., Liu, S., Jin, J., Tu, W., Zhu, X., and Zhu, E. Align then fusion: Generalized large-scale multi- view clustering with anchor matching correspondences. Advances in Neural Information Processing Systems, 35: 5882-5899 2022b.
Xu, J.. Tang. H., Ren, Y., Peng, L., Zhu, X., and He, L. Multi-level feature learning for contrastive multi-view clustering. In Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp. 16051- 16060, 2022.
Wang. S., Liu, X., Zhu, X., Zhang, P., Zhang. Y., Gao, F., and Zhu, E. Fast parameter- -free multi-view subspace clustering with consensus anchor guidance. IEEE Trans- actions on Image Processing, 31:556-568, 2022c.
Xu, J., Ren, Y., Wang. X., Feng, L., Zhang, Z., Niu, G., and Zhu, X. Investigating and mitigating the side effects of noisy views for self-supervised clustering algorithms in practical multi-view scenarios. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern 2024.
Wang, X., Lei, Z., Guo, X., Zhang, C., Shi, H., and Li, S. Z. Multi-view subspace clustering with intactness- aware similarity. Pattern Recognition, 88:50-63 000-6,22200 2019.
Recognition, pp.
Yan, W., Xu, J., Liu, J., Yue, G., and Tang, C. Bipar- tite graph-based discriminative feature learning for multi- view clustering. In Proceedings of the 30th ACM In- ternational Conference on Multimedia, pp. 1411-3411, 2022.
Wen, J., Liu, C., Xu, G., Wu, Z., Huang, C., Fei, L., and Xu, Y. Highly confident local structure based consensus graph learning for incomplete multi-view clustering. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 15712-157211 2023.
Yang, B., Zhang, X., Nie, F., Wang, F., Yu, W., and Wang, R. Fast multi-view clustering via nonnegative and orthog- onal factorization. JEEE Transactions on Image Process- ing, 30:2575-2586, 2021.
Wen, J., Deng, S., Wong, W., Chao, G., Huang, C.,, Fei, L., and Xu, Y. Diffusion-based missing- -iiew generation with the application on incomplete multi-view cluster- ing. In Forty-first International Conference on Machine Learning. PMLR, 2024a.
Yang, H., Gao, Q. Xia, W., Yang, M., and Gao, X. Mul- tiview spectral clustering with bipartite graph. 1EEE Transactions on Image Processing, 31:3591-3605, 2022.
Wen, J., Xu, G., Tang, Z., Wang, W., Fei, L., and Xu, Y. Graph regularized and feature aware matrix factorization for robust incomplete multi-view clustering. IEEE Trans- actions on Circuits and Systems for Video Technology, 34 55:3778-3744, 2024b.
Yang. X., Jiaqi, J., Wang, S., Liang, K., Liu, Y., Wen, Y., Liu, S., Zhou, S., Liu, X., and Zhu, E. Dealmve: Dual contrastive calibration for multi-view clustering. In Pro- ceedings of the 31 ACM international conference on multimedia, pp. 1337346,770 2023a.
Wu, H., Huang, S., Tang, C., Zhang, Y., and Lv, J. Pure graph-guided multi-view subspace clustering. Pattern Recognition, 136:109187, 2023.
Yang, X., Liu, Y., Zhou, S., Wang, S., Tu, W., Zheng, Q.. Liu, X., Fang, L., and Zhu, E. Cluster-guided contrastive graph clustering network. In Proceedings of the AAAI con- ference on artificial intelligence, volume 37, pp. 10834 10842, 2023b.
Xia, W. Zhang, X., Gao, Q. Shu, X., Han, J., and Gao, X. Multiview subspace clustering by an enhanced tensor nuclear norm. IEEE Transactions on Cybernetics, 52(9): 8962-8975, 2022.
Yang, X., Min, E., Liang, K., Liu, Y., Wang, S., Zhou, S., Wu, H., Liu, X., and Zhu, E. Graphlearner: Graph node clustering with fully learnable augmentation. In Proceedings of the 32nd ACM International Conference on Multimedia, pp. 5517-55266 2024.
Xia, W., Gao, Q., Wang. Q., Gao, X., Ding, C., and Tao, D. Tensorized bipartite graph learning for multi-view clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):5187-520) 2023.

---

From Spectrum-free towards Baseline-view-free: Double-track Proximity Driven Multi-view Clustering
Yu, H., Liang, K., Hu, D., Tu, W., Ma, C., Zhou, S., and Liu, X. Gzoo: Black-box node injection attack on graph neural networks via zeroth-order optimization. IEEE Transac- tions on Knowledge and Data Engineering, 2024a.
Zeng, P., Yang, M., Lu, Y., Zhang, C., Hu, P., and Peng. X. Semantic invariant multi-view clustering with fully in- complete information. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46(4).2139-215 2023.
Yu, H., Ma, C., Wan, X., Wang, J., Xiang, T., Shen, M., and Liu, X. Dshield: Defending against backdoor attacks graph neural networks via discrepancy learning. Net- work and Distributed System Security Symposium, NDSS, 2025a.
Zeng, P. Yang, M., Lu, Y., Zhang, C.,, Hu, P., and Peng, X. Semantic invariant multi-view clustering with fully in- complete information. IEEE Transactions on Pattern Analysis and Machine Intelligence, 46(4):2139-2150, 2024.
Yu, S., Dong, Z., Wang, S., Zhang, P., Zhang, Y., Liu, X. Guan, N., Li, T., and Cheung, Y.-m. Simple yet effective incomplete multi-view clustering: Similarity- level imputation and intra-view Sybrid-group prototype construction. In The Thirleenth International Conference on Learning Representations.
Zhang, Fu, H., Hu, Q., Cao, X., Xie, Y., Tao, D., and Xu, D. Generalized latent multi-view subspace cluster- ing. IEEE Transactions on Pattern Analysis and Machine Intelligence, 42(1):86 19086699-6600 2020a.
Zhang, Fu, H.,, Wang, J., Li, W., Cao, X., and Hu, Q. Tensorized multi-view subspace representation learning. International Journal of Computer Vision, 128(8):2344- 2361, 2020b.
Yu, S., Wang, S., Wen, Y., Wang, Z., Luo, Z., Zhu, E., and Liu, X. How construct corresponding anchors for incomplete multiview clustering. IEEE Transactions Circuits and Systems for Video Technology, 34(4):2845- 2860, 2023a.
Zhang, C., Cui, Y., Han, Z., Zhou, J. T., Fu, H., and Hu, Q. Deep partial multi-view learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(05); 2402-2415, 2415, 2022.
Yu, S., Dong, Z., Wang, S., Wan, X., Liu, Y., Liang, W., Zhang, P.. Tu. W., and Liu, X. Towards resource-friendii extensible and stable incomplete multi-view clustering. In Proceedings of the s1 International Conference Machine Learning, pp. 77415-57440, 2024b.
Zhang, H., Nie, F., and Li, X. Large-scale clustering with structured optimal bipartite graph. IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(8):9950- 9963, 2023.
Yu, S., Wang, S., Dong, Z., Tu, W., Liu, S., Lv, Li, P., Wang, M., and Zhu, E. A non-parametric graph clustering framework for multi-view data. In Proceedings of the AAAI conference artificial intelligence, volume 38, pp. 16558-16567, 2024c.
Zhang, P., Pan, Y., Wang, S., Yu, S., Xu, H., Zhu, E., Liu, X., and Tsang, I. Max-mahalanobis anchors guidance for multi-view clustering. In Proceedings of the AAAI Con- ference on Artificial Intelligence, volume 39, pp. 22488- 22496, 2025.
Yu, S., Wang, S., Zhang. P., Wang, M., Wang. Z., Liu, Z., Fang. L., Zhu, E., and Liu, X. Dvsai: Diverse view- shared anchors based incomplete multi-view clustering. In Proceedings of the AAAI Conference on Artificial In- telligence, volume 38, pp. 16568-16577, 2024d.
Zhang, Y., Wang, S., Liu, J., Yu, S., Dong, Z., Liu, S., Liu, X., and Zhu, E. Dleft-mkc: Dynamic late fusion multiple kernel clustering with robust tensor learning via min-max optimization. In The Thirteenth International Conference on Learning Representations.
Yu, S., Zhang, P., Wang. S., Dong, Z., Yang. H., Zhu, E., and Liu, X. Differentiated anchor quantity assisted in- complete multiview clustering without number-tuning tuning. IEEE Transactions on Cybernetics, 54(11):7044-7037 2024c.
Yu, S., Liu, S., Wang, S., Tang, C., Luo, Z., Liu, X., and Zhu, E. Sparse low-rank multi-view subspace clustering with consensus anchors and unified bipartite graph. IEEE Transactions Neural Networks and Learning Systems, 36(1): - 2025b.
Yu, W., Xing, L., Nie, F., and Li, X. Multi-view fuzzy clustering based on anchor graph. IEEE Transactions on Fuzzy Systems, 2023b.
12

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
A. Notations
For more clarity, we summary the utilized symbols and their corresponding meaning, as shown Table 8.
Table 8. The description of symbols used in this article
Symbol
Meaning
n
the number of samples
m
the number of anchors the number of views the number of clusters the data dimension view p the data matrix view p the anchor matrix on view the permutation matrix on view p
V
k
d
X Rd
Ap E Rd xm
Тр E R
B, E Ro*k
the basic coefficient matrix on view p the cluster indicator matrix the anchor self-expression matrix view the degree matrix of Sp view p the view weighting vector the anchor graph on view p the Laplacian matrix about S,
R4xn
Sp Rmxm
D Rmxm
a E R
Zp E Rmxn
Lg R m
Ep E R mxX
B,B.C т T.Sp AJA, A SpS,T
F2 TR mmxm
Go Rmxm
H. R
M, E Rmmm
B,
Jp Ruxm
AJ P
Op Rmxm
IAJJA,
Z Baxk
2 o X p A,T.B
W E R
M B. T A A, ABBLL..........LLLLLLLLLLL
B. Brief Introduction of Seventeen Comparison Methods
To demonstrate the strong points of the proposed DTP-SF-BVE, we select seventeen remarkable MVC methods as the baselines. Their brief introduction as follows,
1. FMR (Li al., 2019): This method utilizes kernel dependence measure instead of projecting original samples to enhance the correlation between different views, and highlights the comprehensiveness of potential representations through subspace reconstruction.
2. PMSC (Kang et al., 2020a): This method merges view information in level of partition spaces via ensemble learning, and integrates consensus clustering and graph generation maintain the consistence among views.
3. AMGL (Nie et al., 2016); This method assigns a group of weights for all graphs to increase the diversity automatically, and reformulates conventional spectral partitioning procedure into a convex problem so as to generate the optimal solution.
13

---

From Spectrum-free towards Baseline-view-rre: Double-track Proximity Driven Multi-view Clustering
4. MSCIAS (Wang et al., 2019): This method maximizes the dependence between intact points by constructing informative affinity matrix, and avoids view information imbalance by guiding intactness-awant relationship construction using HSIC criterion.
5. MVSC (Gao al., 2015): This method conducts subspace clustering action on each view concurrently to explore specific characteristics, and employs an indicator matrix that is shared for all the views to preserve the cluster consistence.
6. MLRSSC (Brbić & Kopriva, 2018): This method generates a common similarity matrix with low-rank and sparsity properties to learn joint subspace representations, and utilizes the kernel extension skill to optimize the objective in Hilbert space.
7. MPAC (Kang al., 2019): This method aligns each partition alternatively using a permutation matrix to formulate agreement cluster indicator, and performs graph learning and data partitioning jointly in common framework to facilitate each other.
8. MCLES (Chen et al., 2020): This method tries to capture global structure by exploring embedding representations in latent space, and concurrently learns the cluster labels and similarity matrix without requiring subsequent spectral grouping procedure.
9. FMCNOF (Yang al., 2021): This method integrates matrix factorization skill and bipartite graph construction together to improve the computational overhead, and embeds the factor matrix into cluster matrix avoid extra k-means operation.
10. PFSC (Lv et al., 2021): This method finds a common partition by collaboratively learning multiple basic partitions to improve the robustness to noise, and jointly performs basic partition generation and unified graph learning to achieve mutual co-evolution.
11. SFMC (Li et al., 2022a): This method coalesces view-specific costs to seek for a joint graph that is compatible among views, and indicates clusters straightforwardly by employing connectivity constraint the joint graph.
12. MSGL (Kang et al., 2022): This method discriminates landmarks by building a dictionary matrix to decrease the cost of graph generation, and discovers a graph with explicit components to preserve the data manifold.
13. FPMVS (Wang et al., 2022c): This method designs group of space-guided projection matrices to alleviate the dimension inconsistency in common space, and determines the contribution each individual view to the unified graph in a learnable manner.
14. UOMVSC (Tang et al., 2023): This method unifies the spectral embedding and spectral discretization via one-pass strategy to alleviate the information loss caused by the two-step process, and approximates the rank of affinity graph through the inner product of embedding matrices.
15. PGSC (Wu et al., 2023): This method exploits the connectivity and sparsity of each similarity graph to achieve the pure graph with a block- diigonal structure, and assigns labels directly by enforcing it including corresponding connection components.
16. OrthNTF (Li et al., 2024b): This method establishes an orthogonal non-negative tensor factorization scheme to directly consider the cross cross-correlation between views, and extracts complementary information hidden in multi-view samples through tensor regularization.
17. FASTMI (Huang et al., 2023): This method achieves multi-stage mergence by building view-wise relations using random view grouping, and utilizes a graph partitioning mechanism to generate basic clusterings for each view group.
To further demonstrate the advantages of DTP-SF-BVE, we also compare it with several deep learning based competitors. Their introduction is as follows,
1. ADAGAE (Li et al., 2022b): This method utilizes a graph auto-encoder to extract the potential high-level information behind data and the non-euclidean structure, and avoids the collapse by building the connections between sub-clusters before they become thoroughly random in the latent space.
14

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
2. DEMVC (Xu et al., 2021a): This method generates the embedded feature representations by deep auto-encoders, and adopts the auxiliary distribution generated by k-means to refine the deep auto-encoders and clustering soft assignments for all views.
3. MFLVC (Xu et al., 2022): This method jointly realizes view-specific reconstruction objective and semantic consistency objectives by learning diverse levels representations in fusion-free way, and utilizes the common semantics generate the clustering labels.
4. DSMVC (Tang & Liu, 2022b): This method concurrently exploits complementary information and discards the meaningless noise by automatically selecting features to reduce the risk clustering performance degradation caused by view increase.
We also compare DTP-SF--VE with popular alignment methods, and their introduction is as follows,
1. FMVACC (Wang et al., 2022b): This method utilizes feature information and structure information of the bipartite graph generated by fixed anchors to build the matching relationship, and regards the first view of each dataset the baseline view.
2. 3AMVC (Ma et al., 2024a): This method gets rid of prior knowledge by identifying and selecting discriminative anchors within single view using hierarchical searching, and takes the view exhibiting the highest anchor graph quality as the baseline view.
3. AEVC (Liu et al., 2024): This method narrows the spatial distribution of anchors on similar views by leveraging the inter-view correlations to enhance the expression ability of anchors, and treats the view concatenated by column as the baseline view.
C. Brief Introduction of Seven Public Datasets
In experiments, we evaluate the algorithm performance on seven public multi-view datasets, and their brief introduction is as follows,
1. DERMATO: This is skin image dataset and consists of 358 samples. It contains 2 views and 6 clusters. The feature dimensions on each view are 12 and 22 respectively.
2. CALTE7: This an object image dataset and consists of 1474 samples. It contains views and 7 clusters. The feature dimensions on each view are 48, 40, 254, 1984, 512 and 928, respectively.
3. Cora: This citation network dataset has 2708 samples, and includes 4 views and 7 clusters. The feature dimensions on each view are 2708, 1433, 2708 and 2708, respectively.
4. REU7200: This document dataset has 7200 samples, and includes 5 views and 6 clusters. The feature dimensions on each view 4819, 4810, 4892, 4858 and 4777, respectively.
5. Reuters: This is a news article dataset with 18758 samples, and involves views and 6 clusters. The feature dimensions on each view are 21531, 24892, 34251, 15506 and 11547, respectively.
6. CIF10Tra4: This is color image dataset with 50000 samples, and involves 4 views and 10 clusters. The feature dimensions on each view are 944, 576, 512 and 640, respectively.
7. FasMNI4V: This fashion product image dataset with 70000 samples, and involves 4 views and 10 clusters. The feature dimensions each view are 512, 576, 640 and 944, respectively.
D. Other Related Work
To effectively tackle MVC tasks, Chen et al. (2022) utilize the algebraic property to learn a group of orthogonal bases for anchors while preserving the scalability, Qiang et al. (2021) iteratively partition original data into two balanced parts using k-means++ output informative anchors, Zhang al. (2023) integrate anchor selection into the generation of anchor
15

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
graph in which the number of connection components is the same that of clusters to explicitly explore cluster structure, Li et al. (2024c) devise a pre-defined prior matrix for view-wise anchors to regularize their order and utilize graph matching model to handle unpaired data, Yu et al. (2023b) combine membership learning and the construction of anchors to decrease the disagreement between views, and improve the clearness of cluster grouping via trace norm regularizer. Lao et al. (2024) choose to jointly construct multiple sets of anchors for hasic clusterings so as to form discriminative subspace representations.
Orthogonal to them, Xu et al. (2021b) optimize a view-common variable and view-specific variables by introducing variational auto encoder into MVC to regulate consecutive visual characteristics of multiple views, Cui et al. (2024) highlight consistent representations from the perspective of information theory and decrease the view redundancy by minimizing the representation lower bound, Zhang et al. (2022) reach to the balance between complementarity and consistency by encoding view information using an adversarial strategy and utilize a parameter-free loss to complete the formation of structured representations while avoiding over-fitting, Fu et al. (2024) excavate potential structure distributions among samples in a generative manner and utilize anchor graphs to guide the learning process by generating structured spectral embedding using graph convolution network. By virtue of tensor tool, Li et al. (2024a) orthogonally project anchor graph into potential label space to explore the cluster distribution and alleviate the loss of spatial structure information caused by projection transformation via tensor regularization. Long al. (2024) form embedding tensor by stacking embedding features of all views together to simultaneously explore the inter-view and intra-view correlations, and utilize the uniformity between semantics by employing an unified constraint to guarantee the smoothness of embedding.
To enhance the block structure of anchor graph, Qin et al. (2022) integrate multiple similarity matrices into one by introducing semi-supervised information and concurrently perform self-mapping and backward encoding via reconstruction. Nie et al. (2024) conduct number limitations on each cluster by combining min-cut and size constraints to enhance the flexibility and decrease the parameter sensitivity, and decompose lower constraints and upper constraints respectively via augmented Lagrangian multiplier strategy. Wen et al. (2024b) enhance the robustness by reducing the negative impact of noisy features and redundant information using feature weighting constraints, and utilize graph-embedded learning to maintain the structure characteristics. Huang et al. (2022) construct various metrics by randomizing exponential similarity in metric subspace rather than original space to improve the diversification of similarity matrices, and probe into the spatial characteristics of clusters via entropy criteria. Zeng et al. (2023) capture unified semantics by climinating the discrepancy across views using the semantically invariant -nnvaiiaanll distribution hidden within views, and alleviate the impact of defective instances via distribution transformation skills. The approaches, such as (Lu et al., 2024; Yu et al.; Sun al., 2024; Yu et al., 2024a; Xu al., 2021a; Yang et al., 2023b; Yu et al., 2025a; Liang et al., 2024b), have been also studied.
E. More Illustrations for the Objective Function
For the methodology, we here provide more details to explain the reasons for the design of each component.
First of all, to exploit the geometric characteristics between anchors, inspired by the concept of subspace reconstruction, we introduce self-expression learning for anchors. Specially, we utilize the paradigm A| S, to explicitly extract the global structure between anchors.
After obtaining the anchor-anchor characteristic Sp E m we need to integrate that into anchor-sample so as to exploit the manifold features inside samples. To this end, we adopt the idea of point-point guidance to adjust the anchor graph. Note that the rows of anchor graph Zp E Rm*x correspond to anchors, and thus we utilize the element [Spla, to guide [Zpla, and [Zpla.e & t= ===1,, 1,...., n. which can be formulated as Σij=1 [Zpli. [Zpl,, [Sples and aims at restricting similar features to maintain the consistency.
Then, to alleviate the anchor misaligmment, considering that the nature of misalignment is that the order of anchors on different views is not identical, we alleviate the misalignment issue by rearranging anchors. Particularly, we associate each view with a learnable permutation matrix T, to freely transform anchors in the original dimension space according to the characteristics of respective view. In addition to not involving selecting the baseline view, our mechanism also can coordinate with anchors in the unified framework and thereby facilitates the learning of anchors. Correspondingly, the anchor matrix Ap is reformulated as A, T, The self-expression term ||A, A and the reconstruction term [X,, - A, are reformulated as A A sll 11 and |X,, - 11, respectively.
Subsequently, considering that the variance arises from the construction of spectrum, we avoid forming spectrum and choose to directly learn the cluster indicators. In particular, factorize the anchor graph as basic coeffi-
16

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
cient matrix Bp and consensus matrix C, and utilize binary learning to optimize the consensus matrix. Therefore,
we have that the term | ppTpZpli is reformulated as X - A,T,B,CII F. The point-point guidance term [Sp.. which can be equivalently
Σx==1 Zpla,: (Sp is reformulated as شi [| B, L. 1 Sp. D, - diag(E (Spla 1 i 1, , .m). transformed as the matrix trace form of Tr D, This paradigm not only makes the consensus matrix C successfully represent the cluster indicators, but also provides a common structure for anchors on all views, inducing them rearranging towards the corresponding matching relationship.
At the last, due to views generally having different levels of importance. assign a weighting variable to each view to adaptively adjust its contributions. Accordingly, the term XXp A B, is further reformulated as of X A A,T,B,
Based on the above analysis, we have that the objective is formulated as Σp-1 - A,T,B,CCI + A A, - A T,
The first item in Eq. (3) aims at building the similarity via minimizing the reconstruction error. The second item represents the self-expression affinity of aligned anchors. The third item plays a role in feeding anchor-anchor characteristics into anchor-sample.
Further, for the feasible region, the constraints {a 1 =1 a, 0} aim at doing normalization and meanwhile avoid trivial solutions. [B, B, I aims at learning discriminative basic coefficients. (T = 1, T, = 1, Tp E {0, 1}mxm} aims at rearranging anchors and meanwhile guarantees not to change the values of anchors. 1, .......... n, Ce {0, 11,xx]]_ guarantees that there only one non-zero element each column, i.e., one sample belongs to only one cluster. { 1, S, 2 0, Ei= [Splet = 0} guarantees expressing oneself through other anchors while avoiding using oneself express oneself.
& Ci,j = 1,j
F. The Influence of Anchor Number
In experiments, we set the number of anchors to be equal to the number of clusters. The reasons are as follows.
When updating the variable Tp the objective function Tr (T p G, (AH, p + M,M, - 2AS,) - TTTTT which the form of A BAC + A.D. Besides, the feasible region {T 1 = 1, T,1 = 1, 1,T, E {0, 10mmm is discrete. These cause this optimization problem being hard to solve. To this end, we adopt traversal searching on one-hot vectors to obtain the optimal solution. The traversal searching operation takes (mm!) computing overhead where m is the number of anchors. Too large m will induce intensive time cost. Therefore, in all experiments, we set m to the number of anchors k. More genius solving schemes could be further investigated in the future.
G. Derivation Details
In this section, we provide a more detailed derivation procedure about the minimization of the loss function Eq. (3).
Update Ap: When updating Ap, Eq. (3) equivalently becomes
min of A,T,B,CIE + A,T,SP1I
(17)
Due to the independence of views, anchor sets on different views are also independent of each other. Accordingly, we can cquivalently transform Eq. (17) as
min 1X, A,T, + - A,T,S,
This is an unconstrained optimization problem, and according to the derivative value of zero, we can obtain
a (A, x,) ( T. +A A,T, (T - T,
a E.E.E,E - X.E + F
(18)
A E. + FF,F a.X,ETT
17

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
where Ep E R T,B,C, Rm T pyp Tp is a permutation matrix, and thus is invertible. Further, according the property of permutation matrix that its inverse is equal to its transposition, i.e., Tp T have that T is also permutation matrix, and consequently can be seen as series of elementary transformation operations. Based on the fact that elementary transformation does not change the rank of matrix, we have rank Fp)) = rank(Fp)) Additionally, rank FP)) rank(I - Sp). Since Sp is an anchor self-expression matrix and its diagonal elements are zero, we have rank(1 Sp) = m. That is, its rank is full. Thus, we have (F,) = m. It is full rank and accordingly invertible. So, F, F is also invertible. Further, the eigenvalue of F F is greater than O, the eigenvalue of E, is greater than or equal to O, and thus the eigenvalue of E, + FF, is greater than O. Consequently, the item E, + FT is invertible. Based on th above analysis, we can get that A, - x,E E, + F Update Tp When updating Tp. Eq. (3) equivalently becomes
E P I A,T,B, ( +A A,T,
(19)
s.t. T I 1, T. E {0. 1m
Due to Tр being performed on respective view, separately optimize each Tp. Thus, Eq. (19) be equivalently written
min Tp
X, A,T, + AI, - A,
s.l. 1 1, E {0, 1
After expanding the objective using the trace operation and deleting irrelevant items, we can get
min a X A,T,B,CCI + AI,,T - S,S
min Tr A CC B. 2 c C + I
(20)
TA,T,S,S,T,A, 2TA,T,S,TTA
According to the fact that T is permutation matrix, we have T I. Additionally, considering that Σ Di=1 Cij
1.J E 21. n). C E {0, 1] 1,x",, we have that CC a diagonal matrix, and its diagonal elements are 1 -=1 Caj+ i= 1,2, .. k. Further, combined with Bp being orthogonal, obtain Tr B, Tr (CCT) Ei, Ciyy Based on these analysis, we can get
min T,
- AIA, A,T,S,
= min Tr 20 ST C B + A A,T,
4, A, B, - 2XT, A, A,A,A,AAAA T,S.))
(21)
min Tr T G.T.H. a T G,T,M, 2XT, S. -2
T,
min Tr (T) G (ΛH a 2XT p G. 20 Jp).
T
where Gp E R = AJ Hp E R S,ST p. M. T = B,CC B Jp E mm = =, Combined with the feasible region in Eq. (19), we can determine the optimal solution of Tp via traversal searching using
[e1, ...... .... ,e,
em] where e; is the one-hot vector. Kindly note that the size of Tp m x m and m is generally small, performing traversal searching on T, will not ineur significant computing costs.
Update By: When updating B,, Eq. (3) equivalently becomes
a X A,T,B, ST
min B,
(22)
s.l. B B, = I
18

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
Since the basic coefficient matrices (Bp) different views independent of each other, can equivalently transform Eq. (22) as
min B₁
2 T,, LBTT(B,,,,BCCC)
(23)
B, I
Expanding the objective and then deleting irrelevant items, we can obtaín
min a X, A, A,T,B, B BBB B
(24)
min Tr (o To B B, B, A, 20 A X B + L,B,LB,,CC'
Bp
Since CC T is diagonal and Bp is orthogonal, we can further have
min 2
X A.T,B,CCI BTB
B
min Tr (BB (JB - Q,
Q, 2 A A C, mi ( (oL, + a0 B,CC - CX, A,T, Bp
(25)
where Op mm -
Considering that the feasible region B 0, 1,2, .......kiii...... i 1,2,. k, we can solve B, column by column. Thus, have
p Bp = I can equivalently divided into [Bpl. Bppk and [Bp]. BBpp.
min Tr (B (SL, + Q B,CC - cx A,T,Bp)
min
B p 15: (SL, a Q B, CC 1.s + 2x A, Tpl jn [Bplas
[Bpla
(26)
min B
J (oL, + a Qp) BBla + cx, Ap ApTpl,,
BBl..
1=1
where the objective is quadratic.
can be equivalently writ- ten as [BpT Bpl- - = 0. Bp B Bph. = 0, = 1,2,... ,k, # j can be written as [Bp].1: [Bpl.s.'.. ... Bpl-s-1, [Bpla-i.... [Bpka]] [Bpl. 1x-1) Apparently, the constraints are also quadratic.
Besides, the constraint [B [Bpl. Bpl.
Consequently, the optimization problem about B, can equivalently transformed as
min Bal.d
C2. (BL + + Bpla + [-- 2a, CX A,Tp), [Bpla
s.t. [BIT.] mmxm Bpks O, [Bp] [Bl..+1. [B]],*]
[Bpl. Bpl...
[Bpl- 0(1-1)x1-
This is QCQP optimization problem, and can be solved in O(m)) computing complexity.
Update Sp: When updating Sp, Eq. (3) equivalently becomes
min > A, A,T,S, + LTBBLB,,,CCC Sp
(27)
s.t. 8 = 1. 2 0. Σ
S Spks = 0.
19

---

From Spectrum-free towards Baseline-view-free: Double-track Proximity Driven Multi-view Clustering
Expanding the objective, we have
min X A - A,T
Sp
+ min TF (AA P T A 2AA,T,S, T,,,
Sp
+AA, p T T , + D,B, SBB,B,,CC')
min Tr (Λy s, T. A 2AA,T, SB,,B,C
Sp
(28)
min Tr (AS, p T, A T,, 2AT A A, - BB, BB,CCBBBSs) Sp
min Tr (AS) S - S, - 3M,
3 Mp Sp
+ min'I Tr Sp S, p + -Qp 2X
Sp
where Op E 1 = -T M, E m CC B..
Noticed that the feasible region is for each column of Sp. consequently, we can equivalently rewrite the constraints in the form of columns. That is, we can transform S, = S, 2 0, 1 (Splet = 0 as [Sp].]] 1 1, 1,(sp 2 0, [Spli =
j,j = 1, 2, , Further, we can transform [Sp 0,j 1,2, as e] [Splas = 0, = 1,2, ... m,, where ej is the one-hot vector.
Based on these, for the objective function, we can further have
min Tr Sp
S SpSp +2
2g
Mp
Sp
2X
(29)
3
min 2 s 1plsp +
-Opp-
M
[Sp
2X
Therefore, the optimization problem about Sp, can be equivalently written as
min pp 2 S Splis +
Qp 2 J::
Mp
[
(30)
s.t. [Sp p 1 = 1,0 = /Sp e, [Splis = 0, = 1,2, m,
which is QP problem, and can be solved within O(m²) computing complexity.
Update C: When updating C, Eq. (3) equivalently becomes
min
2
X
(31)
s.t.
Cūj = 1,j = 1,2,..., Ce {0, kkxm
For the objective function, we have
min
2 X -
min TF
c
Σ
a
A,
> a J A,,,B, 3C Σ B B L.B.C.C
C
p=1
p
p=
(32)
min Tr
CT
Σ
a
T + BB L.B,
C-2 Σ
a 2x A
T.T.BBCC
p=1
min (C WC ZC).
C
20

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
where W E R =
Ap A,, 2 L. Z B = =2 p= 0 - Bpp
The constraints mean that there is only non-zero element in each column of C, and consequently we can optimize C by column. We can get
min Tr (C WC 2CC = min C. WC., Zy.:C...
(33)
C
Further, the item C WC., indicates that it takes a diagonal element of W, and Z,. C., indicates that it takes a element of Zj:-- Thus, we can determine the corresponding index of minimum by
arg min Wt Zz,,, = 1,2, k.
(34)
Then, the value C. can be determined by assigning Cre as while assigning other elements of C.5 as O.
Update a: When updating a, Eq. (3) equivalently becomes
min Σ Ix - APT,B,CI
s.t. T 1=11:,220 0.
Considering that the term -
by |X,, A, is constant with respect a, can solve a using Cauchy
inequality. Specially, we can get that the optimal solution is ap = Ep
H. More Conclusions for Table 1
1. On CALTE7, MFLVC receives better clustering results in NMI, probably because it achieves reconstruction and consistency by learning features at multiple levels rather than at single level for each view, and utilizes the consensus semantics shared in all views and semantic labels decrease the view-private unfavorable influence.
2. FPMVS achieves 0.29% increasement in terms of ACC on CIFI0Tra4, and possible reasons are that it employs group of projectors to maintain the anchor dimension consistency and extracts consensus multi-view isomeric features by utilizing an unified graph structure with cluster distribution constraints.
3. On Cora in Fscore, MSCIAS slightly surpasses with 0.72%, which is mainly because it enforces encoded similarity to maximally depend on the potential matae-ssapless through HSIC criterion and utilizes the local connectivity of intact space to eliminate outliers and enhance the distinguishability of similarity.
4. For SFMC, it makes preferable results on REU7200 in NMI, main reasons of which could be that it integrates connectivity constraint into the learning of joint graph to reflect cluster distribution and adaptively adjusts the graph contributions on different views in self- eeff-suprrvised weighting way.
5. PMSC, AMGL and MLRSSC express inferior performance in certain scenarios, the reasons of which could be that PMSC reaches the consensus clustering under the premise of the basic partition realizing the ground truth and meanwhile equally treats every view, the factors generated by the cluster indicator with orthogonal constraints in AMGL impair the discriminability of some graphs, and MLRSSC linearly combines the generated representation matrices and only utilizes truncation operation to determine the penalty parameters.
Other Ablation Studies
In the paper, rather than treating views equally, adopt square weighting scheme to adaptively combine views together. To validate the effectiveness of this strategy, we conduct the comparison experiments with equal view weighting (EVW). The results are summarized in Table 9, where AVW denotes the clustering results based on our adaptive view weighting. Obviously, AVW receives more desirable results than EVW most cases, which suggests that the adaptive view weighting strategy recommendable. Additionally, we also plot the learned view weights, as shown in Fig. 3. It can be seen that indeed assigns different weights to measure the contribution between views.
21

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
Table 9. The effectiveness of view weighting
Metrie
Ablation
DERMATO
CALTE7
Cora
REU7200
Reuters
CIF10Tra4
FasMNI4V
EVW
84.59
76.73
44.94
16.82
47.84
25.26
52.69
ACC
AVW
85.47
80.66
52.44
26.22
54.26
26.83
57.36
EVW
89.44
42.88
33.06
3.60
29.54
15.20
56.84
NMI
AVW
89.97
45.25
43.70
6.25
31.87
15.64
59.21
EVW
86.59
71.80
35,36
28.77
42.32
18.03
46.90
Fscore
AVW
87.92
78.12
41.12
28.55
44.84
20.64
51.37
1
View
View
0.9
View
View
View
View
0.8
0.7
0.6
0.4
0.3
0.2
0.1
0
DERMATO CALTE7
Cora
REU7200
Reuters
CIF 10F10 FasMNI4V
Figure 3. The learned view weights seven public datasets.
Besides, unlike current techniques generating anchors via random sampling or heuristic searching. which leads to anchors being separated from subsequent procedures like graph learning and spectrum construction, we integrate anchors into objective optimization framework to make them able to interact with other parts and thereby facilitate each other. To investigate its effectiveness, we organize corresponding ablation experiments and present the comparison results in Table 10, where HS denotes the clustering results based on anchors generated by heuristic searching while LA denotes the results based on our anchor learning. easy to observe that LA outperforms HS with noticeable margins, which illustrates that the anchor learning strategy is functional and can provide more pleasing clustering results.
Table 10. The effectiveness of learnable anchor
Metrie
Ablation
DERMATO
CALTE7
Cora
REU7200
Reuters
CIF10Tra4
FasMNI4V
HS
65.64
64.59
30.24
16.68
27.20
24.08
47.21
ACC
LA
85.47
80,66
52.44
26.22
54.26
26.83
57.36
HS
69.84
37.95
33.54
1.06
1.43
12.98
47.07
NMI
LA
89.97
45.25
43.70
6.25
31.87
15.64
59.21
HS
69.33
61.54
30.40
24.43
35.25
18.03
41.43
Fscore
LA
87.92
78.12
41.12
28.55
44.84
20.64
51.37
22

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
J. More Illustrations for Table 7
The selection of baseline view not only brings complicated solving procedure but also affects the clustering performance. If the baseline view is not well selected, the graph structure will be inaccurately fused. Unlike this, we do not require the baseline view, and can automatically rearrange anchors according respective view characteristics. Besides, also coordinate with anchors in the unified framework and thereby facilitate the learning of anchors, which makes view information interact across different levels.
K. The Effectiveness of Anchor Self-expression
The seff-expression affinity learning utilized to construct the sample-sample affinity with full size in subspace clustering. Inspired by this, we explicitly extract the global structure between anchors via self-expression learning, and meanwhile feed that into anchor-sample so to better exploit the manifold characteristics hidden within samples. (Kindly note that in this work, did not calculate the sample-sample relations through self-expression learning.) In addition to this, our work also designs a joint-alignment mechanism which does not involve the selection of the baseline view and meanwhile can cooperate with the learning of anchors. Moreover, a solving scheme with linear complexity enables our framework to effectively tackle MVC tasks.
Anchor self-expression learning can help extract the geometric characteristics between anchors, and meanwhile facilities the learning of anchors owing to the joint-ptiimzzation mechanism. To validate this point, we organize four groups of comparison experiments, i.e., No self-expression No leanring (NSNL)), No self-expression Having learning (NSHL), Having self- expression + No learning (HSNL), Having self-expression + Having learning (HSHL, i.e., Ours). The comparison results are reported in Table 11.
Table 11. The effectiveness of anchor self-expression
Dataset
DERMATO
CALTE7
Cora
REU7200
Reuters
CIF10Tra4
FasMNI4V
ACC(%)
NSNL
60.42
43.88
27.96
14.33
25.32
19.73
41.27
NSHL
71.51
49.05
30.35
16.75
47.05
26.69
52.15
HSNL
65.64
64.59
30.24
16.68
27.20
24.08
47.21
HSHL
85.47
80.66
52.44
26.22
54.26
26.83
57.36
NMI(%)
NSNI
64.37
35.68
5.88
1.01
1.38
12.57
44.79
NSHL
83.97
40.21
6.02
2.53
23.19
15,48
58.13
HSNL
69.84
37.95
33.54
1.06
1.43
12.98
47.07
HSHL
89.97
45.25
43.70
6.25
31.87
15.64
59.21
Fscore (%)
NSNL
63.76
48.43
28.79
23.42
33.87
16.86
37.64
NSHL
73.79
51.25
30.42
28.54
43.04
17.70
46.77
HSNL
69.33
61.54
30.40
24.43
35.25
18.03
41.43
HSHL
87.92
78.12
41.12
28.55
44.84
20.64
51.37
As seen, HSNL is consistently preferable than NSNL, and HSHL consistently preferable than NSHL. These demonstrate that the anchor self-expression can facilitate the clustering performance improvement. Additionally, NSHL is consistently preferable than NSNL, and HSHL consistently preferable than HSNL. These illustrate that the anchor learning can help increase the clustering results. Therefore, we conclude that the anchor self-expression learning enhance the quality of anchors to increase the clustering results.
L. The Effectiveness of View-specific Anchors
The works based on feature space fusion or shared representations usually extract a group of unified anchors rather than multiple groups of view-specific anchors to construct the similarity relationship. Although avoiding alignment, this paradigm
23

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
could not effectively exploit complementary information between views due the unified anchors being shared for all views. To further illustrate this point, organize the comparison experiments between unified anchors (UA) and view-specific anchors (VSA, i.e., ours). The results are presented in Table 12.
Table 12. The effectiveness of view-specific anchors
Dataset
DERMATO
CALTE7
Cora
REU7200
Reuters
CIF10Tra4
FasMNI4V
ACC(%)
UA VSA
81.23
71.33
49.73
24.97
48.46
22.36
53.38
85.47
80.66
52.44
26.22
54.26
26.83
57.36
NMI(%)
UA VSA
82.76
42.26
39.87
6.03
29.89
13.43
51.97
89.97
45.25
43.70
6.25
31.87
15.64
59.21
Fscore (6)
UA
80.64
69.72
42.28
25.21
46.13
19.58
52.21
VSA
87.92
78.12
41.12
28.55
44.84
20.64
51.37
As seen, our results more preferable than the counterparts adopting unified anchors, the reason of which could be that the view-exclusive complementary representation information (view-specific (aligned) anchors contain) outweighs the view-common consensus representation information (unified anchors contain).
Table 13. Single-view experimentl results in ACC
Dataset
DERMATO
CALTE7
Cora
REU7200
Reuters
CIF10Tra4
FasMNI4V
FMR
66.50(#5.19)
21.59(#044)
24.08(=0.38)
PMSC
64.83(#8.99)
27.93(=0.86) 18.18(+1881)
19.24(#0.53)
AMGL
22.09(+0.16) 17.00(+0.05) 14.67(=0.17) 39.72(+1.35)
20.32(#0.27)
MSCIAS
64.35(+8.27) 40.63(+2.993)
30.88(+1.30)
19.25(+0.93)
25.38(+1144))
MVSC
57.08(+9.81) 45.54(+2.03)
MLRSSC
31.01(+0.00) 50.14(=0.00)
30.20(=0.00)
16.92(+0.00)
MPAC
61.14(+0.00) 41.72(+0.00)
32.48(=0.00)
18.40(+0.00)
MCLES
64.47(++4.27) 46.33(+2.58)
30.47(+1.32)
FMCNOF
52.51(++4.93)
48.51(+4.22)
24.34(+2.43)
18.67(+2.38)
19.55(+1.86) 31.52(-2.21)
ADAGAE
PESC
63.80( +6.28)
50.62(+4..7)
SFMC
65.88(+0.00)
50.24(+0.00)
30.07()0.00)
15.75(+0.00)
20.53(+0.00)
22.08(+0.00)
MSGL.
29.61(+1.03)
17.99(+0.88)
23.83(+0.64)
21.48(-0.57)
DEMVC
FPMVS
68.46(++7.24)
29..8
32.05(++191)
19.55(+0.19)
22.90(+2.15)
22.37(+0.62) 51.97(=3.26)
3АMVC
MFLVC
UOMVSC 66.20(+0.00) 3..04(10.00)
30.45(+0.00)
09.000
26.28(10.00)
PGSC
64.94(17.61) 36.37(+4.52)
31.27(+224)
18.07(10.74)
22.15(+0.62)
DSMVC
AEVC
OrtNNT
FMVACC
FASTMI
+4.53)
37.46(+1.933)
21.79( 21.73(+0.11) 14.52(=1.21)
53.12(=4.21) 28.03(+0.00) 25.73( 56.92(-0.00)
23.21(=1.05)
Ours
66. 66.70(+0.00)
02.04(=0.000 3..08
20.06(+0.000)
M. The Effectiveness of Single View Scenarios
Except for multi-vien scenarios, sometimes we also may encounter the datasets containing only one view. To validate the ability to tackle single view scenarios, we conduct clustering operation on one view rather than on all views of datasets mentioned earlier. The experimental results are summarized in Table 13, 14 and From these tables, we can draw
24

---

From Spectrum-free towards Baseline-view-free: Double-track Proximity Driven Multi-view Clustering
Table 14. Singee-vieewceppermeennaa
experimental results in NMI
Dataset
DERMATO
CALTE7
Cora
REU7200
Reuters
CIFI0Tra4
FasMNI4V
FMR
76.63(+3399)
1.38(+0.18)
5.12(=0.17)
PMSC
80.21(#5.10)
34.04(=0.71)
6.12(=0.83)
1.43(+0.12)
AMGL
3.07(+0.24)
34.34(+1.28)
0.83(=0002)
0.72(+0009)
0.89(=0.03)
MSCIAS
75.79(++4.77)
35.42(+1.66)
9.45(=0.46)
1.69(+0.277
1.22(+0.17)
MVSC
92.19(+111.8))
25.22(+1.225)
MLRSSC
0.54(+0.00)
0.73(+0.00)
0.48(=0.00)
0.56(++0.00)
MPAC
77.69(+0.00) 29.88(+0.00)
9.30(=0.00)
1.29(+0.00)
MCLES
78.19(+3344)
27.61(+1887)
8.4((-0.93)
FMCNOF
42.62(++4.22)
9.42(+1.23)
4.58(-0.79)
1.1((+0.17
9.36(+2.32)
32.07(+33.26
ADAGAE
PFSC
76.00(+2.445
32.47(+1.85)
SFMC
62.77(+0.00)
10.33(+0.74) 27.46(+0.00)
6.01(=0.00)
2.37(40.00)
1.44(=0.00)
9.44(=0.00)
MSGL
1.1((40.08)
1.02(=0.06)
7.64(20.68)
DEMVC
FPMVS
77.14(+5.56)
34.02(+1.85)
9.28(=1.64)
2.44(#0.24)
11.73(43.38)
10.99(=0.82) 53.71(=2.18)
3АMVC
MFLVC
UOMVSC 77.0000
28.86(=0.00)
8.95(=0.00)
2.83(+0.00)
9.45(=0.00)
PGSC
69.96(14.66)
22.46(+2.37)
1.43(=0.32)
2.03(#0.26)
0.92(=0.088)
DSMVC
AEVC
OnhNTF
EMVACC
FASTMI
7-215(+2.10))
32.49(+1.29)
13)13)
3.03(=0.66)
13.48(=0.64) 10.12(+1.58))
58. 18.14(=1.73)
Ours
80.52(+0000)
36.59(=0000)
10.04(=0.00) 2.50(+0.00)
1.37(=0.00)
15.09(=0.00)
61.44(00.00)
Table 15. Single-view experimental results in Fscore
Dataset
DERMATO
CALTE7
Cora
REU7200
Reuters
CIFIOTia4
FasMNI4V
FMR
66.40(14.64)
22.72(10.15)
19.09(10.23)
PMSC
68.06(18.46) 46)
37.68(11.18)
27.42(12.21)
21.34((1102))
AMGL
19.17(10.28)
37.52(10.89)
22.53(110.00) 24.77(10.11)
28.53(10.211)
MSCIAS
67.19(18.74)
40.65(13.12)
23.41(11.52)
21.07((10:41)
30.72(11.77)
MVSC
55.63(1110.99) 44.64(12.87)
MLRSSC
33.39(+-0.00) 50.64(+0.00)
30,40(+0.000)
23.21(+0.01)
MPAC
70.02(+00.00) 41.65(+0.00)
26. 26.100+0.00)
21.94(+0.00)
MCLES
67.27(+4469) 48.97(+2.84)
30.34(+1.73)
FMCNOF
46.98(++3.88)
46.35(++4.1)
20.05(++2.23)
21.68(+2.79)
17.00(+1.17) 26.54(+1.89)
ADAGAE
PFSC
68.99(+4.95)
48.47(+3.26)
SFMC 60.29(+0.00)
47.32(#00.0)
30.35(+0.00)
20.64(+0.00)
29.04(+00.00)
15.28(=0.00)
MSGL
31.90(+0.97)
22.48(+0.37)
28.97))
16.14(=0.32)
DEMVC
FPMVS
68.31(16.97)
49.97(12.50)
26.33(10.70)
20.33(11.15)
31.8((11.)))
17.54(10.35) 45.93(12.06)
3АMVC
MFLVC
UOMVSC (7.900(1..00)
24. 24.1001) 38.64(10.00)
22.78(10.00)
31.91(10.00)
PGSC
63.05(16.85)
25.01(13.77) 37.62(14.23)
22.53(11.24)
32.42(111.33)
DSMVC
AEVC
OrtiNTF
FMVACC
FASTMI
69.65(14.79)
19.24(#1.49) 11.50 50.41(20.00)
19.27))
15.35(=0.69) 30.45(=0.71)
45.68(+2.35)
Ours
69.70(+0.00)
32.00()0.00)
25.26(11.000
35.02(=10.00
17.86(=0.00) 49.77(+0.00)
that ADAGE, DEMVC, 3AMVC, MELVC, DSMVC, AEVC, OrthNTF and FMVACC are powerless against single view scenarios, which is mainly because they generally need utilize the information of other views help optimize. FMR, PMSC,
25

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
AMGL, MSCIAS, MVSC, etc, able work properly with single view scenarios, nevertheless, they generally produce inferior clustering results in most situations. By comparison, besides being able operate properly single view scenarios, our DTP-SF-BVF also generate desirable results. Accordingly, DTP-SF-BVF enjoys wider serviceability.
Table 16. The effectiveness of view information gathering
Clustering Results
Dataset
Metric
V1
V2
V3
V4
V5
V6
Ours
ACC
60.06
66.76
85.47
DERMATO
NMI
56.66
80.52
89.97
Fscore
49.99
69.70
87.92
ACC
35.01
52.04
48.71
53.87
39.42
46.40
80.66
CALTE7
NMI
17.84
36.59
34.1 14
48.75
41.74
39.98
45.25
Fscore
33.35
50.41
47.41
53.85
48.15
47.42
78.12
ACC
30.24
35.78
30.17
29.73
52.44
Cora
NMI
10.22
10.04
12.24
11.49
43.70
Fscore
30.39
32.40
30.36
29.91
41.12
ACC
21.24
20.06
24.06
19.61
20.83
26.22
REU7200
NMI
1.27
2.50
4.81
1.39
1.40
6.25
Fscore
23.67
25.26
25.41
27.01
24.90
28.55
ACC
46.65
28.03
27.24
27.80
47.62
54.26
Reuters
NMI
23.47
1.37
1.10
1.06
20.83
31.87
Fscore
42.71
35.02
35.24
35.16
43.67
44.84
ACC
21.89
25.73
23.25
22.05
26.83
CIF10Tra4
NMI
10.01
15,09
12.35
12.50
15.64
Fscore
16.46
17.86
15.87
16.34
20.64
ACC
41.76
56.92
46.71
52.32
57.36
FasMNI4V
NMI
49.81
61.44
53.26
54.86
59.21
Fscore
37.26
49.77
42.80
45.04
51.37
N. The Effectiveness of View Information Gathering
Compared to single view datasets, multi-view data can provide more comprehensive and detailed descriptions for the same instance and thereby facilitates more accurate representations for better clustering. validate the effectiveness of DTP-SF-BVF in gathering the information from multiple views, on the basic of Section M, we conduct clustering individually each view of multi-view datasets mentioned earlier and compare the generated single-view clustering results and multi-view clustering results, shown in Table 16, where VI V6 denote the results based on view ~ 6 respectively and 'Ours' denotes the results based on all views. As seen, multi-view clustering results outperform single-view counterparts with remarkable margins in most cases, which highlights that our DTP-SF-BVE VF VF is able to effectively gather multi-view information for preferable clustering. The reason of some sub-optimal results could be that the quality of certain views relatively poor and disorganize the cluster structure.
O. Time Overhead Proportion
To further dissect the performance of the proposed DTP-SFFBVE, we count the time overhead proportion of each optimization variable, as shown in Fig. 4. From these figures, we can observe that on CALTE7 and Cora datasets, B, and Tp occupy most of the overall optimization time, which is mainly because the number of clusters is slightly larger and accordingly the traversal searching and QCQP searching consume relatively more time than other parts. On REU7200 and Reuters, the time overhead of C and a holds dominant position, possibly because the higher data dimension exacerbates the computing
26

---

From Spectrum-free towards Baseline-view-rre: Double-track Proximity Driven Multi-view Clustering
9.82%
10.93%
18.13%
20.92%
26.42%
33.00%
5.16%
Tp
S,
12.76%
2.12%
5.00%
6.10%
39.73%
14.99%
25.26%
32.78%
(a) CALTE7
(b) Cora
(c) REU7200
7.99%
11.32%
11.28%
23.51%
7.88%
1.59%
6.5.1
1.27%
27.715%
16.26%
15.67%
5.56%
11% 3.80%
57.14%
(d) Reuters
(e)
CIF10Tra4
FasMNI4V
Figure 4. The time consumption ratio of different optimization variables.
7 x10
3.2
1.308
1.306
6.99
3-
1.304
1O555
Lo55
Loss
6.98
2.8
1.302
1.3
2.6
1.298
6.96
1.296
10
7
Iteration
Iteration
Iteration
(a) CALTE7
Cora
(c) REU7200
x107
6.52 X106
8.5 x10
5.69
5.685
6.6
5.68
6.58
LD55
5.675
5.67
6.56
7.5
5.665
5.66
5.655
6.5
10
20
30
Iteration
Iteration
Iteration
(d) Reuters
(c) CIF10Tra4
(f) FasMNI4V
Figure 5. Changes in the loss function.
burden of W. and the coefficient bp. When dealing with CIF10Tra4 and FasMNI4V, the time overhead of updating Tp and is larger than that other variables. Possible reasons are that the cluster number and the feature dimension on
27

---

From Spectrum-free towards Baseline-view-rre: Double-track Proximity Driven Multi-view Clustering
04
0.5
0.4
ACC
0.2
0
10
10
10
10P
10
100
TO!
2
To!
2
1o
2
1oa
10a
10p
10
B
2
21
2-
(a) CALTE7
(b) Cora
(c) REU7200
0.6
03
0.8
ACC
0.4
ACC
02
ACC
0.4
100
TOP
10P
A
1o7
T0?
A
fo7
70P
22
A
107
10a
22
10*
22
1oa
2
10a
2
3
(d) Reuters
(c) CIF10Tra4
(f) FasMNI4V
Figure Sensitivity of the parameters À and 8 in terms of ACC.
0.5
0.08
0.4
0.4
0.06
NMI
NMI
0.04
0.2
0.2
0.02
10
10
101
10
10T
2
10a
101
24
70e
22
10a
2
20
2
B
X
10²
10a
2
2F
B
2-
24
2
(a) CALTE7
(b) Cora
(c) REU7200
02
0.8
NMI
NMI
0.1
NMI
0.4
0.1
0.2
0
10
10
10
100
700
10P
A 70
102
20
22
for
TO2
2
2
A
for
10a
2
24
1pa
24
22
B
T0a
24
2
3
(d) Reuters
(e) CIF10Tra4
FasMNI4V
Figure 7. Sensitivity of the parameters À and 3 in terms of NMT.
these two datasets are relatively larger and accordingly induces much time overhead. Especially, Tp takes the most time expenditures, which is mainly due the searching on a set of one-hot vectors. Although the time consumption proportion
28

---

From Spectrum-free towards Baseline-view-fre: Double-track Proximity Driven Multi-view Clustering
between optimization variables is diverse in different cases, combined with Fig. we have that the overall time consumption of our DDP-SF-BVF is competitive.
P. Convergence
In addition to owing linear complexities, our DTP-SF-BVF also convergent. To demonstrate this point, we plot the changes in loss function with respective to the number of iterations, as shown Fig. 5. One can observe that the function loss is monotonically reducing after iterations and gradually reaches to a steady state, which gives evidence that the proposed DTP-SF-BV is convergent.
Q. Sensitivity
In our DTP-SF-BVE method, there involve hyper-parameters À and 3. We conduct fine tuning for them in aaper-parametrss À and B, ,10] and [2 2. ,21 respectively. To investigate the sensitivity easy to see that with given we plot the clustering results under each parameter combination, as shown Fig. and 3, the clustering results are not dramatically changed in most cases. So, we can conclude that the proposed DTP-SF-BVF not fairly sensitive to A. Moreover, combined with Table 1, we have that within broad range of parameters, the generated clustering results are still comparable. Thus, we can summarize that the proposed DTP-SF-BVE is somewhat robust to hyper-parameters.
0.4
0.4
Fscore
Fscore
Fscore
0.5
0.2
0.2
10
10
10
10P
109
10P
td"
21
10"
21
to
2
10a
20
102
20
102
2
10
2
to
2
B
103
2
24
24
2
(a) CALTE7
(b) Cora
(c) REU7200
Fscore
0.4
Fscore
0.2
Fscore
0.4
0.1
0.2
10
1011
100
10P
A
107
102
22
A
107
102
2
A
1o!
102
2
2a
10
22
10a
10
2
3
(d) Reuters
(c) CIF10Tra4
(f) FasMNI4V
Figure Sensitivity of the parameters À and in terms of Fscore.
R. Potential Improvement Directions
In this work, generate anchors via learning strategy, nevertheless, do not explicitly consider the spatial distribution of anchors. Given the fact that the role of anchors aims at approximately characterizing the overall samples, generating the anchors that are with similar distributions to original data could further enhance the clustering performance. Besides, it needs to perform searching on one-hot vectors when updating the permutation model, which could bring additional computing overhead, and thus designing other talented solutions will further accelerate its running speed.
In addition to these, the tensor Schatten p-norm is usually regarded as good means exploit the complementary information
29

---

From Spectrum-free towards Baseline-view-rre: Double-track Proximity Driven Multi-view Clustering
between views. Our model present dose not involve the Schatten p-norm, and adopts the shared cluster indicator matrix capture view complementary information. Including the Schatten p-norm could help further enhance the clustering ability of our model in the future.
30


Reviews:
[{"review_id": "Osb9jkSCUa", "reviewer": "Reviewer_ywBk", "summary": "The paper builds up double-track proximity for multi-view clustering to investigate the manifold structure among samples. In particular, it encodes anchor-anchor relation into anchor-sample similarity using self-expression learning and topology learning concurrently.  It relieves the restriction of baseline-view by assigning a matrix variable consisting of 0 and 1 to each view, and transforms anchors generated on each view via joint-learning to reshape them. Beyond these, a binary strategy is adopted to take shape the cluster indicator, and meanwhile the cluster indicator connects all views and the anchors on them.  A six-step optimization scheme with linear complexity effectively minimizes the loss 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:** The parameter searching may undermine the practical capability.\n\n**A1:** Thanks! $\\lambda$ governs the trade-off  between error reconstruction and anchor self-expression, while $\\beta$ adjusts the cross-view consistency guidance. They collaboratively modulate the model's capacity. Despite the different searching ranges, our model, as evidenced by the results in Table 1, with the suggested ranges, achieves more competitive data clustering under multiple datasets with diverse scales, which reveals that our model is suitable for various scenes.\n\nThe parameter searching indeed compromises practical applicability to some extent,  particularly in resource-constrained scenarios. Developing a dynamic parameter derivation mechanism based on data characteristics will be a compelling research direction, and we will conduct systematic investigations in future work. \n\n**Q2:** The geometric properties of feasible region about permutation should be systematically characterized. \n\n**A2:** Its feasible region consists of discrete elements, 0 and 1. There is only one 1-element in every column and row. These guarantee that the permutation only rearranges anchors in their original space, and does not alter the values of learned anchors.\n\nBesides, its objective is $\\operatorname{Tr}\\left(\\mathbf{T} _p^\\top \\mathbf{G} _p \\mathbf{T} _p \\hat{\\mathbf{A}} _p + \\mathbf{T} _p^{\\top} \\hat{\\mathbf{B}} _p\\right)$ where \n$\\hat{\\mathbf{A}} _p=\\lambda\\mathbf{H} _{p}+\\boldsymbol{\\alpha} _p^2\\mathbf{M} _p-2\\lambda\\mathbf{S} _{p}^{\\top}$ and$\\hat{\\mathbf{B}} _p=-2\\boldsymbol{\\alpha} _p^2 \\mathbf{J} _{p}$. It exhibits quadratic operation about permutation elements and inherent non-separability, which causes difficulties in optimizing. Note that the permutation is with the size of $m\\times m$, we can utilize one-hot vectors to constitute it. \n\n**Q3:** A fundamental exploration for baseline-view-free mechanism.\n\n**A3:** This approach rearranges anchors without requiring any baseline views. Given that the misalignment arises from the differing order of anchors, we link each view to a learnable permutation, which allows for the flexible transformation of anchors within the original space. Moreover, it is compatible with anchors in an unified framework, and collaborates with their learning process. Selecting a baseline view introduces complicated solving procedure. An improperly chosen baseline view will also lead to inaccurate graph structure fusion. In contrast, we do not rely on any baseline views and automatically rearranges anchors. Besides, it is also proven to be with linear complexity, and hence does not harm the overall efficiency. \n\n**Q4:** Do the cluster indicators affect the overall efficiency? How to mitigate these factors? \n\n**A4**: Combined with Eq.(13), it mainly involves $\\mathbf{W}$ and $\\mathbf{Z}$. So, updating the cluster indicators will take $\\mathcal{O}(d _pmk+d _pk^2+km^2+k^2m+nd _pm+nm^2+nmk)$ cost. Generally, $m$ and $k$ are largely smaller than $n$. $d_p$ is a constant and irrelevant to $n$. As a result, the cost on cluster indicators is linear to $n$. So, it does not affect the overall efficiency. \n\nBesides, the coefficients related to $n$ are $m$, $k$ and $d_p$, while $k$ is an inherent characteristic of datasets, and $m$ is set equal to $k$ in experiments. So, we can decrease the feature dimension to further improve the efficiency. \n\n**Q5:** What is the reason for no-spectrum strategy spending less time than traditional scheme? \n\n**A5:** The spectrum generation process and subsequent embedding grouping result in extra computational expense.\nBesides, we utilize binary learning to directly formulate cluster labels. This only involves the element comparing between two small-sized matrices $\\mathbf{W}$ and $\\mathbf{Z}$.\n\n**Q6:** How is the performance under anchors with the same dimension.\n\n**A6:** Thanks! Via projection, we indeed can make all anchors have the same dimensions (SD). However, even with the same dimensions, due to anchors still being  generated on each individual view, the anchor order across views could remain inconsistent. So, the spatial misalignment still exists. The following table summaries the comparisons. DD is the results under anchors with original diverse dimensions. \n\n|Dataset|DERMATO|CALTE7|Cora|REU7200|Reuters|CIF10Tra4|FasMNI4V|\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|SD|74.51|65.89|47.43|24.13|52.84|23.14|54.92|\n|DD|**85.47**|**80.66**|**52.44**|**26.22**|**54.26**|**26.83**|**57.36**|\n|NMI||||||||\n|SD|76.82|37.64|38.62|5.24|28.61|12.73|56.94|\n|DD|**89.97**|**45.25**|**43.70**|**6.25**|**31.87**|**15.64**|**59.21**|\n|Fscore||||||||\n|SD|75.34|66.82|35.12|25.63|41.63|18.83|45.62|\n|DD|**87.92**|**78.12**|**41.12**|**28.55**|**44.84**|**20.64**|**51.37**|\n\nThe paradigm using anchors with SD produces inferior results, the reasons of which could be that the projecting operation causes information loss and degenerates multi-view diversity, weakening the performance.", "claims_and_evidence": "The main claims are that the anchor-sample plus anchor-anchor, alignment without reference, and no-spectrum contribute to producing more distinctive clusters. They are reflected by the comparative experimental results in Table 1 and multiple ablation results respectively. The running time comparison Figure 2 supports its linear time complexity.", "methods_and_evaluation_criteria": "The proposed method displays the effectiveness in dealing with MVC problems, and demonstrates advantages against 17 clustering methods.", "theoretical_claims": "Yes", "experimental_designs_or_analyses": "The designs are valid, and provide evidence for verifying the effectiveness of the presented method, together with thorough analyses.", "relation_to_broader_scientific_literature": "The idea of embedding anchor-anchor relation into anchor-sample similarity using point-to-point guidance after reshaping is worthy of recommendation in multi-view clustering literature.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "-  The overall structure of this paper is cohesive, enabling readers to easily trace the progression. \n\n-   Extensive comparing experiments, detailed discussions, and rounded ablations reinforce the contributions.  \n \n\n**Weaknesses:**  \n\n-  The demand about the parameter searching may undermine the practical capability.  There are two parameters requiring manual tuning. Moreover, according to the guideline in Section Q, their suggested ranges are also different. This may bring inadaptability for the model when encountering certain scenes with vague clusters. \n\n\n-  The permutation learning procedure necessitates additional methodological clarification, as the determination of $\\mathbf{T}_p$ relies on  traversal exploration according to the feasible region. Hence, the geometric properties of feasible region should be systematically characterized. \n\n-  The baseline-view-free approach achieves  enhanced performance metrics compared to conventional approaches. Although empirical evidence currently supports this observation, a fundamental exploration of underlying mechanisms is warranted to strengthen the validity of this advantage.", "other_comments_or_suggestions": "**Grammars:**\n\n-  In line 198, ‘This spectrum-free model directly output ….’---> outputs;\n\n-  In line 200, ‘$\\boldsymbol{\\alpha}$ play…’---> plays;", "questions_for_authors": "1. About the complexity, combined with the theoretical expressions, it seems that the computing of cluster indicators takes more time consumption. Does it affect the overall efficiency?  Is there any other way to mitigate this factor?\n\n2. What is the reason for no-spectrum strategy spending less time consumption than traditional scheme? \n\n3. In conjunction with the statement ‘one can project all anchors into a common space to make them have the same dimension’, by projecting raw data, we will own the anchors with the same dimension. Does the spatial misalignment emerge between anchors in this case? Furthermore, would this yield enhanced performance compared to the proposed DTP-SF-BVF method?", "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": "M78tUK20AG", "reviewer": "Reviewer_5v1e", "summary": "This paper develops a multi-view clustering algorithm named DTP-SF-BVF to address the problems: (1) current methods usually focus only on the anchor-sample proximity and fail to take into account the anchor-anchor relationship; (2) they require to select the baseline view; (3) existing spectrum paradigm induces clustering variance. DTP-SF-BVF leverages double-track proximity to extract both anchor-anchor and anchor-sample characteristics hidden into data, and devises a permutation mechanism for each view, eliminating the need for selecting the baseline view. Further, it chooses to directly formulate the label indicators through a consensus structure, and in turn the consensus structure bridges anchors on all views.  Extensive experiments validate DTP-SF-BVF’s effectiveness.", "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": "**Q1:** Can the binary be further refined? How is the performance under orthogonality?\n\n**A1:** Thanks. Orthogonal constraints (OC) could deteriorate semantic topological continuity and limit the model's expression ability. Moreover, they will change the value and distribution of anchors. The following experiments further illustrate the distinction. BE is the results based on binary elements.\n\n|Dataset|DERMATO|CALTE7|Cora|REU7200|Reuters|CIF10Tra4|FasMNI4V|\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|OC|83.46|78.46|52.37|**26.31**|52.79|27.12|55.42|\n|BE|**85.47**|**80.66**|**52.44**|26.22|**54.26**|**26.83**|**57.36**|\n|NMI||||||||\n|OC|87.13|43.62|41.42|6.14|**31.91**|14.38|55.23|\n|BE|**89.97**|**45.25**|**43.70**|**6.25**|31.87|**15.64**|**59.21**|\n|Fscore||||||||\n|OC|83.26|73.59|40.16|26.97|42.78|18.94|50.67|\n|BE|**87.92**|**78.12**|**41.12**|**28.55**|**44.84**|**20.64**|**51.37**|\n\nBE outperforms OC in most cases, the reasons of which are that BE does not change anchor characteristics, and only rearrange them within the original space.\n\n**Q2:** Somewhat sensitive to hyperparameters. Exploring potential implications of anchor noise.\n\n**A2:** The parameter $\\lambda$ is responsible for striking a balance between the reconstruction loss and the regularization of anchor self-expression, whereas $\\beta$ modulates the degree of inter-view consistency. Inaccurate calibration may result in diminished performance or instability, particularly in the presence of noisy anchors.\n\nAnchor noise: (1) It could potentially compel the model to assimilate the noise patterns, precipitating an over-fitting condition; (2) It could potentially render the loss surface irregular, complicating the optimization process; (3) It could predispose the model to gravitate towards spurious correlations.\n\nSeveral potential strategies could be employed: (1) Implement a pre-filtering mechanism or confidence-based thresholds; (2) Develop a multi-anchor voting mechanism; (3) Integrate prior knowledge to dynamically adjust hyper-parameters; (4) Conduct dataset pre-processing to eliminate instances of anchor noise.\n\n**Q3:** How does self-expression improve the anchor quality? Still functional under anchors with other means?\n\n**A3:** Self-expression learning helps extract the geometric characteristics between anchors, and facilities the learning of anchors owing to the co-optimized mechanism. OSE-L and WSE-L are to validate the effectiveness of self-expression.  OSE-L and WSE-L are the results without/with self-expression under no learning.\n\n|||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|OSE-L|60.42|43.88|27.96|14.33|25.32|19.73|41.27|\n|WSE-L|**65.64**|**64.59**|**30.24**|**16.68**|**27.20**|**24.08**|**47.21**|\n|NMI||||||||\n|OSE-L|64.37|35.68|5.88|1.01|1.38|12.57|44.79|\n|WSE-L|**69.84**|**37.95**|**33.54**|**1.06**|**1.43**|**12.98**|**47.07**|\n|Fscore||||||||\n|OSE-L|63.76|48.43|28.79|23.42|33.87|16.86|37.64|\n|WSE-L|**69.33**|**61.54**|**30.40**|**24.43**|**35.25**|**18.03**|**41.43**|\n\nThe following WSE+L and WSE-L are to show the effectiveness of learning.'+L' denotes having learning. \n\n|||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|OSE+L|71.51|49.05|30.35|16.75|47.05|26.69|52.15|\n|WSE+L|**85.47**|**80.66**|**52.44**|**26.22**|**54.26**|**26.83**|**57.36**|\n|NMI||||||||\n|OSE+L|83.97|40.21|6.02|2.53|23.19|15.48|58.13|\n|WSE+L|**89.97**|**45.25**|**43.70**|**6.25**|**31.87**|**15.64**|**59.21**|\n|Fscore||||||||\n|OSE+L|73.79|51.25|30.42 |28.54|43.04|17.70|46.77|\n|WSE+L|**87.92**|**78.12**|**41.12**|**28.55**|**44.84**|**20.64**|**51.37**|\n\nBesides, OSE-L and WSE-L show that under the anchors constructed by sampling, our mechanism is still functional. \n\n**Q4:** How does the model perform under varying anchor quality and graph level? \n\n**A4:** In the paper, we introduce a view-aware weighting to harmonize anchor significance on each  view.  OVA denotes the results without view-aware weighting. \n\n|||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|OVA|84.59|76.73|44.94|16.82|47.84|25.26|52.69|\n|NMI||||||||\n|OVA|89.44|42.88|33.06|3.60|29.54|15.20|56.84|\n|Fscore||||||||\n|OVA|86.59|71.80|35.36|**28.77**|42.32|18.03|46.90|\n\nPerhaps, an instance-aware adaptive weighting paradigm  could be more advisable. Implementing anchor-wise importance calibration may yield more superior results  since anchors on the same one view also could exhibit diverse importance.  \n\nAbout the performance at the graph similarity level, we first generate view-specific graph and then construct fusion (FGTC) to integrating complementarities .\n\n|||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|FGTC|80.17|73.98|48.21|22.12|54.13|22.79|53.89|\n|NMI||||||||\n|FGTC|82.74|38.97|39.26|5.83|27.89|11.79|58.93|\n|Fscore||||||||\n|FGTC|80.87|70.46|33.72|21.63|42.87|18.84|47.73|\n\nEvidently, our model gathering information at the cluster-level receives better results in most cases.", "claims_and_evidence": "Yes, the motivation of double-track proximity, baseline-view-free, and spectrum-free paradigm is clearly claimed.", "methods_and_evaluation_criteria": "Yes, it is suitable for multi-view clustering tasks.", "theoretical_claims": "The solving procedure is presented detailly.", "experimental_designs_or_analyses": "I checked the experimental results and discussions.", "relation_to_broader_scientific_literature": "This paper constructs a multi-view clustering method that contains both anchor-sample and anchor-anchor proximity and is without involving spectrum and baseline-view. It considers anchor-anchor characteristics and directly outputs clustering results without variance.", "essential_references_not_discussed": "The references are discussed sufficiently.", "other_strengths_and_weaknesses": "**Strengths:**\n1. The idea is well-motivated.  The double-track proximity paradigm effectively enhances the clustering accuracy via utilizing a more comprehensive view of data representation.  \n2. Baseline-view-free and spectrum-free schemes highlight the flexibility and reliability of DTP-SF-BVF for clustering data from diverse sources. \n3. The organized experiments are sufficient.\n\n**Weaknesses:**\n1.  The binary elements inherent in the alignment model introduce complexities. Consequently, the question arises as to whether this model can be further refined. Specially, the orthogonality possesses the capability to rearrange anchors whilst preserving the irrelevance.  So under such circumstances, does the model's performance witness an enhancement?\n2. The performance appears to be somewhat sensitive to the fine-tuning of hyperparameters, such as on CALTE7. It would be valuable if the authors could delve deeper into the potential implications of anchor noise and provide additional insights to mitigate its influence.  \n3. A central concept in this paper involves leveraging self-expression to derive anchor proximity and thereby boost the clustering outcomes. However, how does this contribute to improving the anchor quality? Moreover, if constructing anchors through other means, whether this remains functional?  \n4. Although the model incorporates anchor relations, it appears to fall short in capturing deeper cross-view complementarities. When encountering the scenarios where anchor quality exhibits substantial variability across different views, how does the model perform? When integrating complementarities at the graph similarity level, how is the performance?", "other_comments_or_suggestions": "No other comments or suggestions.", "questions_for_authors": "Please see the weaknesses.", "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": "awFcvIOl1S", "reviewer": "Reviewer_Les7", "summary": "In this paper, the authors concentrate on three key issues in multi-view clustering field:  the neglect of anchor-anchor geometric proximity, the reliance on the baseline view for anchor alignment, and the instability caused by spectrum. Firstly, the authors adopt a self-expression subspace skill to explicitly exploit anchor characteristics and feed them into similarity graph via topology learning to explore manifold structure inside samples. Then, they introduce a joint permutation mechanism, eliminating the requiring of baseline view and concurrently working with the generation of anchors. Furthermore, they design a consensus discrete structure, which skips the spectrum, to directly produce cluster labels and meanwhile provide a common link to facilitate anchor transformation.", "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": "**Q1:** The introduction about the computational cost of permutation seems a bit concise.\n\n**A1:** More details are provided here.  Due to $\\mathbf{A}_p\\in{d_p\\times m}$, $\\mathbf{S}_p\\in{m\\times m}$, $\\mathbf{B}_p\\in{m\\times k}$, $\\mathbf{C}\\in{k\\times n}$ and $\\mathbf{X}_p\\in{d_p\\times n}$, building $\\mathbf{G}_p$, $\\mathbf{H}_p$, $\\mathbf{M}_p$ and $\\mathbf{J}_p$ will require $\\mathcal{O}(d_pm^2)$, $\\mathcal{O}(m^3)$, $\\mathcal{O}(mkn+m^2n)$ and $\\mathcal{O}(d_pmn+mnk+m^2k)$ cost. Due to $\\mathbf{T}_p\\in{m\\times m}$ consisting of 0 and 1, conducting traversal searching on one-hot vectors will take $\\mathcal{O}(m!)$.  So, updating the permutation takes $\\mathcal{O}(d_pm^2+d_pmn+m^3+mkn+m^2n+m!)$.\n\n**Q2:** In Algo 1, the stopping condition is missing.\n\n**A2:** Thanks! We run the Algo 1 under $f(t)-f(t+1)<=10^{-3}f(t)$. $f(t)$ is the objective value at $t$-th iteration.\n \n**Q3:** Employing a table format to introduce datasets.\n\n**A3:** Good advice! We will present that in a table format to enhance its visual appeal.\n\n**Q4:** Under consensus anchors, how is the performance? Is the permutation necessary?\n\n**A4:** The paradigm based on consensus anchors extract a set of common anchors rather than multiple sets of view-specific anchors to build similarity. Although avoiding alignment, due to the lack of view-unique characteristics, this could not extract sufficient diverse features. To further demonstrate this, we organize  experiments. CA and VA are the results based on consensus and view-related anchors respectively.\n\n|Dataset|DERMATO|CALTE7|Cora|REU7200|Reuters|CIF10Tra4|FasMNI4V|\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|CA|81.23|71.33|49.73|24.97|48.46|22.36|53.38|\n|VA|**85.47**|**80.66**|**52.44**|**26.22**|**54.26**|**26.83**|**57.36**|\n|NMI||||||||\n|CA|82.76|42.26|39.87|6.03|29.89|13.43|51.97|\n|VA|**89.97**|**45.25**|**43.70**|**6.25**|**31.87**|**15.64**|**59.21**|\n|Fscore||||||||\n|CA|80.64|69.72|**42.28**|25.21|**46.13**|19.58|**52.21**|\n|VA|**87.92**|**78.12**|41.12|**28.55**|44.84|**20.64**|51.37|\n \nThe reason of this could be that the view-exclusive representation that view-related (aligned) anchors contain outweighs the view-common information that consensus anchors contain.\n\n**Q5:** From Eq.(8) to Eq.(9), why does it hold?\n \n**A5:** For the objective, we have\n$\\operatorname{Tr} \\left(\\mathbf{B} _p^{\\top}\\mathbf{H}_p\\mathbf{B} _p \\mathbf{C}\\mathbf{C}^{\\top} \\right)  \\Leftrightarrow [\\mathbf{B} _p^{\\top}] _{j,:} [\\mathbf{H}_p\\mathbf{B} _p \\mathbf{C}\\mathbf{C}^{\\top}] _{:,j}  \\Leftrightarrow  [\\mathbf{B} _p^{\\top}] _{j,:}\\mathbf{H}_p\\mathbf{B} _p[\\mathbf{C} \\mathbf{C}^{\\top}] _{:,j}$ and $\\operatorname{Tr}\\left(\\boldsymbol{\\alpha} _p^2\\mathbf{C}\\mathbf{X} _p^{\\top}\\mathbf{A} _p \\mathbf{T} _p \\mathbf{B} _p\\right) \\Leftrightarrow \\left[\\boldsymbol{\\alpha} _p^2\\mathbf{C}\\mathbf{X} _p^{\\top}\\mathbf{A} _p\\mathbf{T} _p\\right] _{j,:}[\\mathbf{B} _p] _{:,j}$ where we omit the $\\min$ operator and $\\mathbf{H}_p=\\beta\\mathbf{L _s}+\\boldsymbol{\\alpha} _p^2\\mathbf{Q} _p$. $\\mathbf{C}\\mathbf{C}^{\\top}$ is diagonal, and we have $[\\mathbf{B}_p^{\\top}] _{j,:}\\mathbf{H}_p\\mathbf{B} _p[\\mathbf{C}\\mathbf{C}^{\\top}] _{:,j} \\Leftrightarrow [\\mathbf{B} _p] _{:,j}^{\\top}\\sum _{i=1}^n \\mathbf{C} _{j,i}\\mathbf{H}_p[\\mathbf{B} _p] _{:,j}$. \nFor the feasible region, $\\mathbf{B} _p^{\\top}\\mathbf{B}_p=\\mathbf{I} _k$ can be divided into \n$[\\mathbf{B} _p] _{:,j}^{\\top}[\\mathbf{B} _p] _{:,j}=1$ and $[\\mathbf{B} _p] _{:,j}^{\\top}[\\mathbf{B} _p] _{:,i}=0,i=1,2, \\cdots,k,i\\neq j,j=1,2,\\cdots,k$. Further, $[\\mathbf{B} _p] _{:,j}^{\\top}[\\mathbf{B} _p] _{:,j}=1$ can be written as $[\\mathbf{B} _p] _{:,j}^{\\top}\\mathbf{I} _{m\\times m}[\\mathbf{B} _p] _{:,j}-1=0$. The $[\\mathbf{B} _p] _{:,j}^{\\top}[\\mathbf{B} _p] _{:,i}=0,i=1,2,\\cdots,k,i\\neq j$ can be written as $\\left[[\\mathbf{B} _p] _{:,1},[\\mathbf{B} _p] _{:,2},\\cdots,[\\mathbf{B} _p] _{:,j-1},[\\mathbf{B} _p] _{:,j+1},\\cdots,[\\mathbf{B} _p] _{:,k}\\right]^{\\top}[\\mathbf{B} _p] _{:,j}=\\mathbf{0} _{(k-1)\\times 1}$.\n\n**Q6:** How to set $m$?\n\n**A6:** We set it equal to the number of clusters.\nDuring updating $\\mathbf{T}_p$, the objective form is $\\operatorname{Tr}\\left(\\mathbf{T} _p^\\top\\mathbf{B}\\mathbf{T} _p\\mathbf{C}+\\mathbf{T} _p^{\\top}\\mathbf{D}\\right)$. Besides, the feasible region is discrete. These  cause it being hard to solve. To this end, we adopt the traversal searching on one-hot vectors to obtain the optimal solution. This takes $\\mathcal{O}(m!)$ computing cost. Too large $m$ will induce intensive time.\n\n**Q7:** Why taking less time than CS? Possible reasons?\n\n**A7:** CS needs to first form spectrum and then conduct embedding partitioning. These two procedures induce extra computing cost. On the other hand, we directly generate labels via binary learning. The labels can be get in a closed-form solution. It only compares the diagonal elements of $\\mathbf{W}$ and the row elements of $\\mathbf{Z}$. The element scale is $k$, which is small-sized.", "claims_and_evidence": "Yes. The claims in this paper are verified by experiments and discussions.", "methods_and_evaluation_criteria": "Yes, It makes sense.", "theoretical_claims": "The Appendix provides the update rules and derivations for variables.", "experimental_designs_or_analyses": "Yes. The experimental designs follows the commonly-used settings and the results are also reliable.", "relation_to_broader_scientific_literature": "This paper integrates the proximity between anchors into anchor graph to more adequately   characterize the manifold structure inside samples, and designs a spectrum-free multi-view clustering paradigm without requiring baseline-view.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n\n(1) The analysis about experimental results is in-depth. \n\n(2) The linear time and space complexities of designed model guarantee its practicality. \n\n(3) The review about existing literatures is comprehensive. \n\n\nWeaknesses:\n\n(1) During optimizing the permutation variable on each view, the introduction about its computational cost seems a bit concise, specially, the searching on one-hot vectors. \n\n(2) In Algorithm 1, the stopping condition is missing.  It would be helpful for authors to explicitly describe it. \n\n(3) To improve the accessibility and visual appeal of the dataset introduction, employing a table format would be highly advantageous.", "other_comments_or_suggestions": "See the following questions.", "questions_for_authors": "(1) Rather than employing view-related anchors to extract data features, when adopting consensus anchors, the misalignment could not exist since all views share a common group of anchors. Under this situation, how is the performance? Is the permutation necessary?\n\n(2) From Eq.(8) to Eq.(9), why does this transformation hold? \n\n(3) How to determine/set the value of $m$? The parameter $m$ may affect the clustering performance since it leads to graphs with diverse scales.  \n\n(4)  According to the ablation results about the spectrum-free strategy, one can obtain that it takes less time cost than CS. why? what are possible reasons for this phenomenon?", "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": "NGAKZQHUFw", "reviewer": "Reviewer_Mxef", "summary": "This work proposes a spectrum-free and baseline-view-free multi-view clustering method with double-track proximity, DTP-SF-BVF.  It aims at improving the clustering stability and alignment flexibility as well as the anchor itself characteristic exploration. Unlike current methods that usually overlook the proximity relationship between anchors, this work utilizes self-expression learning and point-point topology learning to capture that. To get rid of the limitation in baseline view, this work designs a learnable permutation strategy to jointly reorder anchors according to respective view characteristics. To alleviate the variance impact, this work avoids the formulating of spectrum. Experiments on multiple benchmark datasets demonstrate the effectiveness of the presented DTP-SF-BVF.", "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:** Final steps for deriving clustering and variable initialization.\n\n**A1:** After getting $\\mathbf{C}$, we derive the clustering by sequentially identifying the row numbers where the element 1 is located. We initialize $\\mathbf{A}_p$, $\\mathbf{T}_p$, $\\mathbf{B}_p$ and $\\boldsymbol{\\alpha}$ with random matrix, unit matrix, orthogonal matrix and $1/v$. For $\\mathbf{C}$, we create a zero matrix and then randomly assign a single 1 to each column. For $\\mathbf{S}_p$, we assign elements ranging from 0 to 1 column by column and ensure that the diagonal elements remain 0 and the sum of each column equals 1.\n\n**Q2:** Explanations for sub-optimal results.\n\n**A2:** Thanks! MSCIAS achieves a marginally superior Fscore (0.72\\%) on Cora, possibly because of the introduction of HSIC and the employment of local connectivity. FPMVS achieves a 0.29\\% accuracy gain on CIF10Tra4 owing to the adoption of orthogonal projection ensembles. SFMC integrates a structural coherence and self-supervised attention. MFLVC introduces hierarchical feature learning and consensus semantics.\n\n**Q3:** Why not learn a unified anchor graph? Performance.\n\n**A3:** Anchor graphs on respective views may be more conducive to expressing the characteristics of their own views. A single graph structure may not adequately capture the intrinsic characteristics across all views. To confirm this, we conduct relevant experiments. IAG and UAG are the results based on individual and unified anchor graph respectively.\n\n|Dataset|DERMATO|CALTE7|Cora|REU7200|Reuters|CIF10Tra4|FasMNI4V|\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|UAG|83.58|74.38|49.43|25.44|51.81|23.27|53.31|\n|IAG|**85.47**|**80.66**|**52.44**|**26.22**|**54.26**|**26.83**|**57.36**|\n|NMI||||||||\n|UAG|85.37|43.92|42.07|6.17|29.74|13.93|54.37|\n|IAG|**89.97**|**45.25**|**43.70**|**6.25**|**31.87**|**15.64**|**59.21**|\n|Fscore||||||||\n|UAG|83.72|70.43|**42.63**|25.79|**46.83**|19.18|**52.93**|\n|IAG|**87.92**|**78.12**|41.12|**28.55**|44.84|**20.64**|51.37|\n\n**Q4:** Interpretation about the clustering outputs and the practical significance.\n\n**A4:** Our model introduces a alignment mechanism without the necessity of selecting a baseline view. It is adept at synchronizing with the anchor generation, and adeptly rearranges anchors within their original space, well preserving the data diversity. Our model  incorporates the geometric properties among anchors into the anchor-sample similarity, more thoroughly uncovering the manifold structure inherent into samples. Besides, our model engages in the direct learning of consensus cluster indicators, consolidating  multi-view information at the cluster-label level. Further, our model is equipped with linear complexity. Owing to these, our model yields stable results and is adept at handling large-scale scenes.\n\n**Q5:**  Restriction factors for the proposed method.\n\n**A5:** The computing cost is $\\mathcal{O}(m^2nv+dnm+m!v+m^3kv)$. In general, $m$, $v$ and $k$ are greatly smaller than $n$. $d$ is a constant and irrelevant to $n$. The computing cost is linear with respect to $n$. Compared to the square and cube, the factorial about $m$ needs more cost. Too large $m$ will induce expensive cost. In addition, the space overhead is $\\mathcal{O}(nk)$. So, the proposed method is not limited by its space overhead. Further, combined with the sensitivity study, the performance is relatively robust to hyper-parameters. So, the method is mainly influenced by the anchor number.\n\n**Q6:** Exposition on the alignment criticality.\n\n**A6:** The works based on shared features (SF) typically derive  consensus anchors rather than view-tailored  anchors to establish similarity. This harnesses the complementary. The following  experiments further show this.\n\n|||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|SF|81.23|71.33|49.73|24.97|48.46|22.36|53.38|\n|NMI||||||||\n|SF|82.76|42.26|39.87|6.03|29.89|13.43|51.97|\n|Fscore||||||||\n|SF|80.64|69.72|**42.28**|25.21|**46.13**|19.58|**52.21**|\n\nIn our model, the alignment mechanism builds pure self-expression affinities. If not aligning, the structure of  anchor-anchor affinity would be chaotic, which will in turn impair  the anchor-sample proximity, hindering the clustering performance. The following experiments validate this point. WOA is the results without our alignment. \n\n|||||||||\n|:---:|:---:|:---:|:---:|:---:|:---:|:---:|:---:|\n|ACC||||||||\n|WOA|80.73|76.59|31.65|16.67|45.29|25.91|53.68|\n|NMI||||||||\n|WOA|82.53|39.55|35.41|3.32|24.77|15.30|56.47|\n|Fscore||||||||\n|WOA|79.47|72.23|30.69|21.14|42.59|17.90|47.41|\n\nIn cases where the consistent information predominates over the complementary, it may be feasible to establish unified anchors to formulate similarity and thus potentially circumvent or mitigate the need for alignment. Nonetheless, given the inherent complexity of multi-view data, accurately quantifying the complementary and consistent information is usually a challenging task.", "claims_and_evidence": "The claims are clear.", "methods_and_evaluation_criteria": "The proposed DTP-SF-BVF method can effectively handle the multi-view clustering problem and be applied to large-scale scenes.", "theoretical_claims": "I have checked the derivation procedure.", "experimental_designs_or_analyses": "I have checked the soundness of experimental designs.", "relation_to_broader_scientific_literature": "The work proposes an interesting multi-view clustering, which is with spectrum-free and baseline-view-free as well as double-track proximity properties.  Besides these, it is with linear time complexity and linear space complexity.", "essential_references_not_discussed": "None", "other_strengths_and_weaknesses": "Its strengths are as follows, \n\n-  The writing is easy to follow, and the organization is clear.\n\n-  The solving procedure presented in the paper is detailed.  \n\n-  Experiments conducted are sufficient and reveal the effectiveness of proposed method from multiple perspectives. \n\n\nIts weaknesses are as follows, \n\n-   Clearly outlining the final steps for deriving clustering results and specifying the initialization settings of relevant variables are crucial for ensuring a thorough understanding of the algorithm's workflow. \n\n-   Offering theoretical explanations for the sub-optimal results generated would significantly deepen the understanding of the algorithm's underlying mechanisms and limitations.", "other_comments_or_suggestions": "None", "questions_for_authors": "-   The current methodology involves constructing individual anchor graphs for each view, while why not directly learn a unified anchor graph matrix? After aligning, could this potentially provide a more superior clustering accuracy? \n\n-   While the experimental results in the paper are impressive, the interpretability of the clustering outputs and the practical significance could benefit from additional exploration. The authors could provide valuable insights by discussing the interpretation of clustering results in real-world applications, highlighting the method's potential for broader impact. \n\n-  When clustering different kinds of datasets, which factors are more prone to restrict the proposed algorithm?  dataset size, feature dimension, or hyper-parameter? \n\n-  While this approach demonstrates potential advantages, it is noteworthy that several recent MVC techniques have effectively circumvented the need for alignment by leveraging shared features. Whether are there particular scenarios where the necessity for such alignment could be reduced? The authors are encouraged to provide a more detailed exposition on the alignment criticality.", "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 give positive scores. The idea is well-motivated. The overall structure and experiments of this paper are sufficient. The review of existing literature is comprehensive.
Decision: Accept (poster)