From Dictionary to Tensor: A Scalable Multi-View Subspace Clustering Framework with Triple Information Enhancement
Authors: Zhibin Gu
Songhe Feng
Venue: NeurIPS 2024 poster (2024)
Keywords: Multi-view clustering
Low-rank tensor representation
Anchor hypergraph Laplacian regularization
Primary area: learning_theory

Abstract: While Tensor-based Multi-view Subspace Clustering (TMSC) has garnered significant attention for its capacity to effectively capture high-order correlations among multiple views, three notable limitations in current TMSC methods necessitate consideration: 1) high computational complexity and reliance on dictionary completeness resulting from using observed data as the dictionary, 2) inaccurate subspace representation stemming from the oversight of local geometric information and 3) under-penalization of noise-related singular values within tensor data caused by treating all singular values equally. To address these limitations, this paper presents a \textbf{S}calable TMSC framework with \textbf{T}riple inf\textbf{O}rmatio\textbf{N} \textbf{E}nhancement (\textbf{STONE}). Notably, an enhanced anchor dictionary learning mechanism has been utilized to recover the low-rank anchor structure, resulting in reduced computational complexity and increased resilience, especially in scenarios with inadequate dictionaries. Additionally, we introduce an anchor hypergraph Laplacian regularizer to preserve the inherent geometry of the data within the subspace representation. Simultaneously, an improved hyperbolic tangent function has been employed as a precise approximation for tensor rank, effectively capturing the significant variations in singular values. Extensive experimentation on a variety of datasets demonstrates that our approach surpasses SOTA methods in both effectiveness and efficiency.

From Dictionary to Tensor: A Scalable Multi- View Subspace Clustering Framework with Triple Information Enhancement
Zhibin Gu Songhe Feng 2:3+
College of Computer and Cyber Security, Hebei Normal University, China
2 Key Laboratory of Big Data & Artificial Intelligence in Transportation (Beijing Jiaotong University), Ministry of Education, China School of Computer Science and Technology, Beijing Jiaotong University, China {guzhibin, shfeng)eb edu. cn
Abstract
While Tensor-based Multi-view Subspace Clustering (TMSC) has garnered sig- nificant attention for its capacity to effectively capture high-order correlations among multiple views, three notable limitations current TMSC methods necessi- tate consideration: 1) high computational complexity and reliance on dictionary completeness resulting from using observed data as the dictionary, 2) inaccurate subspace representation stemming from the oversight of local geometrie informa- noise-related singular values within tensor data
tion and 3) under- pendder-pnalization caused by treating all singular values equally. To address these limitations, this pa- per presents a Scalable TMSC framework with Triple infOrmatioN Enhancement (STONE). Notably, an enhanced anchor dictionary learning mechanism has been utilized to recover the low-rank anchor structure, resulting in reduced computational complexity and increased resilience, especially in scenarios with inadequate die- tionaries. Additionally, we introduce an anchor hypergraph Laplacian regularizer preserve the inherent geometry of the data within the subspace representation. Simultaneously. an improved hyperbolic tangent function has been employed as a precise approximation for tensor rank, effectively capturing the significant varia- tions in singular values. Extensive experiments on variety of datasets show that the STONE outperforms SOTA approaches in both effectiveness and efficiency.
Introduction
Data clustering, fundamental technique within the domains of machine learning and computer vision, aims to partition an unlabeled dataset into discernible subgroups characterized by substantial internal similarity (15). In practical scenarios, objects are frequently characterized by multitude of properties or data originating from various sources 6.D). For instance, medical analysis, imaging data from modalities such as X-ray, CT, and MRI play crucial role in diagnosis and disease monitoring. These diverse features, representing various aspects the same object, collectively constitute multi-view view data. Multi-view clustering (MVC), which endeavors to harness the abundant information inherent in multi-view data to enhance the quality clustering, has emerged as a highly esteemed research avenue [10-12] Existing MVC methods can be broadly categorized into four groups based on the underlying learning mechanisms: matrix factorization-base [3.15) bubspa approaches [16-T8]. graph-based approaches [19]21), and kernel-based
approaches
approaches [22 24 Among these, subspace approaches are highly regarded for their straightforwwrd implementation and excellent results.
"Conesponding author
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Multi-view subspace clustering is oriented towards the incorporation of diverse constraints within subspace representations to acquire a consensus one that conducive to clustering [25 For instance, Cao et al. [26] and Li et al.
30 proposed the utilization of the Hilbert-Schmice independence criterion as a dependency measure, aiming to capture diversity and consistency from multi-view data, respectively. Pan and Kang [28) integrated a contrastive loss regularization into the consensus subspace representation, encouraging the proximity of similar samples and the separation of dissimilar ones. In addition, Huang et al. [32 facilitated the extraction of valuable consensus representation by assuming cross-view view sparsity of inconsistent components in multi-view data. Nevertheless, these methods are confined to investigating only the linear affinity relationships between data pairs within individual views and do not capitalize on the higher-order correlations among data points across multiple views. This limitation results in suboptimal clustering performance. As result, tensor- based multi-view subspace clustering methods (TMSC) have remained the focus of sustained attention in recent years. Typically, these TMSC methods consolidate various subspace representations into 3D tensor, subsequently applying global structural constraints to uncover the complex, nonlinear relationships among data points across different views [9] For example, Xie et al. [4 advanced cross view consistency exploration by employing Tensor Nuclear Norm (TNN) on the rotated tensor. Jia et al. [36] characterized the intra-view and inter-view relationships of data points by applying symmetric low-rank constraints to the frontal slices and structured sparse low-rank constraints to the horizontal slices. Furthermore, Guo et al. [38] and Sun al. [39 introduced the logarithmic Schatten-p norm and the arctan rank norm as compact surrogates for tensor rank, aimed at capturing
distinctive information from tensor singular values.
Despite the noteworthy clustering quality achieved by the TMSC methods described above, there remains considerable potential for further enhancements across four critical dimensions. First, many existing methods exhibit quadratic or even cubic time and space complexities, which restricts their scalability for large-scale datasets. Second, previous techniques have utilized the given feature matrix as the dictionary for subspace recovery. However, this method requires that the feature representations to include a sufficient number of uncontaminated sampled points; otherwise, the resulting subspace representation may not accurately capture the affinity relationships among the data points. Third, traditional approaches often emphasize the low-rank structure of tensor representations to investigate high-order nonlinear correlations among data points actoss various views, while frequently neglecting the intricate local geometric correlations within individual view. Finally, many methods impose equal penalties on the singular values of tensor data, which may lead to excessive penalization of larger singular values while under-penn peddee-en smaller ones, resulting in suboptimal tensor representations. This issue arises from the differing significance of singular values in tensor data, where larger singular values indicate valuable features and smaller ones are often associated with noise.
Drawing from the principles and justifications discussed previously, this paper proposes a Scalable TMSC framework with Triple infOrmatioN Enhancement (STONE). First, STONE employs an enhanced anchor dictionary representation mechanism instead of the traditional self-representation to learn a subspace representation. This approach effectively reduces computational complexity and enhances the stability and robustness of the algorithm in situations where dictionary are insufficient or corrupted. Additionally, we introduce an anchor hypergraph Laplacian regularization to guide the learning of target anchor tensor representation, facilitating the simultaneous utilization of high- order correlations among data points across views and geometric correlations among data points within each view. Furthermore, refined hyperbolic tangent rank is developed as non-convex low-rank regularization for tensor data, enabling the STONE model to effectively distinguish the distinct physical meanings of various singular values. Compared to existing TMSC methods, the contributions of this paper can be outlined as follows:
We introduce an enhanced anchor dictionary representation strategy to recover the anchor subspace representation. mitigating the high computational complexity of self-representattinn methods and improving accuracy under dictionary under-sampling
We develop a refined Hyperbolic Tangent Rank (HTR) as a precise approximation to the tensor rank. In contrast to TNN, HTR allows for variable penalties on individual singular values, facilitating a thorough exploration of differences among different singular values.
We utilize anchor hypergraphs that encode geometric manifold correlation to regularize the target tensor representation, allowing for the simultaneous utilization of high-order correlations across different views and the complex relationships within each view.
2

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We present an iterative optimization algorithm along with analyses of its complexity and convergence. Comprehensive experimental results demonstrate that the STONE model excels in both clustering performance and efficiency.
2 Theoretical Foundation
Let X x=(xx (x1,..... xn) E RRdx denotes a dataset comprising n instances, with each instance repre- sented by d-dimensional feature vector. Low-Rank Representation (LRR) [40] aims to recover subspace representation by employing the feature matrix X as a dictionary, which can be mathemati- cally described as follows:
min Z.E
Z + E.1. s.t. XZ + E,
(1)
where Z E R represents the subspace representation, which regularized with the nuclear norm || ll. to ensure a low-rank structure. The reconstruction error is denoted by E E Rdx and is constrained by the nnom to promote sparsity. The parameter a serves as a balancing factor.
LRR has proven its effectiveness in uncovering the spatial structure of data patterns 41H433 yet hinges on critical requirement: the data matrix X must contain sufficient number of data points sampled from the subspaces. Otherwise, a potential solution to lq.. (1) could be the identity matrix, which hinders the implementation of low-rank representation (LRR). To address this issue, Liu and Yan [44] proposed that, alongside the given data X, there exists a set of unobserved data points Y in the dictionary representation, which acts an ideal supplement X. This strategy known as the latent low-rank representation model (LatLRR), which helps to mitigate the impacts of insufficient and corrupted observational data. Its mathematical definition as follows:
min |Z|l + | 2, s.t. X X; YJZ E E,
where Y E IR represents the unobserved feature representation, which is concatenated with X along the columns form a complete feature representation serving as the dictionary. For practicality, 44 relaxes Eq. (2) into the following nuclear norm minimization problem to approximate the unobserved data and learn an accurate subspace representation:
min 2.P.E
Z | |P|+ +a 2, X
where P E IRd denotes an intermediate result, which is obtained through the skinny SVD theory, which serves as tool for feature extraction 44. Emphasizing our focus on clustering, the subsequent discussion revolves around the subspace representation Z, and the nuclear norm on P will be relaxed to the Frobenius norm a convex surrogate for low-rank constraint that adheres to the block diagonal condition 45.
3
The Proposed Method
3.1 The STONE Model
Consider a dataset containing samples and views, denoted as {X")*, where X" Rd. represents the v-th view feature, and d, indicating the corresponding dimension. The objective of the TMSC method is to organize multiple view-specific subspace representations into 3-D low-rank tensor, with the aim of unveiling higher-order correlation information spanning multiple views. Formally, the general mathematical expression of TMSC is as follows:
min (Z. FE
R + af( E }) + (TZ"))
s.t. Vv. X = X"Z" +E" ",, 2 = (2' (Z.....Z'')
where Z" E R represents the subspace representation of the v-th view, and Z E R is 3-D tensor formed from the collection {Z") ) with & acting as the tensorization operator. R(.)
is used for compact approximation of the tensor rank, while C(-) is tailored to capture noise. T(:)
represents the structured constraint applied to the subspace representation Z" and are two trade-off parameters.
3

---

Although model (4) effectively captures the high-order consisteney of data points across different views, it has two notable limitations regarding its mechanism of using the observed data dictionary for constructing the tensor representation. First, the time and space complexity of Model (4) becomes quadratic or even cubic, which restricts its scalability to large datasets. Second, requires the feature representation matrix X" to contain sufficient number of uncontaminated sampled data points; otherwise, the learned subspace matrix Z" may manifest as the identity matrix, hindering the effectiveness of the LRR method [44).
To overcome these limitations, we introduce the Enhanced Anchor Dictionary (EAD) representation strategy for recovering anchor subspace representations, EAD first selects a set of distinctive samples from the available data to form an anchor dictionary (i.e., A " Rd,xi, I is the number of anchors), enabling the recovery of a subspace representation Z" < Rx that is smaller in size. This approach helps alleviate the issue of high computational complexity. Additionally, inspired by Latl [44), EAD integrates the observed anchors A with the unobserved sampled data Y into a comprehensive dictionary (i.e., [X"; Y effectively mitigating problems arising from sampling of feature characteristics in the anchor dictionary. As a result, we formulate TMVC framework induced by
EAD as follows:
min R(2) + aF((P (C((E +) {2" A P EE)
s.t. Vv, X" = A + P"X E" E,, (A") A" 1I,
(5)
Z = == (Z.,, Z P (P' ,..." )
where Z" Ruxi and P 12d "dd" and presented in tensor forms as E E Rxxmxxn and P Rdd xmxd. F(-) is constraint on P.
xd" denote the anchor subspace and projection matrix, respectively,
Notably, the anchor matrix A E Rdd is subjected to orthogonality constraints to ensure optimal distinguishability. a, B and 7 are three trade-off parameters.
To delve deeper into the valuable information embedded multi-view data and refine the qual- ity of the anchor tensor representation obtained in the Model (5), tailored constraints iintluding Hyperbolic Tangent Rank, Linear Weighted Frobenius norm, and Anchor Hypergraph Laplacian Regularization applied to Z, P, and Z", respectively. These constraints are clearly defined as follows:
Definition 1. For a tensor Z n. X the Hyperbolic Tangent Rank (HTR) is defined as follows:
ny
(0 e (-88)((8)) (8.2)
Z
HTT
113
HTR
T1
(0) +e 8C3 (8.3)
(6)
where 8 8-0, 0, h min(ni n2). 3 denotes the k-th frontal slice of Z and C/jy is the representation of the Fourier domain obtained by the tensor-SVD (i.e., 2 =B
Definition 2. For multiple matrices {P")m their Linearly Weighted Frobenius (LWF) norm is defined as follows:
m
m Σ 5" P |
PP LWW = Σ
1 LWW
(7)
1=1
where 1|1 denotes the Frobenius norm of matrix, and - 52, "..,, represents the weighted coefficient vector, with each weight empirically set to 1.
Definition 3. For the given tensor Z BBxxxxn, its Anchor Hypergraph Laplacian Regularization (AHR) is defined as follows:
Z AHR
IZ IAHHR IT(Z''
where Lý, denotes the anchor hyper-Laplacian matrix constructed based the anchor hypergraph (V, Q, W) (with V, Q and W denoting vertices, hyperedge set, and weights, respectively). Specificully, LE = D (DD Here, D De and W being degree matrices with diagonal elements vertex degrees, hyperedge degrees and hyperedge weights, respectively. R defines vertex-hyperedge relationships, where r" (v,e) e) = if the w-th vertex the e-th hyperedge. otherwise 0 50).

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By unifying Eqs. - (8), we formulate the objective function for the STONE model as follows:
min PM A E.E
2 |TTR + a IWF + 3
B +
7
AHR
s.t. Ve, X AZ") P E"
E E
(9)
) A' - v 7m), P
where E E ;....; E is derived by horizontally concatenating elements along the rows of [E"). In the end, by utilizing the k-means clustering algorithm on the left singular vectors of the connectivity matrix
J vm | zn E gaxim, we achieve the clustering partition results [5T].
Remark 1. [hhy STONE outperforms other self- representation methods?] Unlike previous TMSC methods 38 [2 the STONE model employs the
A E 7
EAD strategy instead of self-representation recover subspace representations, combining the benefits of anchor representation and latent low-rank representa- tion (LatLRR) for the preservation of both accuracy and efficiency. Notably, illustrated in Figure I the efficacy of EAD stems from its thoughtful design: the utilization of the anchor representation enables the EAD model to recover the subspace representation of size n ensuring linear scalability for extensive datasets. Additionally, the introduction of the LatLRR mechanism permits both the observed anchor vectors and the unobserved sampled data function as dictionaries, safeguarding the recovered anchor tensor representation against deficiencies in insufficient dictionaries.
X" c 25
ETe R
X R4
Figure 1: Schematic of Enhanced Anchor Dictionary Representation (EAD).
Remark 2. [Why STONE outperforms other tensor rank methods?] The STONE focuses on using the hy- perbolic tangent tensor rank as a low-rank structural reg- ularization constraint for tensor representation, defined as 6x
f(z) -d 4-0 Since HTR non-convex function with adjustable slopes, can delve into the distinct phys- ical meanings of different singular values in tensor data, thereby enhancing the representation capability of tensors. Analysis of Figure2
a clear superiority of the HTR approximating tensor rank compared TNN [34] and TLS,N [38), particularly for values nearing zero and rel- atively large singular values. Specifically, as æ approaches 0, funR (z) is considerably greater than and log(1 + r²); on the other hand, as z increases, fHIR (x) approaches 1, The STONE method adaptively applies appropriate strong and weak penalties both small and large singular values, preserving valuable information while also demonstrating robustness against noise. Furthermore, when x 0, [(x) = O, which consistent with the true tensor rank.
Figure 2: Tensor Rank Approximation:
HTR vs. TNN and TLS,NN
3.2 Optimization
To tackle the objective function, we start by introducing auxiliary variables S and (Q"), }. which ensure that all variables in Eq. (9) become separable, as follows:
Tr TQQ")L 0 -
S+ y
min (2" P",A'" "..."...QQQH ".A"""""EE.. E.S
|S IITR to
P P
B
2,1
H 22
X" A'
I Q"
P
u=1
(10)
where y, } and {H)} are the Lagrange multipliers, and 6, 61 and 62 signifies the penalty coefficients. Then, the optimization of the STONE objective function can be streamlined into six sub-problems labeled as [Z"}. (A*). [P"]. (Q"). E and S for individual optimization. Given space limitations, the comprehensive optimization procedures and pseudocode are outlined in the A.
of the supplementary materials.
5

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3.3 Convergence Analysis
The validation presented in Theorem T establishes the reliability of the optimization algorithm's convergence, while Appendix A.2 of the supplementary materials offers an in-depth exploration of the underlying details.
Theorem 1.
The sequence generated by the employed optimization algorithm, denoted as G [Z] E, Pr. Ay. H H a Siliz, adheres to the following two fundamental principles:
The set { bounded;
Any accumulation point of the sequence {9r)mm is KKT point of Eq.(
3.4 Complexity Analysis
The computational requirements of STONE are split into two primary areas: optimizing variables and performing clustering. At the outset, the process involves updating several key variables-Ljj Z", E A", Q" S -with their respective time complexities being Ol' log(!), (nnl=+nld") (ndd") ) + ruld)), O'nld" + +1dd",, ( ( log(mn) + nmm²)). ft). In the following phase, the computational complexity is given by O(nlm + ndmax). This indicates direct proportionality to the sample size n. Furthermore, the memory complexity of the STONE model, expressed as O(nlm + ndmax), also maintains linear growth pattern with respect n.
3.5 Comparison with Previous Studies
In recent years, various tensor- based multi-view clustering algorithms, such as MVSC-TLRR 36 TLS,NM-MSC (38), SSG-TAR [3)), NOODLE (53), ASR-ETR [33], and EDISON [54], have been proposed to explore high-order correlations among views by pursuing a global low-rank structure in tensor representations. However, our STONE model significantly differs from these methods. For instance, unlike MVSC-TLRR, NM-MS, SSG-TAR and NOODLE, our STONE model differs by enhancing computational efficiency through the construction of anchor subspace representations rather than relying traditional subspace representations. Moreover, unlike the ASR-ETR, which lowers computational complexity through anchor dictionary representation, our STONE model builds this by using the EAD strategy address challenges related to insufficient data sampling. Additionally, STONE employs anchor hypergraph Laplacian regularization rather than anchor Laplacian regularization in ASR-ETR, which further enhances the accuracy of subspace representations. In contrast to the EDISON, designed for incomplete multi-view data, our approach not only has differences in dictionary representations due to variations in data completeness, but also employs distinct non-convex functions to regularize the singular values of tensor data during the recovery of compact tensor representations.
Experiment
In this section, we present comprehensive experiments to evaluate the performance of the STONE model. Due to space constraints, a portion of the experiments is presented here, with additional experiments detailed in the Appendix A.3 33 of the supplementary materials.
4.1 Experimental Setup
Datasets: For the clustering experiments, we em- ploy eight datasets: NGs, BBCSport, HW, Scene 15, MSRCVI, Caltech101 -al- ALOI- 100, and CIFARIO. More detailed descriptions of these datasets can be found in T
Table 1: Overview of Statistical Features for Eight Datasets.
Datasets
Type
Samples
Clusters
Views
NGs
Text
500
3
BBCSport
Text
544
2
HW
Digit
2000
10
2
Baselines: Ten SOTA methods, including eight
Scene15
Scene
4485
15
3
salllow-based models and two deep learning mod-
MSRCV1
Object
216
-7
5
els: SMVSC (2021) [55], SFMC (2022) [56]. GMC
Caltcch101 all
Object
9144
102
6
(2019) 577. MSC3D (2023) (58). MVCtopl (2022)
ALOI-100
Object
10800
100
CIFARIO
Object
50000
59
MVSCTM (2022) (60), ETLMC (2019) (61.

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Table 2: Clustering Performance Comparison Across Eight Datasets (Mean Standard Deviation).
Viris
S1-ket
SPMC
KSS
ViCCsp
TIGI
N0TNC
GET
SWON
00.
1210121000
052431000
NG6
F18
0. 130ke 636
031 063
0.30n 600
048hw0 503
2500
1766-9100
a 0.00e0 606
0 03000030
6 677600000
08800 001
1.0000000
033551.00
4333
020142.00 000
033583 503
0551
040100 002
12000390
46X
039269 00)
1. 609
09770 103
1433:8100
6:580000
06300 00
BBCSpert
P0R
145
0.3350 005
0.3902000
014400 203
0463U0000
0000
06360 D01
0199950000
0.7118
0.3003
072269 203
030350014
33775000
0.15501800
0 018860000
650220000
0.399 00)
ACE:
0.9
088360 201
5263.0100
0.826 301
NW
PUK
10799001000
032210 00
015331000 50
015300 103
65018000
E3012000
0.220 00
ARI
0-63be1 E30
0333ed 960
01 500
010 Do1
1563-0000
0.7554 D01
N00
(1300w
500
0333
0ER Fon
1100-0160
0.3960
(1788wr 001
n
B. not
277-5100
030 201
All
0025501005
01700 905
0.291100 600
GOV D01
0545a0000
010750 D00
100
(138502.00
037 500
0.211200
034220 203
LJKUU 891
638720000
04Pu0 503
12001300
MISRCVI
PUR
1170001100
418790 301
0SPA 500
02AM0 201
Dassanate
DA0CaCAEO
541522660
0.3770 201
ABL
1445241000
034900 500
03630000 000
0SAND 201
CISTURINO
(3662000)
0.300 D00
NMI
06:35110 060
0 (522m 20
0 010000 1o1
861760100
(1938110
000
072840330
C46120000
ALOI-II
PUR
077140 200
C-TRLDIKD
13355
6.3
030001109
023691 600
1 (10 D00
0 11900 101
014/80000
021550000
0 681
NMA
03000 000
0336 200
0.7860 D01
C185-0100
QM
0M
6:7744000
Calcchol--dl
PUK
0.3033 203
GM
0M
0112200000
0.235 903
n1 017301007 000
1200ml
100
0M
0M
611140000
106
0377343 901
08
CM
QM
0M
00
037540 D01
0199333
0U57748000
0M
CM
CM
OM
0337 903
01977201000
(U77505.000
0M
OM
OM
OM
903
180
ACC
ACC
(a) NGs
(b) HW
(c) MSRCVI
Figure 3: Impact of Parameter ó on the STONE Model.
TBGL (2023) 62
63 GCFAgg (2023) [64] along with spectral clustering with the best view best 665) used for comparison.
MFLVC (2022)
Evaluation Metries: To provide a comprehensive evaluation of clustering quality, we employ five metrics, namely ACC, NMI, PUR, F-score, and ARI. Better clustering quality is indicated by higher values of these metrics.
Implementation Overview: For the comparative methods, the parameters are fine-tuned in ac- cordance with the instructions presented in the respective literature, and the optimal outcomes are reported. For the STONE model, there are five parameters that necessitate adjustment. To be specific, the intrinsic parameter 8 and the number of anchor points e are tuned individually within the ranges [0.1, 0.5, 1, 1.5, 5] and [c. 2c, .... 7c], respectively. The three balancing parameters a, 8, and are finely tuned within the range [le-5,1e-5.... le+1] using a grid search strategy. To maintain rigor, we perform each experiment total of 10 times, and we present both the mean results and the standard deviations for comparison. The experimental procedures for the shallow learning model are implemented using MATLABB 201 8a on a computer featuring 3. 3.70GHz 10-10900k CPU and 64GB RAM. Conversely, the deep learning model experiments are facilitated by PyTorch 1.12, deployed on an RTX 4060 GPU.
4.2 Comparison of Clustering Performance and Efficiency
The clustering performance and computational efficiency of the proposed STONE model are demon- strated separately in this subsection.
Performance Assessment: To validate the effectiveness of our STONE method, we evaluate its clustering performance on eight datasets and compared it with ten SOTA methods across five metrics.
7

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Table 3: Efficiency Comparison of Different Methods Datasets with over 4,000 Samples.
Datasets
SMVSC
SEMC
GMC
MSC'D
MVCtopl
MVSCTM
ETLMSC
TBGL
MELVC
GCEAgg
STONE
Sccne15
19.79
23.91
57.14 14
174.1
131.91
482.22
639.95
1279.9
107.59
145.63
6.52
ALOI-100
1013.9
400.85
Callech101-all
247.29
165.36
398.94
856.3
5704.9
1169.2
OM
OM
1212.5
589.01
285. 11
CIFAR1O
867.26
4251.3
OM
OM
OM
OM
OM
OM
1257.08
1978.88
860.8
(a) NGs
(b) HW
(c) MSRCVI
Figure 4: The Influence of Anchor Quantity on STONE Model Performance.
The results are summarized Table where the highest and second second-highest values are marked with bold and underlined, respectively. The acronym 'OM' signifies occurrences of out-of-memony errors. From Table 2 we can draw the following three findings:
1) Our STONE model exhibits excellent clustering performance across all datasets, significantly surpassing competitors in certain scenarios. For instance, on the Scene 15 dataset, STONE outperforms the second-ranked method, ETMSLC, across five performance metrics (ACC, NMI, PUR. F-score, and ARI) with improvements of 13%, 9.5%, 9.1 %. 12.7 % and 13.7 %, respectively. Moreover, STONE demonstrates ideal clustering performance on the NGs, BBCSport, HW, and MSRCV1 datasets, These results indicate that the STONE model effectively uncovers higher-order correlations among multiple views as well as the geometric manifold information within each view, contributing to improved clustering outcomes.
2) Methods based on tensor constraints often outperform those based matrix constraints in terms of clustering performance. This enhancement primarily attributed to the ability of tensor-based approaches to impose low-rank constraints at the tensor level, effectively capturing the inherent high-order correlations multi-view data. In contrast, matrix-based methods generally focus only on linear correlations within individual views.
3) In comparison to prominent deep learning models, such as MFLVC [63] and GCFAgg 64 the STONE method demonstrates superior performance in most scenarios. This suggests that shallow learning models can still produce more effective clustering results than deep learning methods in multi-view tasks by cleverly extracting the rich information contained within multi-view data.
Efficiency Assessment: To demonstrate the efficiency of the STONE method, record its running time on datasets containing over 4000 instances and compared it with other benchmark methods. The comparison results summarized in Table 3 Notably, STONE exhibits significant efficiency in this comparison. For instance, on the ALOI-100 dataset, our method runs in 66.57 seconds, whereas the anchor tensor-induced model TBGL takes over 26000 seconds, which considerably longer than the STONE model. The STONE method achieves higher efficiency due to its innovative combination of anchor point dictionary representation learning and anchor hypergraph Laplacian regularization. This approach selectively incorporates a small subset of the most discriminative anchor points, ensuring faster computational efficiency while preserving precise clustering performance.
4.3 Parameters Analysis
In the STONE model, there are five parameters, including the built-in parameter ö, the number of anchors and three balancing parameters a, 3, and 7. This subsection investigates the impact of these parameters on the STONE model. Specifically, the parameters 8 and are treated as independent variables for individual tuning, while the balancing parameters are adjusted pairwise using grid search strategy.

---

g
g
(a) HW
(b) HW
(c) HW
Figure 5: Sensitivity Analysis of the STONE Model to the Balance Parameters a, / and 7.
Impact of the Built-in Parameter ō: HTR is utilized as non-convex penalty term for the singular values of tensor data, dynamically controlling the degree shrinkage applied to different singular values by adjusting the parameteras 6. We explore the impact clustering results for datasets NGs, HW, and MSRCVI by adjusting the parameter ō across the values [0.1, 0.5, 1, 1.5, 2, 5J. This variation enabled Figure Clearly, alterations in the value of 5 lead to fluctuations in the clustering results, driven by the varying contraction degree of across different singular values.
assess its impact
the clustering outcomes, with the results detailed in
Influence of the Number of Anchors: this subsection, we study how the number of anchors affects the performance of STONE, with anchor counts ranging from [c, 7c] and a step size of As illustrated in Figure 4 the clustering performance of STONE shows a fluctuating pattern as the number of anchors varies. Interestingly, the performance curve does not monotonically increase with the number of anchors, which means that choosing a smaller number of discriminant anchors is preferable choosing larger number of non-discriminant anchors. In addition, the best clustering quality can be obtained by using or 2c anchor points in STONE, which shows that the coordination between the EAD and the AHR improves the discrimination of the anchor points.
Sensitivity Analysis of Balancing Parameters: To assess the importance of the balancing parameters a, B. and in the STONE model, we implement grid search strategy across the range of [le-5, le+1] to optimize these parameters. Figure demonstrates how the model's performance changes with various combinations of these parameters, highlighting fluctuations in clustering performance based on the chosen values. Notably, optimal performance can be achieved through careful tuning, suggesting that the modules within the STONE model can effectively coordinate their importance to extract valuable information, thereby enhancing clustering performance.
4.4 Convergence Behavior
This subsection provides an experimental validation of the convergence the STONE model, utilizing two key metrics: reconstruction error (RE), defined as RE = >
T-Zz)T-Po - (X-A)"(Z")T -E" and matching error (ME), represented as ME = 2 S oc. The iterative trends observed on NGs, HW, and MSRCVI datasets, as depicted in Figure 6 demonstrate that both RE and ME exhibit rapid convergence within iterations, followed by stabilization. This outcome substantiates the robust convergence properties of the STONE method.
-
4.5 Ablation Study
Comprehensive ablation experiments are car- ried in this subsection to systematically as- sess the contributions of various modules within the STONE model. Here, we assign the val- ues of the balancing parametersset
Table 4: Analysis of STONE Model Ablation.
Ditasets
NGs
MISCVVI
HW
LEAD
Eax
EADH
ACC
NMI
ACC:
NMI
ACC
NMI
0.458
0.291
0.148
0.030
0.988
0.974
B, and Y governinnn the loss terms LEAD, LRE. and CahR. respectively- O, essentially isolating and removing each loss term separately from the STONE model. The experimental results for the NGs, MSRCVI and HW datasets are presented in Table 4 with checkmarks denoting the considera-
0.208
0.012
0.205
0.033
0.977
0.951
0.148
0.974
0976
0.534
0.397
0.786
0.631
0.983
0.971
0.158
0.311
0.571
0.381
0.854
0.841
1.000
1.000
1.000
1.000
1,000
1.000
tion of the corresponding loss. The best best-performing results are indicated in bold. Table reveals

---

Table 5: Comparison STONE and STONE-v1 across Different Datasets.
Datasets STONE-11
NGs
BBCSport
HW
Scone
MSRCVI
ALOI-100
CallOl-all
CIFARTO
0.37900.000
0.64820.000 1.000000000
0.74010.000
0.46920.000 0.629s0.000 1,00000.000
0.31900.000 0.81440.000
0.60010.000
0.501=0.000
STONE
10000000000
0.97740.000
1.00000.0000
0.650#0.000
0.994g0.000
Heach Enior
Roo Erar
Ruzo cesornatuziin Ert
Vatee Ene
10
Ruze vatruction Err
Marce Ene
10
100
(a) NGs
(b) HW
(c) MSRCV1
Figure 6: Convergence Curves of STONE Three Datasets.
that the clustering performance of degraded models, achieved by removing two submodules from the STONE model, is notably inferior to that of the complete STONE model. This emphasizes the successful collaboration of LEAD,, LRE, and LAHR within the STONE framework, allowing them to synergistically exploit the abundant information embedded multi-view data and attain commendable clustering performance. Additionally, HTR is novel tensor low-rank constraint in our STONE model, aimed at capturing high-order correlations and managing variations in tensor singular values. To assess its impact, we conduct an ablation study comparing the original STONE model with version that excludes the HTR module (referred to as STONE-v1) 1). Table shows the clustering ACC across different datasets, revealing a drop in performance on all datasets when the HTR module is removed. This suggests that the integration of HTR enhances the exploration of high-order correlations, thereby improving the quality of data partitioning.
Conclusion
This paper introduces novel tensor-based multi-view subspace clustering framework that integrates triple information enhancement from dictionary to tensor representation. Through the design of the enhanced anchor dictionary representation, hyperbolic tangent rank, and anchored hypergraph Laplacian regularization, our model extensively investigates valuable insights within multi view data. Experimental results demonstrate that the STONE model outperforms SOTA models on eight datasets in terms of both effectiveness and efficiency.
Acknowledgements
This work was supported by the Beijing Natural Science Foundation (No. 4242046) and the Funda- mental Research Funds for the Central Universities (No. 2022JBZY019).
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A Appendix/ / supplemental material
This supplementary material offers a comprehensive elaboration on the optimization steps presented in the main manuseript, well as the validation of the theorems. We also include additional experimental results.
A.1 Optimization of the Algorithm
As detailed in the main text, auxiliary variables S and {Q") are introduced to facilitate the indepen- dent optimization of each variable.
min [Z..P..A SS.P..A.. A...).. J.E.
S
HTR
a
1 P + +B +
2
61 m
Y E
+ 2 1 S+
X AZ
p"X"
12
(11)
0
2
62 +
Σ
Z Q"
+
Y
2
62
Next, we utilize the ADMM algorithm 66 individually optimize each variable as follows:
{Z"} Subproblem: Under the assumption that all other variables remain constant while Z varies, the optimization in Eq (11) simplifies to single-variable optimization problem. To find the optimal
14

---

solution, we take the partial derivative of Eq (11) with respect to Z" and set it to zero, resulting in the following solution:
Z" (0S" -Y" (X") A" (E A e1 (X"))("")))) A"
(12)
+ (A'H))**++0
A"
HD) x [(0 + <2) + 61 (A"))
(P") Subproblem: In this case, treat only P as variable. To find the optimal solution, set the first derivative of Eq. (11) with respect to P to zero, resulting in the following expression:
P ( X (Z") - E Hi)
(13)
x (X*) (al + x (x
E Subproblem: In similar vein, assuming that all variables in Eq (11) are constant except for E, we can reframe the optimization problem for E as follows:
3
1
min E
E 2.1 + 2
E E-R
(14)
where R. is constructed by horizontally stacking X A²(Z")
P"X
Hi
The optimal solution
e1
for E can then be derived by solving Eq (14) follows:
R. 1R,
R
R
E
RI
(15)
0,
otherwise
{A*} Subproblem: When the other variables are held constant and A is treated as the variable, the optimization problem for A " can reformulated as follows:
max Tr((A") TK), K), S.L. (A (A") ( A =I A"
(16)
In Eq (16), we define K E (X" P"X" E Hi decomposition (SVD). From the results of the SVD, A = BDD where B and represent the left and right singular vector matrices, respectively.
IZ" JZ and apply singular value determine that the optimal solution for
[Q"} Subproblem: In the scenario where Q" is the sole variable in Eq (11), we take the partial derivative of Eq (11) with respect to Q" and set it to zero, leading us to express the optimal solution for Q" in the following form: = (E2Z" (2 +22
(17)
S Subproblem: When S is the only variable, the optimal value of S can be redefined as a tensor hyperbolic tangent rank optimization problem in the following mathematical form:
0
2 S y 1
min S
| HTR +
(18)
2
To find the solution for Eq. (18) with respect to S, we first introduce the following theorem related to the tensor optimization problem:
Theorem 2. Let G E fg*1xngXng be a tensor, and consider its t-SVD (tensor singular value decomposition) expressed as g = 13 C D We will analyze the following tensorial hyperbolic tangent rank minimization problem:
min ||ITT|HHTRRRRRR ||TR + AS-111?.
(19)
The optimal solution for Eq. (19) takes the following mathematical form:
S BB- ifff(( 0, DT.
(20)
where (C( 313) is a tensor in which all frontal slices are diagonal matrices, and (c (1) (1))) meet the following condition:
(c(
(k) (ii) (
min (( (2(1)))++ TT(())
1
(21)
=20
where f(x)
-00 dx.
15

---

Algorithm 1: Algorithm for solving STONE model
Input:
Multi-view data (X*) V==1: trade-off parameters o, 9 7, cluster number and anchor
number l.
Output: Clustering results
Initialize {Z", P 1", ",," E", Hi. Hb 10 with zero matrix, S =y- 0, c2 0 10
1/ 2, Hmman = Omax = 10 = 10-7
2 while not converge
Compute hyper-Laplacian matrices (E from {A")=
Update !Z" mm by solving Eq. (12);
J==1
Update [P]m == by solving Eq. (13);
Update E by solving Eq. (14);
Update [ )m by solving Eq. (16);
Update Q" [==1 Q by solving Eq. (17);
Update S by solving Eq. (20);
Update y, Hi. H e and by using Eq. (23); 10
11
Check the convergence conditions: X - A"(Z") - E < and
1 S 1oc < K
12 end
13 Output clustering results via k-means on the left singular vector of the concatenated matrix Z.
Eq. (21) incorporates mix of concave and convex functions, which allows for the use difference of convex programming [67]. This technique facilitates obtaining a closed-form solution.
sitrt11
(
(ii)
(ii) Tdf(otte))
(22)
where o = , (c) (1)),, f(z)
-e- te- and iter indicates the iteration count.
Multipliers and the Penalty Parameters Subproblem: Finally, V. Hi. H and 0 are updated as follows:
y = y 0(2 -),,,,, S),
H = H (j X T" - -"),,
НЕ H. (2 - Q²),
(23)
14: min( Emax); 1,2,
T)E
0 min(1 0mxx).
Thus, the solutions for all variables in the STONE model have been optimized. To provide clarity, the complete optimization process is detailed in Algorithm T
A.2 Convergence Proof
The Theorem D presented in the main text ensures the convergence of the optimization algorithm. We will now demonstrate the two conditions specified in Theorem I To begin, we introduce the following lemma:
Lemma 1. In the context of the real Hilbert space H. we define an inner product (-5-) and a norm
1. along with their dual norm || I[duall For any element within the subdifferentaal of the function f(.), denoted y d|x), the subsequent properties are observed: when x is not the zero vector, the dual norm y is exactly 1; when x is the zero vector, the dual norm of y does not exceed 1.
Lemma Consider the function F(X) 5(X), where 8(X) (x) o, (X)) is the vector of singular values derived from the singular value decomposition (SVD) of X E Rmxn, with r being the minimum of m and n. The function f(-) RR"" - is assumed to be differentiable and invariant under permutation of its arguments. The subdifferential of F(X) at the point X can be expressed as:
0F(X)
(DDia((f(((()))
ax
where 0s(6(X)) (Af(m1(2) 0x
(z)) ax
16

---

Proof of the boundedness of the sequence {= In the course of the (t + 1)-th eycle, the mechanism updating Ei+1 ensures it complies with the necessary first-order optimality criteria. Consequently, it follows that:
H | -P4+1X
0 3d E 12,1 +
611
E1+1 (X1+1- -Xi+1-A'(Zia -" - X'
61
(24)
E 1+1 1111122.11111
H4.
From Eq. (24), we can derive the following:
H Hi lll 11]
(25)
where Hii and [ correspond to the j-th column of the matrices. Furthermore, taking into account the self-duality property of the lz norm and utilizing Lemma 2, we can conclude that ( I This further establishes the boundedness of the sequence [H], In parallel,
the update for Qi+1 ensures that 22, not only meets but also optimizes the first-order optimality criteria. Hence, it can be inferred that the sequence H 2,141 is bounded as well.
Regarding the sequence (D++1)} ). the update mechanism for S guarantees that S.+1 achieves opti- mality and meets the criteria for first-order optimality. Thus, we have:
0| HTR ----1-
(26)
Furthermore, leveraging the tensor singular value decomposition and Lemma 2, we can derive the following relationship: |0|S |
1 B+ ( D" n
(ifft(Of(CC)
(ff(C)))
(27)
min(n.m))
(O(C))M
n
This observation indicates that S||S f.141 ||ttr has an upper bound, which in turn allows us to deduce that the sequene (V++1) is bounded.
Based on the iterative procedures described in Algorithm 1, we can derive the following inequality relationships:
C(Z E. ++1 P Ai Q S 1+1, HJ H Vr, Gc E1, E2,1) CZZ E,, P., AW. Q S, H H > 61,1, E2,2)
C(Z), E. P AT Q S, - H.
O 8.-1 61,-1,
0, 01-1
|| y 1-1
f1 61,-1
H H Hi,-i
(28)
207_1
2G
2,1 62,11
H H 21
262,-11
Consequently, by adding up both sides of Eq. (28) over the range from f to n, we deduce the following consequence:
C(Z E P A Q S41, H H. ++1
1i Vr, E1,1,
C(Z), Ej P A..
Q S Hi. HE Vo. Po. 61,0, 22,
0, 0,-1
|| V-1 11
207_1
(29)
61,1 61,t11
H H
2 26,,--1
22,1 22,11
H H
2 2
17

---

Given the finite nature of the initial L E, AY, Q1 Hi,0 H Vo. 8o 61.0, 62,,,2,0) eval- uated at the starting points and the boundedness of the sequences (Ve), {H1.) (H2.), along with the boundedness of the incremental sums involving
02-0x-1
-1-01 and 20 E 26, --,
Et=1 20,,1 f we deduce that sequence L remains bounded at iteration & + 1. Additionally,
since the norm [St+1||HT is bounded, it follows that the singular values of St+1 are also con- strained. Continuing with the equation:
|S
|
13.3
min(n1
(1))
(30)
1
Σ J=1
n1
is verified that the sequence (S++1} has finite limits. Moreover, it is evident that the sequences(A (41) (Zi+1). (Qi+1)annd (P+1) also finite.
Continuing from the earlier findings, we can assert that the sequence Gr as yielded by Algorithm 1, is bounded for every component within it.
Proof of convergence accumulation points stationary KKT points: As per the Weierstrass- Bolzano theorem, is guaranteed that the sequence {Ge J2=1 contains at least one accumulation point, which we label as 9 {Z, E. P A V H H2, 2, S - From this, we can infer that: lim [Z) E. P A. Q S, H H H...
(31)
Z" E. P., A., Q S., H...
Adhering the update mechanism fory, arrive at the subsequent expression:
Z++1 S (v+11 - -y,)/0,
(32)
Since O approach infinity as f goes to infinity, and considering that the sequence (V/} is bounded, we can use the properties of limits derive: lim (V++1 - =0
lim Zi+1 S 1o 10
(33)
By applying the analogous update processes for H1 and H2, we can formulate the subsequent relationship:
H
X A
(Zi+)) (Z+1)
P X P
E E+1
61,1
(34)
H. H H
Zi+1 Qi11
6,,t
In the same vein, considering that the sequences (Hi.)) and (HE,) remain bounded, while e1,t and c2, increase indefinitely approaches infinity, we can infer the following conclusions based on limit properties:
H 1,411
H
lim X 10000
1 (Zi+1) P PH+1X
X"
E
lim 1-000
61,t
(35)
H
H
2.e 0.
lim 2it1 - Q.+1 100
lim E-300
22,1
Based on Eqs. (33) and (35), we can deduce that in the limit, is equal to Z., while X" and E" exhibit a specific linear relationship. and Zü equals Q Additionally, since S1+1. E and Q satisfy the first-order optimality conditions, we can conclude that:
0 S | Vt+1 y. Ö||S. ||ITTR
0 E 30 pa E - H = H 1.* = 1
(36)
0 L
H 2 H (""""""
Consequently, the accumulation points the sequence G, produced by Algorithm fulfill the KKT conditions.
18

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100
20
194
200
(a) MVCtopl
(b) MSC
(c) GMC
180
100
400
160
(d) MVSCTM
(e) TBGL
(f) STONE
Figure Contrasting Consensus Affinity Matrices: STONE vs. SOTA on NGs Dataset.
A.3 Additional Experimental Results
This section outlines additional experimental results, including visualizations of block diagonal structures, t-SNE, and some extra experiments mentioned in the text, such as parameter sensitivity, convergence curves, and ablation studies.
Comparison of Block Diagonal Structures in Affinity Matrices: In this section, we examine the block diagonal structures of affinity matrices learned by the STONE model alongside several other state-of-the-an multi-view clustering methods. The comparative results on the NGs dataset illustrated in Figure 7 Notably, the consensus affinity matrix generated by the STONE model clearly displays well-defined block diagonal structure, while those produced by other approaches frequently exhibit many spurious connections. This reinforces the effectiveness of the STONE model in leveraging rich information from multi-view data through the synergistic integration of discriminative anchor point learning, local structural information extraction, and the utilization of tensor data priors, ultimately enhancing clustering performance.
Analysis of Multi- View Advantages Over Single- View: To demonstrate the STONE model's capability in leveraging the rich information inherent in multi-view data, Figure g displays the I-SNE visualizations for each individual view alongside the integrated consensus graph. Notably, the consensus graph presents a more distinct clustering pattern when compared to the individual view graphs, which aids in the more precise segregation the MSRCVI dataset into seven unique classes. This finding highlights the effectiveness of the STONE method in synthesizing multi-view data for improved clustering performance.
Experimental Results for More Datasets: Due to space constraints, the main text presents only partial results for some experiments. Here, provide the complete results for all datasets, including parameter sensitivity analyses, convergence curves, and ablation studies. Specifically, Figure 9 Figure I0 and Figure [I showcase the sensitivity of the STONE model to the parameters 8, the number of anchors I, and the balancing parameters. Figure [ illustrate the convergence curves for eight datasets, while Table Ssmmmarie the ablation studies for the three loss terms CEAD. LRE and CAHR of STONE across all datasets.
19

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(a) The 1st view Z
(b) The 2nd view Z
(c) The 3rd view Z
(d) The 4th view Z
(e) The 5th view Z
(f) The consensus Z
Figure 8: Comparative t-SNE Visualization Analysis: View-Specift Graphs Consensus Graph.
Table 6: Analysis of STONE Model Ablation on NGs and BBCSport Datasets.
Datasets
NGs
BBCSport
LEAD
LRE
LAHR
NMI
PUR
F-score
ARI
ACC
NMI
PUR
F-score
ARI
0.438
0.291
0.438
0.348
0.183
0.800
0.690
0.800
0.691
0.600
0.208
0.012
0.212
0.329
0.000
0.831
0.712
0.831
0.711
0.624
0.596
0.473
0.636
0.532
0.396
0.996
0.987
0.996
0.996
0.995
0.960
0.897
0.960
0.924
0.905
0.818
0.715
0.818
0.691
0.598
0.534
0.397
0.572
0.449
0.274
0.368
0.137
0.474
0.296
0.084
0.458
0.311
0.502
0.404
0.196
0.358
0.134
0.474
0.292
0.083
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Table 7: Analysis of STONE Model Ablation on HW and Scene15 Datasets.
Datasets
HW PUR
Scene 15
LEAD
LRE
LAHR
ACC
NMI
F-score
ARI
ACC
NMI
PUR
F-score
ARI
0.988
0.974
0.988
0.976
0.974
0.589
0.557
0.597
0.480
0.440
0.977
0.951
0.977
0.955
0.950
0.787
0.844
0.802
0.714
0.693
0.988
0.974
0.988
0.976
0.974
0.328
0.297
0.359
0.228
0.168
0.881
0.881
0.794
0.724
0.760
0.655
0.629
0.983
0.971
0.983
0.968
0..65
0.713
0.800
0.764
0.656
0.630
0.854
0.841
0.854
0.786
0.762
0.678
0.807
0.746
0.687
0.663
1.000
1.000
1.000
1.000
1,000
0.977
0.962
0.977
0.958
0.955
20

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100
(a) NGs
(b) BBCSport
(c) HW
(d) Scenel5
200
(e) MSRCVI
(1) -1llech101-all
ALO1-100
(h) CIFARIO
Figure Impact of Parameter on the STONE Model on Eight Datasets.
NGs
BBCSport
(c) HW
(d) Scene
(c) MSRCV1
(f) Caltech101-all
(g) ALO1-100
(h) CIFARIO
Figure 10: The Inffuence of Anchor Quantity on STONE Model Performance Across Eight Datasets.
Table 8: Analysis of STONE Model Ablation on MSRCV1 and ALOI-100 Datasets.
Datasets
MSRCVI
ALOI-100
LEAD
LRE
CAHR
ACC
NMI
PUR
F-score
ARI
ACC
NMI
PUR
F-score
ARI
0.148
0.030
0.171
0.243
0.001
0.611
0.759
0.630
0.470
0.464
0.205
0.033
0.214
0.175
-0.002
0,592
0.757
0.628
0.444
0.438
0.148
0.030
0.171
0.243
0.001
0.488
0.669
0.530
0.234
0.222
0.976
0.946
0.976
0.952
0.944
0.545
0.778
0.614
0.440
0.433
0.786
0.631
0.786
0.630
0.570
0.580
0.776
0.625
0.447
0.439
0.571
0.384
0.571
0.407
0.306
0.557
0.737
0.592
0.391
0,383
1.000
1.000
1.000
1,000
1.000
0.909
0.849
0.7.6
0.733
21

---

(a) NGs
(b) NGS
(c) NGs
(d) BBCSport
(e) BBCSport
(f) BBCSport
(g) HW
(h) HW
(i) HW
ii Scene15
(k) Scene
(I) Scene
(m) MSRCVI
(n) MSRCVI
(0) MSRCVI
(p) Cal101-all
(g) Call01-all
(r) Cal101-all
(S) ALOI-100
(1) ALOI- 100
(u) ALOI- 100
(v) CIFARIO
(w) CIFARIO
(x) CIFARIO
Figure 11: Sensitivity Analysis of the STONE Model to Parameters a, B and y on Eight Datasets.
(a) NGs
(b) BBCSport
(c) HW
(d) Scene
(e) MSRCVI
(f) 11llech101-all
(g) AL.01-100
(h) CIFARI0
Figure 12: Convergence Curves of STONE Model on Eight Datasets.
Table 9: Analysis of STONE Model Ablation Caltech -al and CIFARI0 Datasets.
Datascts
Caltech101-al PUR
CIFARIO
LEAD
CRE
LAHR
ACC
NMI
F-score
ARI
ACC
NMI
PUR
F-score
ARI
0.185
0.368
0.365
0.182
0.168
0.833
0.782
0.833
0.733
0.703
0.475
0.786
0.717
0.345
0.334
0.931
0.876
0.931
0.874
0.860
0.271
0.477
0.467
0.206
0.191
0.994
0.983
0.994
0.988
0.987
V
0.499
0.799
0.753
0.378
0.368
0.830
0.867
0.857
0.829
0.809
0.297
0.531
0.527
0.244
0..29
0.830
0.867
0.857
0.829
0.809
0.516
0.740
0.732
0.405
0.394
0.884
0.821
0.884
0.805
0.783
0.609
0.834
0.815
0.494
0.485
0.994
0.984
0.994
0.989
0.988
22

---

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29


Reviews:
[{"review_id": "mcsCkjTD4M", "reviewer": "Reviewer_6B6d", "summary": "The authors introduce the STONE framework, a Tensor-based Multi-view Subspace Clustering (TMSC) approach, designed to surmount the paramount limitations inherent in contemporary methodologies. By augmenting anchor dictionary learning, they adeptly reconstruct low-rank structures, resulting in a reduction in computational intricacy and bolstering resilience, particularly in scenarios constrained by limited dictionaries. Furthermore, the framework incorporates an innovative anchor hypergraph Laplacian regularizer, ensuring the preservation of data geometry within subspace representations, and leverages an enhanced hyperbolic tangent function for precision in tensor rank approximation.", "questions": "1) Does the dimensionality k of the latent data remain consistent across various views? \n2) The proposed method innovatively employs hypergraph Laplacian regularization to delve into and uncover local intrinsic manifold structures within the data. How does this hypergraph differ from traditional anchor graphs? Additionally, how many anchor points are set in the hypergraph, and is this consistent across all datasets?\n3) The method introduces multiple variables, each requiring careful initialization to ensure the stability and convergence of the optimization process. The authors should clarify the initialization strategies adopted for each variable. How are each of these variables initialized?\n4) In Section Experimental Setup, how is $\\delta$ set across different datasets?", "limitations": "See the weaknesses.", "rating": 7, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1)The motivation of this paper is explicit. The authors strive to elevate cluster performance while minimizing computational complexity by refining anchor dictionary learning, incorporating anchor hypergraph Laplacian regularization, and utilizing an enhanced hyperbolic tangent function for precise rank approximation.\n2)The proposed method stands out from other state-of-the-art (SOTA) techniques, surpassing their effectiveness and efficiency. Notably, the authors have graciously shared the source code, facilitating further exploration and validation of their innovative approach.\n3)The authors' rigorous theoretical analyses, encompassing computational complexity and convergence evaluations, provide a solid foundation for the validity and reliability of their method, further strengthening the overall impact of their research.", "weaknesses": "1) The authors clarify that their proposed method employs an advanced dictionary learning mechanism to delve into and uncover latent data, which refers to information that is inherently present but not directly observable or accessible without specific techniques. However, the exact nature of this \"hidden data\" could be further elaborated to avoid ambiguity. It is crucial to distinguish this concept from missing multi-view data, which pertains to instances where certain views or features of the data are absent or incomplete. The enhanced dictionary learning aims to capture and represent these underlying, yet hidden, patterns and structures within the data, distinct from simply filling in gaps caused by missing views.\n2) There are some typographical errors in the manuscript, such as on line 584 where \"Eq. (11)\" seems to have an incorrect sequence number.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review of our manuscript. We have carefully considered each of your comments and addressed them point by point below:\n\n**Weakness 1:**  The authors clarify that their proposed method employs an advanced dictionary learning mechanism to delve into and uncover latent data, which refers to information that is inherently present but not directly observable or accessible without specific techniques. However, the exact nature of this \"hidden data\" could be further elaborated to avoid ambiguity. It is crucial to distinguish this concept from missing multi-view data, which pertains to instances where certain views or features of the data are absent or incomplete. The enhanced dictionary learning aims to capture and represent these underlying, yet hidden, patterns and structures within the data, distinct from simply filling in gaps caused by missing views.\n\n**A1:** In our study, \"hidden data\" refers to latent structures and patterns within the dataset that are not directly observable. This is different from missing multi-view data, which involves cases where certain views or features are absent or incomplete. Our advanced dictionary learning mechanism aims to reveal these underlying patterns and structures rather than simply addressing gaps from missing views. We appreciate your suggestion for further clarification and will include a more detailed explanation of \"hidden data\" in the revised manuscript.\n\n**Weakness 2:** There are some typographical errors in the manuscript, such as on line 584 where \"Eq. (11)\" seems to have an incorrect sequence number.\n\n**A2:** Thank you for pointing out the typographical errors. We will correct the sequence number for \"Eq. (11)\" on line 584, along with any other errors identified in the manuscript.\n\n**Question 1:** Does the dimensionality k of the latent data remain consistent across various views?\n\n**A3:** The latent data in our model is intended to address the limitations in the original data due to insufficient sampling, serving as an idealized supplement across different views. Theoretically, the latent data could differ between views. However, in our approach, we use skinny SVD theory to model the influence of latent data as a regularization term. We do not explicitly recover the latent data itself. Therefore, the dimensionality k we refer to is symbolic and represents the latent data dimensions rather than an actual recovered value.\n\n**Question 2:** The proposed method innovatively employs hypergraph Laplacian regularization to delve into and uncover local intrinsic manifold structures within the data. How does this hypergraph differ from traditional anchor graphs? Additionally, how many anchor points are set in the hypergraph, and is this consistent across all datasets?\n\n**A4:** **Difference from Traditional Anchor Graphs:** The hypergraph Laplacian regularization used in our method differs from traditional anchor graphs primarily in its ability to capture higher-order relationships among data points. While traditional anchor graphs focus on pairwise relationships between data points, hypergraphs extend this concept to capture relationships among groups of points. This allows our method to uncover more complex local intrinsic manifold structures within the data by considering interactions among multiple points simultaneously. **The number of anchor points** is set to 3, and this value is consistent across all datasets.\n\n**Question 3**: The method introduces multiple variables, each requiring careful initialization to ensure the stability and convergence of the optimization process. The authors should clarify the initialization strategies adopted for each variable. How are each of these variables initialized?\n\n**A5:** In our method, all variables are initialized as zero matrices. This initialization approach is outlined in the pseudocode available in the supplementary materials (see page 16). We will provide a more detailed discussion of this initialization strategy in the revised manuscript.\n\n**Question 4:** In Section Experimental Setup, how δ set across different datasets?\n\n**A6:** Thank you for your question. Initially, we set the value of δ empirically to 1. We then fine-tuned this parameter to determine the optimal values for each dataset. Specifically, δ was set to 1 for the datasets NGs, BBCSport, HW, Scene15, MSRCV1, and ALOI-100. For the datasets Caltech101-all and CIFAR10, we adjusted δ to 0.5 and 0.1, respectively. We will provide a more detailed explanation of these parameter settings in the revised manuscript to ensure greater clarity.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "thgAVYIckM", "reviewer": "Reviewer_QZiQ", "summary": "The paper presents a novel framework, STONE (Scalable TMSC framework with Triple information Enhancement), addressing significant limitations in current Tensor-based Multi-view Subspace Clustering (TMSC) methods. The proposed approach aims to reduce computational complexity, improve subspace representation accuracy, and better handle noise-related singular values in tensor data. Through an enhanced anchor dictionary learning mechanism, an anchor hypergraph Laplacian regularizer, and the use of an improved hyperbolic tangent function, the authors demonstrate superior performance compared to state-of-the-art (SOTA) methods.", "questions": "None", "limitations": "None", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1.By incorporating this regularizer, the method preserves the inherent geometric structure of the data, leading to more accurate subspace representation.\n2. Extensive experimentation on a variety of datasets demonstrates the method's effectiveness and efficiency, surpassing SOTA methods.\n3. The framework's design allows for scalable application to large datasets, making it practical for real-world scenarios with high-dimensional multi-view data.", "weaknesses": "1. In Figure 5, the authors should carefully check the name of each subfigure.\n2. The authors state that A represents l anchors of the v-th view, with orthogonal constraints for optimal discriminability. However, the matrix A is d*l, and the size of constraint A * A’ is d * d， It is not true that features have orthogonality to each other.\n3. In Equation 7, the LWF constraint introduces the weighted coefficient vector to balance the weight of each view, while this coefficient vector is not seen in Equation 10. Thu authors should check it.\n4. The trace of Z in Equation 8 should be added.\n5. In the experiments, specifications need to be given as to what number of anchors is selected.\n6. Although the performance of the proposed method is superior to other compared methods, the number of parameters is too many, including four parameters and a parameter number of anchors. According to the code, we can obtain 8 * 8 * 8 * 8 * 6=24576 combinations of parameters. Authors should optimize the model, not simply add constraints together.\n7. The authors should explain the difference between the proposed model and the existing work ''Anchor Structure Regularization Induced Multi-view Subspace Clustering via Enhanced Tensor Rank Minimization'' published in ICCV.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your detailed review of our manuscript. We have carefully considered all your comments and provide our responses to each point below:\n\n**Weakness 1:** In Figure 5, the authors should carefully check the name of each subfigure.\n\n**A1:** We have carefully checked the names of the subfigures in Figure 5 and can confirm that they are correct. Each subfigure in Figure 5 pertains to the HW dataset and illustrates the cross-validation of three parameters: $\\alpha$, $\\beta$ and $\\gamma$. \n\n**Weakness 2:** Typo in Orthogonality Constraint for $\\mathbf A \\mathbf A^T = \\mathbf I$\n\n**A2:** Thank you for pointing out the typo. The correct constraint should indeed be $\\mathbf A^T \\mathbf A = \\mathbf I$ instead of $\\mathbf A \\mathbf A^T = \\mathbf I$. We will revise the manuscript to accurately reflect this correction and provide a clear explanation of how this constraint impacts feature representations.\n\n**Weakness 3:** Why is the weighted coefficient vector from Eq. (7) not present in Eq. (10)?\n\n**A3:** Each element of weighted coefficient vector in Eq. (7) is indeed set to 1, which is why it is not explicitly shown in Eq. (10). This simplification is discussed in the explanation following Eq. (7) in the manuscript. In the revised manuscript, we will provide a more detailed explanation to clarify this point.\n\n**Weakness 4:** The trace of $\\mathbf Z$ in Eq. (8) should be added.\n\n**A4:** Thank you for your insightful review. It appears that there might be a minor misunderstanding regarding Eq. (8). We believe the issue may concern the missing transpose of $\\mathbf Z$ rather than the trace. We will thoroughly reexamine Eq. (8) to ensure that the transpose is correctly included and make the necessary revisions. \n\n**Weakness 5**: Specifications Needed for Number of Anchors in Experiments.\n\n**A5:** In our experiments, the number of anchors for the Scene15 dataset was set to 2c, while for the other datasets, it was set to c. This is discussed on lines 667-669 of the manuscript. We will provide a more detailed discussion of these settings in the revised manuscript to ensure clarity and completeness.\n\n**Weakness 6:** Although the performance of the proposed method is superior to other compared methods, the number of parameters is too many, including four parameters and a parameter number of anchors. According to the code, we can obtain 8 * 8 * 8 * 8 * 6=24576 combinations of parameters. Authors should optimize the model, not simply add constraints together.\n\n**A6:** Our model indeed involves 5 parameters: 2 built-in parameters ($\\delta$ and $l$) and 3 balancing parameters ($\\alpha$, $\\beta$, and $\\gamma$). For clarity, the source code includes all theoretically possible combinations, but in practice, the number of parameter configurations we explore is much smaller. Specifically, $\\delta$ and $l$ are tuned independently, and we perform cross-validation only for the balancing parameters. $\\delta$ and $l$ are adjusted within the ranges [0.1, 0.5, 1, 1.5, 2, 5] and [1c, 7c], respectively, resulting in a total of 13 combinations. The balancing parameters, $\\alpha$, $\\beta$, and $\\gamma$, are tuned using cross-validation within the range [1e-5, 1e1], leading to 343 combinations (7 × 7 × 7). Thus, the total number of parameter configurations we actually explore is 13 + 343 = 356, which is significantly fewer than the theoretical count of 24,576 combinations (as detailed in the experimental section of the manuscript). Furthermore, as shown in the experimental analysis, when $\\delta$ and $l$ are fixed at $\\delta = 1$ and $l = 2c$, our model achieves optimal and stable performance across all datasets. This indicates that these parameters, even when set to these default values, still provide optimal performance, further reducing the parameter burden.\n\nOverall, we only need to fine-tune the three balancing parameters, which is typical for many SOTA methods. Additionally, our model innovatively integrates EAD, HTR, and AHR into a unified framework, with each component demonstrating significant effectiveness and originality in multi-view clustering.\n\n**Weakness 7:** Difference between the proposed model and the existing work ''ASR-ETR''  in ICCV.\n\n**A7:** ASR-ETR [30] is a highly representative contribution to the multi-view clustering field. There are three fundamental differences between our STONE model and ASR-ETR:\n\n**(1). Dictionary Learning Strategy:** The ASR-ETR model relies on traditional self-representation learning, which recovers subspace representations based on the original anchor representations. It does not address the inaccuracies caused by insufficient sampling of the original anchors. In contrast, our STONE model uses an enhanced dictionary learning strategy that more effectively captures and mitigates the inaccuracies due to undersampling, thereby improving the precision of subspace representations. \n\n**(2). Manifold Regularization:** The ASR-ETR model employs traditional Laplacian manifold regularization, which captures only pairwise manifold relationships. Our STONE model, on the other hand, utilizes hypergraph Laplacian manifold regularization, which captures both pairwise linear relationships and higher-order nonlinear topological relationships among multiple data points. \n\n**(3). Tensor Rank Constraint:** While ASR-ETR applies an Enhanced Tensor Rank constraint for low-rank tensor representation, our STONE model introduces a novel Hyperbolic Tangent Rank constraint. This new constraint is designed to capture more nuanced differences between singular values in tensor data, enhancing robustness and representation accuracy.\n\nOverall, our work builds upon the foundation laid by ASR-ETR, pushing the boundaries of multi-view clustering through these novel contributions in dictionary representation, subspace representation, and tensor representation. We appreciate the opportunity to clarify these distinctions and will provide a more detailed comparison in the revised manuscript.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "pILAo0MbQu", "reviewer": "Reviewer_iXTf", "summary": "The manuscript introduces a novel tensor-based multi-view clustering algorithm designed to address three critical limitations of existing approaches: high computational complexity arising from reliance on complete dictionaries, inaccurate subspace representation due to disregarding local geometric information, and inadequate penalization of singular values associated with noise. To overcome these challenges, the authors propose the STONE framework, which integrates enhanced anchor dictionary learning, anchor hypergraph Laplacian regularization, and an improved hyperbolic tangent function for more accurate tensor rank approximation. Experimental results demonstrate that STONE surpasses current state-of-the-art methods in both effectiveness and efficiency.", "questions": "See the weaknesses.", "limitations": "Recent years have seen the emergence of various tensor-based multi-view clustering methods and anchor-based approaches. It is crucial to compare and contrast these methods in the manuscript to highlight their fundamental differences.", "rating": 7, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "1. The proposed method explores rich information in multi-view data from the perspectives of dictionary representation, subspace representation, and tensor representation to enhance clustering performance, offering an interesting perspective for research in the field of multi-view clustering.\n\n2. The proposed method achieves optimal clustering performance while maintaining high computational efficiency, ensuring its scalability.\n\n3. The authors conducted clustering experiments on 8 multi-view datasets covering various types and scales, yielding compelling results.\n\n4. The experimental and theoretical analyses are comprehensive, providing both empirical and theoretical support for the effectiveness of the proposed method.", "weaknesses": "1. The introduction of Enhanced Anchor Dictionary (EAD) lacks clarity. Could the authors clarify the fundamental differences between EAD, traditional dictionary learning, and anchor dictionary learning? Additionally, how does the methodology incorporate hidden data?\n\n2. In Section 4.3, the authors reference Figure 11 in the first part, but the figure numbering is incorrect and needs to be corrected.\n\n3. From Table 2 in the manuscript, it appears that most methods have a standard deviation of 0.000. Could the authors explain why these methods exhibit no standard deviation?\n\n4. Based on Figure 10, why does the performance of the proposed method not exhibit a linear positive correlation with the number of anchor points? Typically, increasing the number of anchor points should lead to acquiring more valuable information.\n\n5. The pseudocode displayed for Algorithm 1 in the appendix has formula indices that do not correspond with those in the main manuscript.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful and detailed review of our manuscript. We have carefully considered each of your comments and addressed them point by point below:\n\n**Weakness 1:** The introduction of Enhanced Anchor Dictionary (EAD) lacks clarity. Could the authors clarify the fundamental differences between EAD, traditional dictionary learning, and anchor dictionary learning? Additionally, how does the methodology incorporate hidden data?\n\n**A1:** The fundamental difference between traditional dictionary learning, anchor dictionary learning, and our Enhanced Anchor Dictionary (EAD) lies in the dictionary representation used. Traditional dictionary learning employs the observed data itself to recover a subspace representation of size $n \\times n$. Anchor dictionary learning, on the other hand, utilizes a subset of samples from the observed data as anchor points to recover a subspace representation of size $n \\times l$. In contrast, while EAD also uses anchor points, it additionally accounts for the influence of hidden data to recover a subspace representation of size $n \\times l$. This approach not only reduces computational complexity but also addresses the imprecision arising from insufficient sampling in observed data, resulting in a more accurate and efficient representation.\n\n**Weakness 2:** In Section 4.3, the authors reference Figure 11 in the first part, but the figure numbering is incorrect and needs to be corrected.\n\n**A2:** Thank you for pointing out the discrepancy with the figure numbering. We will correct the reference to Figure 11 in Section 4.3 to ensure it aligns with the correct figure numbering in the manuscript.\n\n**Weakness 3:** From Table 2 in the manuscript, it appears that most methods have a standard deviation of 0.000. Could the authors explain why these methods exhibit no standard deviation?\n\n**Q3:** Indeed, most methods, including our STONE algorithm, show a standard deviation of 0.000 across all datasets. For comparison methods like SFMC, GMC, MVCtopl, and TBGL, this is due to their clustering strategies which impose connectivity constraints on consensus graphs, ensuring that connected components accurately reflect the true clustering labels and thereby avoiding the variability inherent in spectral clustering. In the case of our STONE algorithm, the stability observed with k-means clustering is attributed to applying k-means to the left singular vector of the concatenated matrix $\\bar {\\mathbf Z} = \\frac{1}{\\sqrt m} [\\mathbf Z^1,...,\\mathbf Z^m]$, which is already well-separated and stable, leading to consistent clustering results across multiple runs. We will explore this issue further in the revised manuscript.\n\n **Weakness 4:** Based on Figure 10, why does the performance of the proposed method not exhibit a linear positive correlation with the number of anchor points? Typically, increasing the number of anchor points should lead to acquiring more valuable information.\n\n**A4:** Thank you for your insightful question about the performance of our method in relation to the number of anchor points, as illustrated in Figure 10. The lack of a linear positive correlation between performance and the number of anchor points can be attributed to diminishing returns, where the incremental benefit of additional anchor points decreases beyond a certain threshold. Additionally, an excessive number of anchor points can introduce redundancy and noise, potentially degrading performance. Moreover, the quality of anchor points is crucial; additional points that lack discriminative power may not improve, and could even harm, performance. We will elaborate on these factors in the revised manuscript to clarify the observed trends.\n\n**Weakness 5:** The pseudocode displayed for Algorithm 1 in the appendix has formula indices that do not correspond with those in the main manuscript.\n\n**A5:** Thank you for pointing out the discrepancy between the formula indices in the pseudocode for Algorithm 1 in the appendix and those in the main manuscript. We will correct the indices in the pseudocode to ensure they match those referenced in the main text.\n\n**Limitation 1:** Recent years have seen the emergence of various tensor-based multi-view clustering methods and anchor-based approaches. It is crucial to compare and contrast these methods in the manuscript to highlight their fundamental differences.\n\n**A6:** Thank you for your constructive feedback. We will incorporate a comprehensive comparison and discussion of recent tensor-based multi-view clustering methods, anchor-based approaches, and our STONE method in the revised manuscript. This will help to clearly highlight the fundamental differences and provide a thorough understanding of each approach.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "6eiCaVN9Wc", "reviewer": "Reviewer_eQ9V", "summary": "This paper proposes a scalable tensor-based multi-view subspace clustering model by using triple information enhancement, which aims to reduce the computational complexity and the bias from the real rank minimization.", "questions": "1) In Eq. (26), how is the parameter “β derived?", "limitations": "Please refer to the above \"Weaknesses\".", "rating": 3, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "（1）This paper has provided significant proof of the algorithm's convergence.\n（2）Comprehensive experiments have been conducted for the investigation of effectiveness and performance.", "weaknesses": "1) In LatLRR, P is constraint by nuclear norm. In model (5), the authors indicate that P is constraint by weighted Frobenius norm. The authors can explain the reasons.\n2) On line 158, “Z_f^k denotes the k-th frontal slice of Z” should be changed into “Z_f^k denotes the k-th frontal slice of Z_f”.\n3) On line 159, “Z=U_f V_f W_f” should be changed into “Z_f=U_f V_f W_f”. The authors could also distinguish the matrix multiplication and tensor multiplication.\n4) In Eq. (8), “Tr(Z^v L_h^v Z^v)” should be changed into “Tr(〖〖(Z〗^v)〗^T L_h^v Z^v)”. \n5) In Eq. (8), is the hyper-Laplacian matrix L_h^v constructed by anchor subspace Z^v or feature X^v? If L_h^v is constructed by Z^v, the authors could introduce a consistent indicator matrix instead of regularize the Z^v in the trace norm.\n6) In Remark 2, as x→0, these small singular values may be caused by the noise. However, the proposed HTR would amplify these small singular values by f(x), resulting in less robustness.\n7) In Eq. (10), “α/2” should be changed into “α”.\n8) In Eq. (10), “Tr(Q^v L_h^v 〖〖(Q〗^v)〗^T)” should be changed into “Tr(〖〖(Q〗^v)〗^T L_h^v Q^v)”.\n9) On line 233, “d” should be changed into “d_v”.\n10) How does the HTR effect the proposed model? In Ablation Study, the authors could provide more experiments for verifying the effectiveness of the HTR.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate your careful review and insightful feedback on our manuscript. We have thoroughly considered each of your comments and have addressed them in detail below:\n\n**Weakness 1:** Why does LatLRR use the nuclear norm for P, but the proposed method uses the weighted Frobenius norm?\n\n**A1:** We replaced the nuclear norm used in LatLRR with the weighted Frobenius norm in our model because the nuclear norm in LatLRR is intended for feature extraction tasks. Since our manuscript focuses solely on clustering and does not involve feature extraction, applying the nuclear norm to P^v would increase computational complexity without significant benefits. Therefore, we opted to relax the constraint on P^v to the Frobenius norm, which has been shown in previous literature [41-43] to effectively preserve the block-diagonal low-rank structure. Additionally, considering the linear combination of different views, we applied a weighted Frobenius norm to the matrices  {P^v}, with the weights for different views set to 1. This approach is briefly introduced in the final paragraph of Section 2 of our manuscript, and we will provide a more detailed discussion of this choice in the revised manuscript.\n\n**Weakness 2:** On line 158, change $\\mathcal{Z}$ to $\\mathcal{Z}_f^k$.\n\n**A2:** We will correct this typo in the revised manuscript.\n\n**Weakness 3:**  On line 159, $\\mathcal{Z}$ should be changed into $\\mathcal{Z}_f^k$. The authors could also distinguish the matrix multiplication and tensor multiplication.\n\n**A3:** We will correct this typo and clarify the difference between matrix and tensor multiplication in the revised manuscript. Matrix multiplication involves dot products of rows and columns in 2D matrices, while tensor multiplication handles multi-dimensional arrays.\n\n**Weakness 4:** In Eq. (8), $Tr(\\mathbf Z^v \\mathbf L_h^v \\mathbf Z^v)$ should be changed into $Tr({(\\mathbf Z^v)}^T \\mathbf L_h^v \\mathbf Z^v)$.\n\n**A4：** We will correct the typo in Eq. (8) to align with the standard format for Laplacian manifold regularization.\n\n**Weakness 5:**  In Eq. (8), is the hyper-Laplacian matrix $\\mathbf{L}_h^v$ constructed from $\\mathbf{Z}^v$ or $\\mathbf{X}^v$? If from $\\mathbf{Z}^v$, why not use a consistent indicator matrix instead of regularizing $\\mathbf{Z}^v$ in the trace norm?\n\n**A5:** Thank you for your comment. We would like to clarify that in Eq. (8), the hyper-Laplacian matrix $\\mathbf{L}_h^v$ is constructed based on the anchor hypergraph $\\mathbf{S}^v$, which in turn is derived from the anchor representations $\\mathbf{Z}^v$, rather than directly from the anchor subspace $\\mathbf{Z}^v$ or the feature matrix $\\mathbf{X}^v$. The hyperanchor graph Laplacian manifold regularization builds upon traditional Laplacian regularization [47][48] and is designed to capture high-order local manifold information between anchor representations, which is crucial for the regularization of the subspace representation $\\mathbf{Z}^v$. We will provide further clarification on this in the revised manuscript.\n\n**Weakness 6:** In Remark 2, as x→0, these small singular values may be caused by the noise. However, the proposed HTR would amplify these small singular values by f(x), resulting in less robustness.\n\n**A6.** Thank you for your comment. It is true that small singular values may be caused by noise as x→0. However, Remark 2 should clarify that our proposed HTR method does not amplify these small singular values. Instead, the HTR method applies a stronger penalty to these values, which helps to mitigate their impact and improve robustness. The HTR method penalizes small singular values more heavily, thereby reducing their influence rather than amplifying it. This is discussed in detail in the REMARK 2 of the manuscript (including the illustration in Figure 2). We will provide additional clarification on this in the revised manuscript.\n\n**Weakness 7:** In Eq. (10), $\\frac{\\alpha}{2}$ should be changed into $\\alpha$.\n\n**A7:** We will correct this typo in the revised manuscript.\n\n**Weakness 8:**  In Eq. (10), $Tr(\\mathbf{Q}^v \\mathbf{L}_h^v (\\mathbf{Q}^v)^T)$ should be changed into $Tr(  (\\mathbf{Q}^v)^T)\\mathbf{L}_h^v\\mathbf{Q}^v)$.\n\n**A8:** Thank you for bringing this to our attention. We would like to respectfully note that the expression in Eq. (10) follows the standard convention for Laplacian manifold regularization. This format is commonly used in the literature[47][48], and we believe it aligns with established practices. \n\n**Weakness 9:** On line 233, “d” should be changed into “d_v”.\n\n**A9:** We will correct this typo in the revised manuscript.\n\n**Weakness 10:** How does the HTR affect the proposed model? In the Ablation Study, the authors could provide more experiments to verify the effectiveness of the HTR.\n\n**A10:** HTR is a novel tensor low-rank constraint in our STONE model, designed to capture high-order correlations and manage variations in tensor singular values. To assess its impact, we conducted ablation studies comparing the full STONE model with a version excluding HTR (STONE-v1). The results are summarized in the table below:\n\n| **Datasets**      | NGs           | BBCSport       | HW            | Scene15       | MSRCV1        | ALOI-100      | Cal101-all | CIFAR10       |\n|:---------------:|:--------------:|:--------------:|:-------------:|:-------------:|:-------------:|:-------------:|:--------------:|:-------------:|\n| STONE-v1 | 0.379±0.000 | 0.648±0.000 | 0.740±0.000 | 0.629±0.000 | 0.469±0.000 | 0.600±0.000 | 0.319±0.000 | 0.501±0.000 |\n| **Ours**        | **1.000±0.000** | **1.000±0.000** | **1.000±0.000** | **0.977±0.000** | **1.000±0.000** | **0.814±0.000** | **0.650±0.000** | **0.994±0.000** |\n\n**Question 1:** In Eq. (26), how is the parameter $\\beta$ derived?\n\n**A11:** $\\beta$ in Eq. (26) is a typo and should be $\\tau$ as described in Eqs. (20)–(25). $\\tau$ is a non-negative real number, as discussed in Lemma 1. 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Author rebuttal:

Meta-review: The paper is about multi-view subspace clustering, a fundamental and significant problem to study. While the review scores diverge greatly, no critical concerns can be found in the reviews. Reviewer QZiQ and Reviewer eQ9V, who give respectively scores of 4 and 3, only complain minor issues such as typos, which are not difficult to fix. I have also read the paper quickly, and my assertion is the paper contains solid contribution that would be valuable to this conference. I would recommend accepting the paper.
Decision: Accept (poster)