Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
Authors: Qiyu Zhong
Yi Shan
Haobo Wang
Zhen Yang
Gengyu Lyu
Venue: ICML 2025 poster (2025)
Keywords: Multi-view learning
Multi-label Learning
Primary area: general_machine_learning->supervised_learning

Abstract: In multi-view multi-label classification (MVML), each object has multiple heterogeneous views and is annotated with multiple labels. The key to deal with such problem lies in how to capture cross-view consistent correlations while excavate multi-label semantic relationships. Existing MVML methods usually employ two independent components to address them separately, and ignores their potential interaction relationships. To address this issue, we propose a novel Tensorized MVML method named TMvML, which formulates an MVML tensor classifier to excavate comprehensive cross-view feature correlations while characterize complete multi-label semantic relationships. Specifically, we first reconstruct the MVML mapping matrices as an MVML tensor classifier. Then, we rotate the tensor classifier and introduce a low-rank tensor constraint to ensure view-level feature consistency and label-level semantic co-occurrence simultaneously. To better characterize the low-rank tensor structure, we design a new Laplace Tensor Rank (LTR), which serves as a tighter surrogate of tensor rank to capture high-order fiber correlations within the tensor space. By conducting the above operations, our method can easily address the two key challenges in MVML via a concise LTR tensor classifier and achieve the extraction of both cross-view consistent correlations and multi-label semantic relationships simultaneously. Extensive experiments demonstrate that TMvML significantly outperforms state-of-the-art methods.

Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
Qiyu Zhong'
Yi Shan 12 Haobo Wang
Zhen Yang' Gengyu Lyu'
Abstract
Black Myth:
Wukong Tops Million Sales on Steam
Black Myth: Wakong has sold 20 million coples leading distributton platform Steam month after the hit action role-playing released, getting USD961 million
In multi-view multi-label classification (MVML), each object has multiple heterogeneous views and is annotated with multiple labels. The key to deal with such problem lies in how to capture cross- view consistent correlations while excavate multi- label semantic relationships. Existing MVML methods usually employ two independent com- ponents to address them separately, and ignores their potential interaction relationships. To ad- dress this issue, we propose a novel Tensorized MVML method named TMvML, which formu- lates an MVML tensor classifier to excavate com- prehensive cross-view feature correlations while characterize complete multi-label semantic rela- tionships. Specifically, we first reconstruct the MVML mapping matrices as an MVML tensor classifier. Then, we rotate the tensor classifier and introduce a low-rank tensor constraint to ensure view-level feature consistency and label-level se- mantic co-occurrence simultaneously. To better characterize the low-rank tensor structure, de- sign a new Laplace Tensor Rank (LTR), which serves as tighter surrogate of tensor rank to cap- ture high-order fiber correlations within the tensor space. By conducting the above operations, our method can easily address the two key challenges in MVML via concise LTR tensor classifier and achieve the extraction of both cross-view consis- tent correlations and multi-label semantic rela- tionships simultaneously. Extensive experiments demonstrate that TMvML. significantly outper- forms fate-of--hee-r methods.
Figure 1. An example of MVML webpage classification.
1. Introduction
Multi- View Multi-Label Learning (MVML. aims to learn from training data, where each sample is represented by sev- eral heterogeneous features while associated with multiple semantic labels (Lyu al., 2024). Different from single- view multi-label data, MVML data incorporates comple- mentary information from various views, thereby providing more comprehensive descriptions for objects. Such charac- teristic has naturally led to its extensive applications in many complex data analysis tasks (Liu et al., 2024d; Fu al., 2024). For example, in news webpage classification (Figure 1), news webpage can be characterized by text, image and video features, and annotated with multiple class labels such as Wukong, Sales and Steam. MVML provides an effective framework to learn a desired multi-label classifier for bridg- ing heterogeneous features with their corresponding labels and make proper predictions for new webpages.
The key learn from MVML data lies in how to effi- ciently integrate heterogeneous features while comprehen- sively characterize all relevant labels. For example, (Zhao et al., 2023) propose an MVML method called LVSL, which seeks cross-view correlations and multi-label relationships by learning the contribution weights of different views and applying label correlation matrix with low-rank constraint. (Zhong et al., 2024) introduce non-negative matrix factor- ization based MVML method called GNAM, which learns individual and common information across different views and leverages dynamic label correlation matrix to enhance recognition performance. Despite these MVML methods have achieved competitive performance improvements, they still face two main challenges: (1) On one hand, these meth-
'College of Computer Science, Beijing University of Tech- nology. China 'Idealism Beijing Technology Co., Ltd., Bei- jing, China "School of Software Technology, Zhejiang Univer- sity, Ningbo, China. Correspondence to: Gengyu Lyu <lyu- gengyu gmail.com>
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
ods are formulated by matrix theory, which only focus on capturing one- second-order relationships between each pair of views and fail to explore higher-order correlations across the whole view space. (2) On the other hand, they separate the extraction of cross-view consistent correlations from multi-label semantic characterization, and ignore the potential interactions between feature representation and se- mantic expression, inevitably leading to suboptimal results.
& Zhang, 2024). Existing studies mainly employ different strategies to explore label correlations for semantic expres- sion enhancement. For example, (Xie & Huang, 2022) pro- pose partial multi-label learning method named PML-NI, which captures the linear correlations of the multi-label clas- sifier by a trace norm with low-rank properties. (Sun et al., 2022) propose a method called Global-Local Label Corre- lation (GLC), which introduces a label coefficient matrix to exploit global label structures across multiple subspaces and a label manifold regularizer to capture local label corre- lations. (Si et al., 2023) propose a high-rank and high-order method called HOMI, which uses self-representation to keep the rank of the label matrix unchanged and indicate the high-order correlations among labels explicitly.
To address the above issues, we propose a novel Tensorized Multi- Multi-View Multi-Label Classification method via Laplace Tensor Rank (TMvML). which is the first attempt to uti- lize tensorized MVML classifier to achieve the high-order feature correlations extraction and multi-label semantic re- lationships characterization simultaneously. Specifically, we first reconstruct the multi-view multi-label mapping ma- trices into unified MVML tensor classifier to character- ize the structure of multi-dimennsonal data. Then, we ro- tate the tensor classifier and introduce low-rank tensor constraint to capture the high-order information embedded within the mapping tensor, which is achieved by comparing each row (label-specific) and each column (view-specific) of the frontal slices along the third dimension (feature-specifie). To better characterize the low-rank tensor structure, we de- sign novel tensor rank approximation named Laplace Ten- sor Rank (LTR), which preserves larger singular values and discards smaller ones (removed noise information) to obtain an accurate low-rank tensor representation. Extensive exper- imental results demonstrate that our proposed TMvML is significantly superior to other state-of- of-the-art methods. The main contributions of our paper are summarized as:
2.2. Multi-View Learning (MVL)
MVL handles the data with multiple heterogeneous view features by exploring consensus and complementary infor- mation hidden in different views (Jiang al.. 2021; Zhang et al., 2024; Xu al., 2024; 2023a;b). Existing MVL meth- ods can be roughly divided into two types based on whether high-order correlation among multi-view representations is explored, including non-tensor methods and tensor- based methods. Non-tensor methods employ matrix-level con- straints to explore the point-to-point relationship within one view or each pair of views. For example, (Cao et al., 2015) diversify the model structure using the Hilbert-SShhiit In- dependence Criterion (HSIC) on pairs of affinity matrices. (Jiang et al., 2024) fuse feature projection and similarity graph to simultaneously select features and learn a unified graph. The tensor-based methods are born to exploit the high-order correlation among views. (Xie et al., 2018) ex- tend the low-rank constraint from the matrix level to the tensor level by minimizing the Tensor Nuclear Norm (TNN) to capture the high-order consistency. To further enhance the characterization of low-rank tensors, (Ji & Feng, 2025) in- troduce a novel tensor rank approximation called Enhanced Tensor Rank, which is more noisy-robust to explore the high-order consistency among different views.
We propose a novel tensorized MVML classification method named TMvML, which is the first attempt to design a concise low-rank MVML tensor classifier to excavate cross-view feature correlations and character- ize multi-label semantic relationships simultaneously.
To better characterize the low-rank tensor structure, we design a novel tensor rank approximation named Laplace Tensor Rank (LTR), which preserves larger singular values and discards smaller ones to obtain an accurate low-rank tensor representation.
2.3. Multi-View Multi-Label Learning (MVML.)
MVML can be seen as an integration of both MVL and MLL, which needs to address both MVL and MLL issues simulta- neously (Lu et al., 2023; Liu et al., 2024c; Wei et al., 2025). For example, (Tan et al., 2021) introduce an MVML method called ICM2L, which captures shared patterns through common subspace and view-specific features with individ- ual classifiers and introduces label correlation matrix to enhance recognition performance. (Lyu et al., 2024) pro- pose a label-driven view-specific fusion method, which di- rectly encodes individual view features to construct a uni- fied multi-label classifier, and captures label correlations using rranformer-based semantic-aware label graph. (Liu
Extensive comparable results and detailed experimen- tal analysis have shown that our proposed TMvML. method can achieve significantly superior performance against other state-of- ofte-off-the-rr the-art methods.
2. Related Work
2. Multi-Label Learning (MLL)
MLL focuses on learning from data with multiple labels, and its challenge lies in how to completely characterize multi--abell semantic relationship (Liu et al., 2024a; Zhang
2

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Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
al., 2024b) propose an attention-induced MVML method, which weights features by the confidence derived from joint attention, and utilizes the weak label correlation matrix and graph attention network to guide classification.
X
y
shifldim(x..1)
k
3. The Proposed Method
3.1. Notations
Figure 2. The rotated coefficient tensor in our method.
Formally speaking, we use bold-upen sold-uppercase X for matrices, bold-lowercase for vectors and calligraphy letter to denote the tensor X' E [R Xm3, For 3-way tensor X, X is used to represent k-th frontal slice; x, (i, j) and ji. denote the mode-1, -2, -3 fibers; Xy= = f2' I.3) means the Fast Fourier Transformation (FFT) along the third dimension of tensor X. Given an MVML dataset D = {(X", Y) S S Vj. each X" = [x]. X2,.... x=]T E Baxd represents the feature vectors of n instances under u-th view and Y E {0, 1
data, we incorporate it into Eq. (I) and obtain the tensorized MVML objective function as:
V
min W.A.AAP Z
Y Z W" 2 + T(W) + |Ei|2,,
s.t. Vv, X Z"AA" E", A"A"""""''=I (2)
V=
W + W
q is the label matrix.
where the v-th view low-dimensional representation matrix Z" Bank and the basis matrix A Rexd, are obtained by matrix factorization. T(-) is the tensor rank or its ap- proximation. E E Raad, denotes reconstruction error. ((-)) constructs the tensor classifier W by merging different classifiers W E IR X4 to a 3-mode tensor, and then rotate its dimensionality to q V x k, as shown in Figure 2.
3.2. Formulation
In this paper, we propose a tensorized MVML method named TMvML, which formulates low-rank MVML ten- sor classifier to simultaneously extract comprehensive cross- view feature correlations while model complete multi-label semantic relationships. To better characterize the low-rank tensor structure, we develop a new Laplace Tensor Rank (LTR), which serves as a more precise surrogate for tensor rank, enabling a comprehensive exploration of multi-view and multi-label information to improve model performance.
[The superiority of 2()) Directly applying the tensor low-rank constraint is still not sufficiently effective. To better model low-rank constraints at both the label and view levels within the tensor space, we perform rotation operation on the coefficient tensor W & k trans- forming it into W E RRxV xk, as illustrated in Figure 2. This rotation repositions the q x V surface as the frontal slice, enabling the exploration of interactions between differ- ent views and labels by comparing every row (label-specific) and every column (view-specific) of the X V surface. By constraining all the frontal slices of the tensor, the rotated lensor offers deeper insights, enhancing the exploration of the higher-order correlations of views and labels.
Remark 1.
3.2. 1. MVML TENSOR CLASSIFIER
In the task of traditional multi view multi-label learning, the label matrix Y is usually approximated by a linear mapping W" from the feature space X":
min Σ Y X W'I' R(W") W
(L)
3.2.2. LAPLACE TENSOR RANK
where W [wY, w.:-.... T is the v-th view map- ping matrix. Label co-occurrence is also known to be widely presented in multi-label space (Read et al., 201 I), leading to the linear dependency of the feature mapping matrix W Therefore, researchers usually denote R(.)) as the low-rank constraint term to capture such intrinsic property.
To better characterize the low-rank tensor structure, we de- sign novel Laplace Tensor Rank (LTR), defined as follows:
Given tensor W T X then the Laplace Tensor Rank (LTR) is defined as:
Definition 1.
Unfortunately, although Eq. (1) may have explored the multi-label semantic relationships, still treats each view independently, ignoring the feature correlations among dif- ferent views. To address this limitation, not only should we keep the low-rank constraint for each W but also need to further ensure the consensus principle by imposing low- rank across all views. Motivated by the fact that tensor can characterize the low-rank structure of multi-dimensional
| =
LTR
((.))
1- exexp exp
n1
8
(3)
where 0< 5 min(n1, n2) and Sy is obtained by t-SVD of W, = UT in Fourier domain.
3

---

Tensorized Multi-Vicss Multi-Label Classification via Laplace Tensor Rank
unconstrained Lagrangian optimization problem:
True rank
TNN
Rank Value
6-0.1 LTSpN p-1 6-0.1
K [(WW 1== E", (A"))= 1
G, (Z') B",)
Σ S 2 z" 1 + a E 2.11 + + 1911LT
(5)
V
0
+ L || W - G ll
Σ
(B", X - -E"
Singular Value
2
-=1
Figure 3. The comparisons of different methods approximate the true rank function as the singular value increases.
+ ト | x A'A E" 12 + (c,
2
where tenser C' and matrix B" are two Lagrange multipliers. I and p are penalty parameters. Then, we solve the variables in Eq. (5) through the following five subproblems.
Remark 2.
[The superiority of LTR) The approximation 2x
function used by LTR is [1) 1- exp (-)) which is inspired by the Laplace function (Lu et al., 2015). Ba- sically, [(T = satisfied, which is consistent with the true rank function. We compare LTR with other exist- ing methods (i.e., TNN (Xie et al., 2018), L, (Guo et al., 2022) and ETR (Ji & Feng, 2025)). LTR approxi- mates the rank best in Figure 3, especially for larger and near-zero singular values. When x + +00, we obtain fLIN(Z) + 1, which is a better substitute for true rank than TNN, S and ETR. If x + it is easy to prove that EI >> loog11 + xx) >> x, which means L.TR 542
Z' -sbprobbee:: We solve the problem by separating the labeled and unlabeled parts, taking advantage of the diago- nal property of S. For the labeled part, with other variables fixed and taking the derivative of Eq. (5) with respect to Z" to zero, we can obtain:
2Y.(W%)T - W + +BA A ZAAAT E/AT)=
= (6)
0, where the subscript and u indicate variables corresponding to labeled and unlabeled data, respectively. We obtain the updating rule for Z as:
fira(1) Re
achieves a stronger penalization on near-zero singular val- ues. This signifies that [|W] LTR can robustly penalize small singular values ussociated with noise, preserving valuable large singular values and make sure that W has spatial low-rank structure to capture high-order fiber correlations.
z = 2Y + BAA + A T 2W"(W") + ml
(7)
Accordingly, for the unlabeled part, we update Z by:
Z B + A A
3.2.3. THE FINAL OBJECTIVE FUNCTION OF TMvML
(8)
By integrating Eq. (2) and Eq. (3), our proposed TMyML can be formulated as follows:
After obtaining Z and Z the common representation corresponding to all data Z" is obtained as Z" = [Z, ZНЛ.
min W.E.A 2
|S - z""") +o [B||,,1 + 8 [IlLER
E" -supproblbem: With other variables fixed, the subprob- lem for E can be formulated as
s.t. VU, X" = Z"A' + E", A'AA"""= II,
á E 2.1
1 E EI
= (W W E (E...., }
min E
(9)
(4)
where a and/ 3 are two trade-off parameters. The vertical concatenation along the column of eиror matrix, i.c., E [E':E" E :...: E can enforce the column of E" in each view to have jointly consistent magnitude values (Liu et al., 2012). To ensure the multi-view representation is predictive corresponding to the known labels, we design the filtering matrix S & R=x= to select the labeled samples with Sü = the i-th sample is labeled and S₃ = O otherwise.
where É is constructed by horizontally concatenating the matrices X A' + B" together along column. Ac- cording to (Liu et al., 2019), the solution can be achieved by applying the l2 minimization thresholding operator,
|1
E
E
(10)
otherwise.
where E.,, represents the i-th column of É.
3.3. Optimization
G-subproblem: With other variables fixed, the subproblem for G can be formulated as
By utilizing the principles of the alternating direction method of multipliers (ADMM) (Lin et al., 2011), we in- troduce a separable variable to transform Eq. (4) into an
p g (w + - S/p
min 3 ||||| m
LTR
(11)
2

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Tensorized Vilt Multi-Label Classification via Laplace Tensor Rank
Theorem 3.1. Suppose A E IR X with 1-SVD = U S VT and & > 0. The Laplace Tensor Rank Minimiza- tion problem can be described as follows,
Algorithm 1 The Training Process of TMvML
Input:
{X")==i Ju=i: Training examples:
Y: Label matrix;
1 min lilllLTRR + +119-A11 9
(12)
and 3: The trade-off parameters;
t: Number of iterations;
8: Convergence threshold.
Then, the optimal solution G is obtained as,
Output:
G' = U P (St), (13) VT,
(13)
Y: Predicted label matrix.
1: Initialize W", E", A", Z", B", g. C, μ and p;
where ifft(Proxj, fn (s [| 3) R X is an fdiagonal tensor, and prox (S) (i, i)) satisfies,
3: Compute the loss La by Eq. (4);
4: for i = 1,2,..., do
5: for -== 1 2,2...... V do
arg min (x - (i,) +(()), #20
(14)
6: Optimize Z" by Eq. (7) and Eq. (8);
7: Optimize E" by Eq. (10);
Optimize w by Eq. (19);
where f(x)= = exp
Optimize A by Eq. (20);
10:
Optimize B by Eq. (21);
11:
end for
Proof. In the Fourier domain, according to the linearity of FFT and the fact that || +
ni || l the objective function 2 g| All + g LTw can be rewritten as:
12:
Optimize G by Eq. (18);
13: Optimize C. μ and p by Eq. (21);
14: Optimize L, by Eq. (4);
15:
if |C, L S break;
||9 A + +111911.m7 ITR
16: end for
2
|gg- A + Y 1
2ng
(15)
where r = -rrx (S (1, 1)), and iter is the iterations.
W -u-subbrobleem: With other variables fixed and taking the derivative of Eq. (5) with respect W to zero, we have:
( || - A + >||s LTTR)
11
w = (z')² SZ" + +I) (-
(19)
Thus, the original tensor optimization problem can be de- coupled into ny independent matrix optimization problems:
+ + STSY + pG
A R subbprobbe::: With other variables fixed, the subprob- lem for A is formulated as
arg min 05 2
|10 - A$1 + >119
(16)
A arg min Tr((A))TNN MM), Ar(Au)T=I
(20)
Here, k n3, the SVD of A is U4 (V)) Since unitary transformations do not change the singular values, LTR functions that applied to singular values are evidently unitarily invariant functions. According to (Kang et al., 2015), the optimal solution of Eq. (16) is:
where M ( + B -.E"). The optimal solution of A" is (V")",, where U and V are the left and right singular matrix of M Finally, the Lagrange multipliers and penalty parameters are updated as follows,
97 = (rxx,
(17)
B B + + A E²),
C C= C - p(W - -g),
(21)
where (S0 (i, arg mig [xx (T.)
J MiH,, P = 1pP
(x() and f(x) = 1- e
where nae Thp are used to accelcrate convergence.
Given that Eq. (14) in Theorem 3.1 is a combination of con- cave and convex functions, we can utilize DC Programming (Tao& & An, 1997) to derive a closed-form solution:
During the whole model training process, we first initialize the required variables, and then repeat the above steps until the algorithm converges or reaches the maximum iterations. Finally, we make prediction for unseen instances according to Y = & E Z.W". Algorithm summarizes the whole procedure our proposed TMyML. method.
ster+1 = (s)( 3-0f(7iten)
(18)
5

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Tensorized Multi- View Multi-Label Classification via Laplace Tensor Rank
3.4. Computation Complexity Analysis
viation in Table 2, with the best performances highlighted in bold. From the 180 comparisons (6 datasets 6 comparing methods x 5 metrics), we can observe that:
The computation complexity of TMvML is mainly deter- mined by the solution of its five subproblems in section 3.3, where the time complexity spent on {Z", E", G. A", W are +nkd (ndd), O(dgV
Among the six employed datasets, TMvML outper- forms all comparing methods on Emotions, Yeast, Plant and Human datasets. And it also superior to other methods on Espgame and CorelSk datasets in 86.67% and 96.67% cases, respectively.
O(nk² + nkg). O(k² respectively. d is the maximum dimensionality of the multi-view data. In our experiments, since n > g. n > k n > dmax. the overall computation complexity of TMvML is (tnk?), where t is the number of iterations and usually no more than 50 on all datasets.
Among the five evaluation metrics, TMvML achieves the best performance on Average Precision, One Error and Coverage metrics in all cases. And on Ranking Loss and Hamming Loss metries, it also superior to other methods in 94. 44.44 and 91.67% cases.
4. Experiments
4.1. Experimental Setting
To validate the effectiveness of TMvML, we conducted in-depth experimental analysis on six widely-used MVML datasets, including Emotions, Yeast, Corel5k, Plant, Human and Espgame, which can be downloaded from Mulan web- site: http://muulnn.we www.caoooa..o.ororrr..... nwt/ddatases--mlee htmml. Table wwwrrrefoore.eeeedtttttttts--s
Compared with non-tensor methods (such as ICM2L), TMvML performs significant superiority all em- ployed datasets, which is attributed to its employed tensor architecture that can leverage the low-rank ten- sor property to capture higher-order correlations across views and labels.
I summarizes the detailed characteristies of these datasets.
Table 1. The characteristics of our employed datasets: the number of samples (n), views (1), classes (q) and the maximum / minimum dimension of all views (dmax, dmii)-
Compared with methods that employ independent com- ponents to mine view and label relationships, TMvML exhibits excellent performance on all datasets. We attribute such success to our designed tensor classi- fier, which can simultaneously excavate cross-view high-order correlations and characterize multi-label semantic relationships.
Datasets
n
V
dmax
dmin
Emotions
593
2
6
64
8
Plant
978
2
12
400
40
Yeast
2417
2
14
79
24
Human
3106
2
14
400
40
Corel5k
4999
260
4096
100
To comprehensively evaluate the superiority of TMvML., the Friedman test (Demšar, 2006) is utilized the statisti- cal test to determine whether multiple algorithms have the same performance. Table 3 shows the Friedman statistical Tr value for each evaluation metric and the critical value. According to Table 3, all evaluation metrics reject the null hypothesis that "all algorithms perform equally" at a 0.05 significance level. Thus, we choose the post-hoc Bonferroni- Dunn test (Demšar, 2006) to further illustrate the differences among these methods. Figure 4 illustrates the CD diagrams for each evaluation metric, where the average rank of each algorithm is marked along the axis. According to Figure 4, it is clearly observed that our proposed TMvML consistently ranks 1t across all evaluation metrics.
Espgame
20770
268
4096
100
For comparative studies, we employ six state-of-the-art MVML methods, including LrMMC (Liu et al., 2015), SIMM (Wu et al., 2019), FIMAN (Wu et al., 2020), ICM2L (Tan et al., 2021), I), ML-BVAE (Fu et al.. 2024) and IMvMLC (Wen et al., 2024), with parameters configured as recom- mended in their respective literature.
In addition, five widely-used evaluation metries in multi- label learning are employed to measure the performance of algorithms,, including Average Precision (AP), Hamming Loss (HI.), One Error (OE), Ranking Loss (RL.) and Cover- age (COV), where the formal definition of the above metrics can be found in (Gibaja & Ventura, 2015). For each dataset, we randomly selected 70% data for training, 10% data for validation and 20% data for testing.
5. Further Analysis
5.1. Ablation Study
To evaluate the contribution of our proposed Laplace Ten- sor Rank (LTR), we conducted additional ablation stud- ies. Specifically, we compared our proposed TMvML with two variants where LTR in Eq. (4) was replaced by the state-of-the Enhanced Tensor Rank (ETR) (Ji & Feng,
4.2. Experimental Results
To ensure reliable comparisons, we run each algorithm five times and record the average metric results and standard de-
6

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Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
Table Experimental results (meandistd), 1 represents the higher better; J represents lower the better.
metrics
LrMMC
SIMM
FIMAN
ML-BVVE
IMvMLC
ICM2L
TMVML
AP
0.763++0000
0.63440.003
0.806=0.027
0.572:+0.022
0.782=0.021
0.567=:00044
0.811=0.020
Emotions
HL 4
0.216:10.011
0.307:10.004
0.20140.013
0.31710.012
0.330=0.021
0.342:10.007
0.210
OE
0..03
0.5011:0.092
0.258:10.042
0.068
0.303=0.032
0.578:-0.016
0.248
RL
0..01
0.044:10.077
0.161
0.423+0.035
0.183=0.019
0.432:10.004
0.066
COV
2.1994+0.094
0.457-10.051
7.796:+0.189
3.163:+0.22
1.873=0.037
3.091
0.300
metries
LrMMC
SIMM
FIMAN
MI.-BVAR
IMvMICC
ICM21
TMVML.
AP1 t
0.61010.013
0.712100014
0.740=10.007
0.712:10007
0.738=0.006
0.69510.0001
0.771=1.0008
HI +
0.255+0.020 0.241+0.011
0.216+0.004
0.232+0.004
0.313=0.007
0.231+0.001
0.208
Yeast
OE
0.309+00028 0.223
0.257+00013
0.253+00012
0.248=0.009
0.264+0.001
0.215=0.007
RL
0.275+0.011
0.218+00018
0.187+00005
0.204
0.180=0.005
0.213+00001
0.167=0.007
COV
10.9
7.368
6.673.10.07
6.706.10.044
6.687100003
0.470
metrics
LrMMC
SIMM
FIMAN
ML-BVAE
IMvMLC
ICM2L
TMVML
AP
0.215+00010
0.292::0.004
0.430:10:077
0,286+0.000
0.333=0.008
0,210=100000
0.440=000008
Corel5k
HL
0.003+0.000
0.013+00018
0,018=0.000
0,013=00000
0.158=0.008
0.020=0.0000
0.013=10,000
OE
0.776100015
0.614:0.0012
0.489=+0.017
0.626=0.008
0.613=0.019
0.797=000000
0.476=20.13
RL
0.173++0.004 0.160:+00005
0.085++0.000
0.080
0.114=0.003
0.174:10.001
0.000
COV
96.72+1..300
95.99
53.94+
108.7+447992
70.50-2.256
98.383-+0.283
0.266
metries
LiMMC
SIMM
FIMAN
ML-BVAE
IMvMLC
ICM2L
TMVML
AP
0.46410.0016
0.36910.029
0.492-10.030
0.505
0.544=0.017
0.587100047
0.608-0.007
HL 4
0.115:10.02 0.090110.001
0.238:10.011
0.090100001
0.202=0.038
0.920:10.006
0.087:10.044
Plant
OE
0.668110.018
0.80910.037
0.69610.001
0.705
0.6$2-0.027
0.637100064
0.570-10.006
RL
J
0.301+00014 0.378+00025
0.277+00028
0.238+0.012
0.210=0.015
0.853+0.027
0.168=00015
COV
4.256+0.147 4.325
3.216+00.30
2.749+0.141
2.461=0.171
1.715=#0317
0.169=0.013
metrics
LrMMC
SIMM
FIMAN
ML-BVAE
IMvMLC
ICM2L
TMVML
AP
0.170:10.003
0.304:10.002
0.284-10.002
0.257
0.273-0.001
0.226:10.000
0.306
Espgame
HL +
0.008 0.017110.000
0.028:10.000
0.01711.000
0.021
0.027110.00
0.026:10.000
OE
0.992::0.001 0.536:10.004
0.628=t0.004
0.615=0.010
0.615=0.013
0,692:000000
0.506=0.0022
RL
0.410:10.003 0.1644:0.003
0.154:10.002
0.211
0.040=0.000
0.207
0.141=0.002
COV
210.911.0200
1.0.1
102.8+1.183
136.5 1.681
92.21=0.113
143. 124.0.11
0.409=100008
metrics
LiMMC
SIMM
FIMAN
ML-BVAE
IMvMLC
ICM2L
TMVML
AP
0.480+000006
0.495++0.02
0.583+:0015
0.536+0.010
0.600=0.010
0.603--0.034
0.631=00.010
Human
HI.
0.006+00.002
0.085++0.001
0.151
0.085+0.001
0.112=0.006
0.920+00.002
0.083
OE
0.673100.09
0.663.10.034
0.58510.020
0.65910.014
0.578-0.015
0.595110.054
0.542-10.016
RL 0.008
0.26110.046
0.186.10.011
0,18110.008
0.149-0.007
0,879100016
0.136110.004
COV
5.281 10.072
3.79710.640
2.817-10.055
2.666.10.112
2.271
1832.10.211
0.150-10.003
and TMvML-TNN in most cases and achieves the best clas- sification performance in the optimal setting. This phe- nomenon is attributed to the distinct ways these methods treat singular values. While LTR and ETR apply vary- ing penalties to individual singular values, TNN uniformly scales them all, thus overlooking differentiated information between large and small singular values in the tensor data. Notably, compared to ETR, LTR preserves larger singular values to 1, effectively characterizing the low-rank tensor structure. Additionally, since the parameter å affects the different constraints in varying ways, TMvML-ETR may outperform TMvMI. in few cases.
Table Friedman statics Ty each evaluation metric.
Evaluation Metric
TF
critical value (a=0.05)
Average Precision
8.696
Hamming Loss
4.805
2.421
One Error
7.714
Ranking Loss
13.129
Methods: 7, Data sets: 6
Coverage 7.600
2025), and the traditional Tensor Nuclear Norm (TNN) (Xie et al., 2018), denoted as TMvML-ETR and TMyML-TNN respectively. As shown in Figure 5, when varying the pa- rameter across the search range of $10......, ... our TMyML. method consistently outperforms TMvML-ETR
In addition, we further analyze the effect of the parameters on the classification results. As shown in Figure 5, we can observe that has a significant effect. The best classification
7

---

Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
Average Precision
Hamming Loss
One error
TM/ML FIMAN
JCMZL
TMVML
ICMEL
IMMLLC
ML
ICMEL
IMMIC
IMMMLC
FIMAN
SIMM
SIMM
FIMAN
ML-VVAE
UMMMC
ML-VAE
Ranking Loss
Coverage
TMYML
LIMMC
IMYMIC
ML-SWAL
IMMMLC
SIMM
FIMAN
SIMM
(CM2L
ML-EVAE
AMAN
Figure 4. Experimental comparisons between proposed TMvML method and all other comparing algorithms on five evaluation metrics with the Bonferroni- Dunoonnn--Dnnntttt test (CD -3211 0.05 significance level).
800
Precision
Average Procision
0.44
0.78
0.42
Convergence 3
Convergence
E00
0.78
0.4
400
2..44
0.30
TMAML
0.30
TM
200
0.72
TMR TMANL-THN
TMAALLETR
0.34
TMM/LL-TNN
0
16
The number of iterations (a) Emotions
The number of iltrations (b) Core15k
(a) Emotions
(b) Corel5k
Figure 5. The Average Precision of TMvML and TMrML-ETR with varying on Emotions and Corel5k datasets.
Figure 7. Convergence analysis of TMvML Emotions and Core15k datasets.
042
04 change the value of a within the set of {10-1 1 1o...... 10*}
and 33 within the set of (10-5, 10-1 10-1..... 10 According to Figure 6, the performance of TMvML is stable when hyperparameter
675
04
025
67
4
around 102 to 10 and hyperparameter [ is around 10-3 to 10 Such phenomenon illustrates that the performance of our proposed TMvML is stable across broad range of parameters, which also empirically demonstrates its efficiency and robustness.
47
14
5
(a) Yeast
(b) Corel5k
Figure 6. The parameter sensitivity analysis of TMvML under dif- ferent a and configurations on Yeast and Corel5k datasets.
5.3. Convergence Analysis
We further analyze the convergence of the our proposed TMvML. Figure 7 shows the convergence curves of TMvML method on the Emotions and Corel5k datasets. According to Figure 7, the value of the objective function drops sharply at the beginning of the iterations and gradually stabilizes as the number of iterations increases. Similar convergence results also observed in other datasets, which empirically verifies the convergence of TMvML.
results for Emotions are obtained when 10-4, while Corel5k peaks at 5 10 The main reasons of such phe- nomenon lie in that, since 8 determines the strength of the penalty for singular values, and the distribution of singular values varies from one dataset to another, each dataset needs an appropriate parameter to provide better discriminability power for the learned low-rank representation tensor.
5.2. Parameter Sensitivity
6. Conclusion
We study the sensitivity analysis of our proposed TMvML with respect to its two key hyperparameters a and B. We
In this paper, we proposed a Tensorized Multi- View Multi- Label Classification method via Laplace Tensor Rank
8

---

Tensorized Multi- View Multi-Label Classification via Laplace Tensor Rank
(TMvML), which is the first attempt to utilize tensorized low-rank MVML classifier achieve the high-order feature correlations extraction and multi-label semantic correlations characterization simultaneously. To better characterize the tensor low-rank structure, we designed a new Laplace Ten- sor Rank (LTR), which serves as tighter surrogate of tensor rank to effectively capturing high-order fiber correlations. Extensive results demonstrate that our proposed TMvML is significantly superior to other state-of-the-an art methods.
Ji, J. and Feng, S. Anchors crash tensor: Efficient and scalable tensorial multi-view subspace clustering. IEEE Transactions Pattern Analysis and Machine Intelli- gence, pp. 1-17, 2025.
Jiang, B., Xiang, J., Wu, X., He, W., Hong, L.,, and Sheng, W. multi- classification. In Proceedings of the 30th ACM International Conference on Information & Knowledge Management, pp. 3117- 3121, 2021.
Robust adaptive-
Impact Statement
Jiang, B., Wu, X., Zhou, X.,, Cohn, A. G., Liu, Y., Sheng, W., and Chen, H. Semi-supervised multi-view feature selection with adaptive graph learning. IEEE Transac- tions on Neural Networks and Learning Systems, 35(3):
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be
3615-3629, 2024.
specifically highlighted here.
Kang, Z., Peng, C., and Cheng, Q. Robust pea via noncon- vex rank approximation. In IEEE International Confer- ence on Data Mining, pp. 211 1-220, 2015.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (No, 62306020), the Young Elite Scien- tist Sponsorship Program by BAST (No. BYESS2024199), Beijing Natural Science Foundation (No. L244009), the Na- tional College Students' Innovation and Entrepreneurship Training Program of BJUT (No. GJDC2025-01-32) the Ma-
Lewis, A. S. and Sendov, H. S. Nonsmooth analysis of singular values. part i: Theory. Set- Valued Analysis, 13: 213-241, 2005.
Lin, Z. Chen, M., and Ma, Y. The augmented lagrange multiplier method for exact recovery of corrupted low- rank matrices. arXiv preprint arXiv:1009.5055, 2010.
jor Program of the National Social Science Foundation of China (No. 22&ZD147), and the National Key Research and Development Program of China (No. 2023YFB3007000).
Lin, Z., Liu, R., and Su, Z. Linearized alternating direction method with adaptive penalty for low-rank representation. Advances in Neural Information Processing Systems, 24, 2011.
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Tensorized Multi- View Multi-Label Classification via Laplace Tensor Rank
Zhao, D., Gao, Q., Lu, Y., and Sun, D. Non-aligned multi- view multi-label classification via learning view-specific labels. IEEE Transactions on Multimedia, 25:7235-7247, 2023.
Zhong, Q.. Lyu, G., and Yang. Z. Align while fusion: A generalized nonaligned multiview multilabel classifica- tion method. IEEE Transactions on Neural Networks and Learning Systems, 2024.

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Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
A. Proofs of Theorem 3.2
Theorem A.1. Let = g E Ag. W B G6) be the sequence generated by Algorithm 1, then the sequence
{P&] meets the following two principles:
1). (Pk) is bounded;
2). Any accumalation point of {Ps) is a KKT point of Algorithm 1.
To prove Theorem 3.2, first introduce important lemmas.
Lemma A.2. (Lin et al., 2010) Let be real Hilbert space endowed with an inner product (:,)), a norm || with the dual norm || [| and y E 0||=|| where (x)x) is the subgradient of f(-). Then we have [|y]] 1 if # 0, and |yy| Idual
< =.
Lemma A.3. (Lewis & Sendov, 2005) Suppose that F RRXX +
defined as F(X) ʃ o o(X) (x..,, (X)), where X UDiag( (X)))(((())))) V is the SVD of matrix Rmxx min(m, n), and f Rº R
differentiable and absolutely symmetric o (X). Then, the subdifferential of F(X) at X is
( UDiag (0
0X
where
Of(o(X)) af(a)()) (x))
(()))))
(f(a,(z))
0X
0x
Proof. Proof of the first part: On the k + 1 iteration, from the updating rule of Es+1, the first-order optimal condition should be satisfied.
0 ad E 2,
+ pa (E - (X" - Zk+1" "A" + Bz/pz))
(22)
= ad E 11 B
Thus, we have
1 B B 0 E 1
where 11 and Iki are the j-th columns of B and E And the & norm self-dual, so based on the Lemma A.2, we have | [ (1 2 1. So the sequence (B*+1) is bounded.
Then, we prove the sequence [Co41} is bounded. According to the updating rule of G, the first-order optimality condition holds
a 119 C(+1.
(23)
Let 24 S TT be the t-SVD tensor g. According to the Lemma and definition of LTR, we have:
12

---

Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
||
9k++ LTR
||
u * ( * VT
n3
[1S((S))1i
(24)
1 ng min(ai m2))
[0S(5)(.511]
n2
26 min(ni, n2)
82
where the second inequality by the fact 0f(x)) S 5. and (LTR(Z) = exp So || IluTR is bounded, meanwhile the sequence {Ck+1} is also bounded. Moreover, from the iterative method in the algorithm of solving TMvML, we can deduce
d (- our rank approximation function.
Pp (Z W
k+1: E E A 1 9k+1, B C&)
fpecPP (Z) W ky
E A 9k, B Cx)
Cpe--1Pp-11
[Zg. W Eg Ag gk, B B
+ Pk + Pk-1
|C - Cx-1 }
(25)
2P²
14 + pAk-1
Σ [ BE
2/16211
b=1
Thus, summing sides of (25) form h: = to
Cu (Z Wo WRTH K+ E Ag + B Cx)
Lno (Z); Wi, E AW, 91, B.
Pk + Pk-1
1CC C 11
2p
(26)
V
Pk + Mk
Σ
B - B 11
2p
w=1
Observe that
Pk + pk+1 < 00,
Pk + Px+1
< 00
(27)
24
2p
k=1
Note that Cup.po (Z), WT, Ey, AJ, 91, By Co) is finite, and sequence (B:), (Cx). En= all bounded. So Curips (Z W Eg A A Ck+1, By, Ck) is bounded.
0a+ 245_.
and
Σ
Pk+ 2p
are
Notice
13

---

Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
pue.pe (Z W + 1
96+1, B B Cx.
= Bk+1 2,,1 + 3 G + Σ S (Y Z+++ W W 1
(28)
(B X" A E Ex+1)
1 Z A Ex+1l
+ (Cx: Wk+1 GK+1) Pk | 9xtill
and each term of Eq.(28) is nonnegative, due the boundedness of CuR (Zi+1) 1 E Ag Gu+I, Bi, Ck), deduce each term of (28) is bounded. So the boundedness of || l|T implies that all singular values of Ge41 bounded. Furthermore, based on the following equation
min(n1:ng)
- 113
119,
Σ
(S)(.)))²
(29)
113
j=1
we can derive the sequence {9x+1} is bounded, then, it easy to prove the boundedness of (Z++1} and (As+1).
Therefore, conclude that the sequence (Z) E A W By Cs. 9k) generated by the Algorithm 1.
2). Proof of 2nd part: According to Weierstrass Bollzano theorem (Bartle & Sherbert, 2000), there is at least one accumulation point of the sequence (Px [ 20 we denote one of the points as P.. Then we have
lim (Z) Eg, A W B Ca. 9m) = (Z, E,, A. W W,, C,, k
Form the updating rule of C and B", we have the following equations:
X Z k+1 A K+1 Ex+1 = (B - B/)/
W4+1 Gk+1 (Cx+1 Cc)/Pt.
According the boundedness of sequences {Cx} and [Bg], and the fact linu- Me oo, we have:
lim X
Z A
E = lim (B BB)/ = 0, 1+000
k-+00
lim Wk-1 Gu+1 lim (C++1 Cx)/ 0, k
1-+00
then, we can obtain
X ZEA"" E. =0 W. 0.
Furthermore, due to the first-order optimality conditions of E and GR+1, we can deduce:
ad E 11.1 B B ad E 2
39 30119411llr CR+1 = C. = 30 G9 |
Thus, the accumulation point P, of sequence (Pa generated by the algorithm of solving TMvML satisfied the KKT condition.
14

---

Tensorized Multi-View Multi-Label Classification via Laplace Tensor Rank
B. More Experimental Results
This section presents the experimental results for all six challenging datasets, including the ablation study of tensor rotation and Laplace Tensor Rank (LTR).
Ablation Study of LTR: Table (4) reports the results of ablation experiments of LTR conducted across six datasets. Specifically, table (4) compares TMvML with its variant TMvML-TNN, where the LTR was replaced by Tensor Nuclear Norm (TNN) (Xie et al., 201 8) to capture the low-rank tensor structure. Experimental results demonstrate that TMvML consistently outperforms TMvML-TNN across all datasets. This compelling evidence proves that our proposed LTR is more effective than traditional TNN in modeling the complex high-order correlations multi-view multi-label learning tasks.
Table The Performance of TMvML TMvML-TNN
Emotions
Yeast
Corel5k
Plant
Espgame
Human
TMvML
AP
0.811
0.771
0.440
0.608
0.306
0.631
TMvML-TNN
AP
0.738
0.747
0.382
0.601
0.270
0.621
TMvML
Cov 0.300
0.460
0.266
0.169
0.409
0.150
TMyML-TNN
Cov
0.344
0.467
0.279
0.171
0.452
0.162
Ablation Study of Tensor Rotation: Table (5) reports the results of ablation experiments of tensor rotation conducted across six datasets. Specifically, Table (5) compared TMvML with variant that removes the rotation operation (denoted as TMoMooNNRRo).. The results show that TMvML. consistently outperforms TMvMLI-NoRot across all datasets. This significant performance gap highlights the critical role of rotation extraction of both cross-view consistent correlations and multi-label semantic relationships simultaneously.
Table 5. The Performance of TMvML and TMvML-NoRol
Emotions
Yeast
Core15k
Plant
Espgame
Human
TMvML
AP 0.811
0.771
0.440
0.608
0.306
0.631
TMvML-NoRot
AP 0.628
0.733
0.231
0.511
0.191
0.532
TMvML
Cov
0.300
0.470
0.266
0.169
0.409
0.150
TMvML-NoRot
Cov
0.473
0.485
0.350
0.210
0.498
0.185
15


Reviews:
[{"review_id": "7IDorNDpcP", "reviewer": "Reviewer_miAu", "summary": "This paper proposes a Tensorized Multi-View Multi-Label Classification (TMvML) method to address the limitations of existing approaches that independently model cross-view consistent correlations and multi-label semantic relationships in MVML learning. The method reconstructs multi-view multi-label mapping matrices into a tensor classifier, where tensor rotation and low-rank constraints are jointly applied to unify view-level feature consistency and label-level semantic co-occurrence. Moreover, Laplace Tensor Rank is designed as a tight surrogate of tensor rank to capture high-order fiber correlations. The experimental results demonstrate the effectiveness of the proposed framework.", "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": "**W1**: The convergence of TMvML is guaranteed through the validation presented in Theorem 1, with comprehensive and rigorous proof below.\n\n**Theorem 1**: Let $\\\\{\\mathcal{P}_k = ({\\bf Z}_k^{v}, {\\bf E}_k^{v}, {\\bf A}_k^{v}, {\\bf W}_k^{v}, {\\bf B}_k^{v}, \\mathcal{C}\\_k, \\mathcal{G}\\_k)\\\\}\\_{k=0}^{\\infty}$ be the sequence generated by Algorithm 1, then the sequence $\\{\\mathcal{P}_k\\}$ meets the following two principles:\n\n1). $\\\\{\\mathcal{P}_k\\\\}$ is bounded;\n\n2). Any accumulation point of $\\\\{\\mathcal{P}_k\\\\}$ is a KKT point of Algorithm 1.\n\nTo prove Theorem 1, we first introduce two lemmas.\n\n**Lemma 1** [1]: Let $\\mathcal{H}$ be a real Hilbert space with inner product $\\langle \\cdot, \\cdot \\rangle$, norm $\\\\|\\cdot\\\\|$, and dual norm $\\\\|\\cdot\\\\|^{dual}$. For $y \\in \\partial \\\\|x\\\\|$, we have $\\\\|y\\\\|^{dual} = 1$ if $x \\neq 0$ and $\\\\|y\\\\|^{dual} \\leq 1$ if $x = 0$.\n\n**Lemma 2** [2]: Let $F: \\mathbb{R}^{m \\times n} \\to \\mathbb{R}$ be defined as $F(\\mathbf{X}) = f(\\sigma(\\mathbf{X}))$, where ${\\bf X} = {\\bf U} \\mathrm{Diag}(\\sigma({\\bf X})) {\\bf V}^T$ is the SVD of ${\\bf X}$, $r = \\min(m,n)$, and $f: \\mathbb{R}^r\\to\\mathbb{R}$ is differentiable and absolutely symmetric at $\\sigma(\\mathbf{X})$.  Then,\n\n$\\frac{\\partial F(\\mathbf{X})}{\\partial\\mathbf{X}}=\\mathbf{U}\\mathrm{Diag}(\\partial f(\\sigma(\\mathbf{X}))) \\mathbf{V}^T.$\n\nwhere $\\partial f(\\sigma(\\mathbf{X})) = (\\frac{\\partial f(\\sigma_1(x))}{\\partial \\mathbf{X}}, \\dots, \\frac{\\partial f(\\sigma_r(x))}{\\partial\\mathbf{X}}).$\n\n**Proof of the first part**: On the $k+1$ iteration, from the updating rule of $ \\mathbf{E}_{k+1}$, the first-order optimal condition should be satisfied.\n\n$0=\\alpha\\partial\\left\\\\|{\\bf E}_{k+1}^v\\right\\\\|\\_{2,1}+\\mu_k({\\bf E}\\_{k+1}^v-({\\bf X}^v-{\\bf{Z}\\_{k+1}}^v{\\bf A}^v+{\\bf B}\\_k^v/\\mu\\_k))=\\alpha\\partial\\left\\\\|{\\bf E}\\_{k+1}^v\\right\\\\|\\_{2,1}-{\\bf B}\\_{k+1}^v,$\n\nThus, we have\n\n$\\frac{1}{\\alpha}[\\mathbf{B}\\_{k+1}^{v}]\\_{i,j}=\\partial\\left\\\\|\\left[\\mathbf{E}\\_{k+1}^{v}\\right]\\_{:,j}\\right\\\\|\\_{2},$\n\nThe $\\ell\\_{2}$ norm is self-dual, so based on the Lemma 1, we have $\\left\\\\|\\frac{1}{\\alpha}[\\mathbf{B}\\_{k+1}^{v}]\\_{:,j}\\right\\\\|\\_{2}\\geq1$. So the sequence $\\\\{\\mathbf{B}\\_{k+1}^{v}\\\\}$ is bounded.\n\nNext, according to the updating rule of $\\mathcal{G}$, the first-order optimality condition holds:\n\n$\\partial \\\\|\\mathcal{G}\\_{k+1}\\\\|\\_{LTR}=\\mathcal{C}\\_{k+1}$\n\nLet $\\mathcal{U} * \\mathcal{S} * \\mathcal{V}^{T}$ be the t-SVD of tensor $\\mathcal{G}$. Based on Lemma 2 and the definition of LTR, we have:\n\n$\\\\|\\partial\\\\|\\mathcal{G}\\_{k+1}\\\\|\\_{\\text{LTR}}\\\\|\\_F^2\\leq\\frac{e^{2\\delta}\\min(n\\_1,n\\_2)}{\\delta^2 n\\_3^2}$\n\nThus, $\\\\{\\mathcal{C}\\_{k+1}\\\\}$ is bounded.\n\nBased on the iterative method constructed in the algorithm, we can deduce:\n\n$\\mathcal{L}\\_{k}(\\mathbf{Z}\\_{k+1}^v, \\mathbf{E}\\_{k+1}^v, \\mathbf{A}\\_{k+1}^v, \\mathbf{W}\\_{k+1}^v, \\mathcal{G}\\_{k+1}, \\mathbf{B}\\_k^v, \\mathcal{C}\\_k) \\leq \\mathcal{L}\\_{k-1} + \\frac{\\rho\\_k + \\rho\\_{k-1}}{2 \\rho\\_{k-1}^2} \\\\| \\mathcal{C}\\_k - \\mathcal{C}\\_{k-1} \\\\|\\_F^2 + \\frac{\\mu\\_k + \\mu\\_{k-1}}{2 \\mu\\_{k-1}^2} \\sum\\_v \\\\| \\mathbf{B}\\_k^v - \\mathbf{B}\\_{k-1}^v \\\\|\\_F^2$\n\nSumming both sides results in: $\\mathcal{L}\\_k$ is bounded, and consequently, all its components are bounded, including  $\\\\| \\mathcal{G}\\_{k+1}\\\\|\\_{\\text{LTR}}$. Then, the boundedness of $\\\\{\\mathcal{G}\\_{k+1}, \\mathbf{Z}\\_{k+1}, \\mathbf{A}\\_{k+1}\\\\}$ is easy to prove. Therefore, the sequence $\\\\{\\mathcal{P}\\_k\\\\}$ is bounded.\n\n**Proof of the second part:** By the Weierstrass-Bolzano theorem [3], there exists at least one accumulation point of the sequence $\\\\{{\\mathcal{P}\\_k}\\\\}_{k=1}^{\\infty}$, denoted as $\\mathcal{P}\\_*$. Then, we have:\n\n$\\lim_{k\\to\\infty}({\\bf Z}\\_k^v, {\\bf E}\\_k^v, {\\bf A}\\_k^v, {\\bf B}\\_k^v, {\\bf W}\\_k^v, \\mathcal{C}\\_k, \\mathcal{G}\\_k)=({\\bf Z}\\_*^v, {\\bf E}\\_*^v,  {\\bf A}\\_*^v, {\\bf B}\\_*^v, {\\bf W}\\_*^v, \\mathcal{C}\\_\\*,\\mathcal{G}\\_\\*)$\n\nFrom the update rules of $\\mathbf{B}\\_k^v$, $\\mathcal{C}\\_k$, with $\\\\{\\mathbf{B}\\_k^v\\\\}$, $\\\\{\\mathcal{C}\\_k\\\\}$ bounded and the fact $\\lim_{k\\to\\infty}\\mu_{k}=\\infty$, we obtain:\n\n$\\lim_{k\\to\\infty}(\\mathbf{X}^v-\\mathbf{Z}\\_{k+1}^v \\mathbf{A}\\_{k+1}^v - \\mathbf{E}\\_{k+1}^v) = 0 \\Rightarrow \\mathbf{X}^v = \\mathbf{Z}\\_*^v \\mathbf{A}\\_*^v + \\mathbf{E}\\_*^v$\n\n$\\lim_{k \\to \\infty} (\\mathcal{W}\\_{k+1} - \\mathcal{G}\\_{k+1}) = 0 \\Rightarrow \\mathcal{W}\\_* = \\mathcal{G}\\_*$\n\nCombining the first-order optimality conditions of $\\mathbf{E}\\_{k+1}^v$ and $\\mathcal{G}\\_{k+1}$, we take the limit and obtain:\n\n$\\mathbf{B}\\_*^v=\\alpha \\partial\\\\|\\mathbf{E}\\_*^v\\\\|\\_{2,1}, \\quad \\mathcal{C}\\_\\*=\\beta\\partial\\\\|\\mathcal{G}\\_\\*\\\\|\\_{LTR}$\n\nTherefore, the accumulation point $\\mathcal{P}\\_*$ generated by TMvML satisfies the KKT conditions.\n\n[1] The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices. arXiv, 2010.\n\n[2] Nonsmooth analysis of singular values. Part I: Theory. Set-Valued Analysis, 2005.\n\n[3] Introduction to real analysis, Wiley New York, 2000.", "claims_and_evidence": "In this paper, the claims are supported:\n\n1. TMvML’s superiority over SOTA is validated via experiments (Table 2) and statistical tests (Friedman/Bonferroni-Dunn).\n\n2. LTR’s effectiveness is justified theoretically and empirically (Figure 3, Figure 5).", "methods_and_evaluation_criteria": "This paper uses the tensorized classifier for MVML, as tensors naturally model multi-dimensional relationships. The rotation operation cleverly reorients the tensor to align label-view interactions, addressing a key limitation of matrix-based methods. This paper evaluates the proposed framework on widely-used MVML benchmark datasets with five standard evaluation metrics and the results demonstrate the effectiveness of the method.", "theoretical_claims": "Theorem 3.1 is correct and clearly proven.", "experimental_designs_or_analyses": "1. This paper conducts extensive experiments on several datasets, and the experimental results demonstrate the effectiveness of the proposed framework.\n\n2. Ablation studies and parameter sensitivity tests rigorously validate the contributions of LTR and hyperparameter stability.", "relation_to_broader_scientific_literature": "The paper builds on foundational works in tensor-based multi-view learning and multi-label classification, but it uniquely integrates these two paradigms through a unified tensorized framework. It applies the tensor framework to the MVML problem for the first time.", "essential_references_not_discussed": "There are no essential references missing or overlooked in the paper's discussion of related work.", "other_strengths_and_weaknesses": "Strengths:\n\n- Originality: This paper applies the tensor framework to the MVML problem for the first time, advancing the field by unifying cross-view consistency and label semantics in a single tensor classifier. This addresses a critical gap in existing MVML methods, which often handle these aspects independently.\n\n- Experiments: The experiments are sufficient and the effectiveness of the proposed method is substantiated through these experiments.\n\n- Clarity: This paper is well organized and the proposed method is clearly written to understand. All experiments details are provided and the codes are also released.\n\nWeaknesses：\n\n- In Section 5.3, the authors provide a textual analysis of the convergence behavior of TMvML, supported by empirical convergence curves (Figure 7). While the empirical results demonstrate stable convergence across datasets, the paper would benefit from theoretical guarantees to further strengthen the credibility of the optimization process.", "other_comments_or_suggestions": "Please see the points under Weaknesses above.", "questions_for_authors": "Please see the points under Weaknesses above.", "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": "oj7MXogdaU", "reviewer": "Reviewer_d2CW", "summary": "This paper presented a method for Multi-View Multi-Label Learning (TMvML) which utilizes tensorized MVML classifier to achieve the high-order feature correlations extraction and multi-label semantic relationships characterization simultaneously. Moreover, a new Laplace Tensor Rank is designed to characterize a better low-rank tensor structure. Experiments show some good results.\n\n\n## update after rebuttal\nThank you for your response. The new explanations and experimental results have strengthened the evaluation. After reading the authors' response, I would like to raise my rating to \"Accept\".", "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": "Thank you for the feedback on our paper. We appreciate the time and effort you have put into reviewing our work. In this rebuttal, we respond to the concerns raised in the reviews.\n\n**W1(C1):** Existing tensor-based methods have primarily been applied to multi-view clustering tasks for mining higher-order feature correlations, while some matrix-based methods employ low-rank constraints to capture label semantic relevance. To the best of our knowledge, our proposed TMvML represents the first attempt to utilize tensor structures for MVML tasks, designed to simultaneously model both multi-view high-order correlations and multi-label co-occurrence patterns. The tensor formulation provides a natural and effective framework for capturing the intrinsic multi-dimensional relationships in MVML data that conventional matrix-based approaches cannot fully characterize.\n\nAlthough direct comparisons with other tensor methods are unavailable, we validate the effectiveness of our proposed Laplace Tensor Rank (LTR) by comparing TMvML with its variant TMvML-TNN, where we replace LTR with the traditional Tensor Nuclear Norm (TNN) for low-rank tensor structure approximation. Experimental results demonstrate that TMvML consistently outperforms TMvML-TNN across all datasets. This compelling evidence proves that our proposed LTR is more effective than traditional TNN in modeling the complex high-order correlations in multi-view multi-label learning tasks.\n\n|  |    | Emotions | Yeast | Corel5k | Plant| Espgame |Human|\n|-|-|-|-|-|-|-|-|\n|TMvML|AP| **0.811±0.020** | **0.771±0.008** | **0.440±0.008** | **0.608±0.007** | **0.306±0.001** | **0.631±0.010**|\n|TMvML-TNN|AP|0.738±0.014|0.747±0.014|0.382±0.015|0.601±0.017|0.270±0.020|0.621±0.007|  \n|TMvML|Cov| **0.300±0.070** | **0.460±0.002** | **0.266±0.006** | **0.169±0.013** | **0.409±0.008** | **0.150±0.003** |          \n|TMvML-TNN|Cov|0.344±0.032|0.467±0.009|0.279±0.003|0.171±0.012|0.452±0.009|0.162±0.005|\n\n\n\n**W2(C2):**  The introduction of the additional exponential term e^δ in the modified Laplace function, $f_{\\mathrm{LTR}}(x)=1-\\exp\\left(-\\frac{e^\\delta x}{\\delta}\\right)$, provides several key advantages over the original Laplace function, $f_{\\mathrm{Laplace}}(x)=1-\\exp\\left(-\\frac{x}{\\delta}\\right)$.\n\n- The modified function offers enhanced flexibility by allowing dynamic adjustment of the growth rate and magnitude through $e^δ$. When $δ$ is large, $e^δ$ amplifies $x$, making the function grow faster for small singular values. When δ is small, the effect of $e^δ$ is reduced, and the function behaves similarly to the original Laplace function. This adaptability makes the modified function more versatile in handling different data distributions.\n\n- The modified function exhibits faster convergence for large values of $x$ due to the exponential scaling $e^δ$, improving optimization efficiency in tasks involving large-scale data or high-dimensional tensors.\n\nThank you again for this valuable feedback. Please let us know if there is any additional information we can provide to assist with your evaluation. \n\n**W3(Q1):** We agree that theoretical guarantees for convergence are critical for ensuring the reliability and robustness of the optimization framework. We have added formal theoretical proofs ensuring convergence to a stationary point and reported the convergence theorem and its detailed proof in our rebuttal to **Reviewer miAu**.", "claims_and_evidence": "The claims are supported by evidence. TMvML’s superiority is evident in its consistent outperformance of existing methods.", "methods_and_evaluation_criteria": "The method design makes sense. Motivated by the fact that tensor can characterize the low-rank structure of multi-dimensional, tensorized MVML classifier can deal with MVML problem.", "theoretical_claims": "The proofs for theoretical claims in optimization are correct.", "experimental_designs_or_analyses": "The experimental designs and analyses are reasonable, with thorough benchmark datasets of varying scales and complexities. The experiment also performs relevant ablation experiments, convergence analysis and hyperparametric analysis.", "relation_to_broader_scientific_literature": "This approach aligns with recent efforts to enhance tensor rank approximations, but goes further by integrating multi-view and multi-label learning into a unified framework.", "essential_references_not_discussed": "All related works are cited or discussed in the paper.", "other_strengths_and_weaknesses": "**Strengths:**\n\n- The organization of this article is reasonable and well-written.\n\n- The proposed LTR offers a non-convex surrogate for tensor rank.\n\n- Extensive experiments demonstrate TMvML’s superior performance.\n\n**Weaknesses:**\n\n- Recent tensor-based MVML methods should be compared, which can judge whether TMvML’s gains stem from tensorization itself. A direct comparison with recent tensor-based MVML methods would provide clearer insights into the specific contributions of tensorization and help validate the effectiveness of the proposed framework.\n\n- The modified Laplace function used in LTR introduces an additional exponential term $e^{\\delta}$ compared with the original Laplace function. The authors need to elaborate on the specific advantages of this modification.\n\n- The paper lacks a dedicated convergence analysis with formal theoretical proofs. Especially, are there formal proofs or conditions ensuring convergence to a stationary point? Addressing these points would enhance the theoretical rigor of the paper.", "other_comments_or_suggestions": "- Recent tensor-based MVML methods should be compared, which can judge whether TMvML’s gains stem from tensorization itself. A direct comparison with recent tensor-based MVML methods would provide clearer insights into the specific contributions of tensorization and help validate the effectiveness of the proposed framework.\n\n- The modified Laplace function used in LTR introduces an additional exponential term $e^{\\delta}$ compared with the original Laplace function. The authors need to elaborate on the specific advantages of this modification.", "questions_for_authors": "Are there formal proofs or conditions that ensure the convergence to a stationary point?", "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": "7FOJ3lOmz0", "reviewer": "Reviewer_3DbN", "summary": "This paper proposes a method named TMvML for multi-view multi-label learning (MVML). The approach includes a low-rank tensor classifier to capture both consistent correlations across views and modeling complex multi-label relationships. Additionally, a new Laplace Tensor Rank (LTR) is introduced to capture higher-order correlations within the tensor space. This approach leads to significant improvements in MVML, as demonstrated by extensive experiments.", "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": "Thank you for the feedback on our paper. We appreciate the time and effort you have put into reviewing our work. In this rebuttal, we respond to the concerns raised in the reviews.\n\n**W1**: We agree that comparing the proposed Laplace Tensor Rank (LTR) with the widely used Tensor Nuclear Norm (TNN) is essential to highlight the advantages of our method. In fact, we have already performed a comparison of the LTR function with the TNN function and the results are shown in Fig. 3. According to the figure, LTR provides a tighter approximation to the true rank function compared to TNN, especially for larger singular values. Theoretically, LTR’s nonconvex formulation more aggressively suppresses small (noise-corrupted) singular values while preserving larger (signal-carrying) ones, leading to a more accurate low-rank representation. This property is not fully captured by TNN, which tends to over-penalize larger singular values due to its convex nature. \n\nHowever, we fully agree with the reviewer that comparing LTR with TNN in the experiments would better highlight its effectiveness. Thus, we compared TMvML with its variant TMvML-TNN, where the LTR was replaced by TNN to capture the low-rank tensor structure, and we report the results for the numerical experiments:\n\n|  |    | Emotions | Yeast | Corel5k | Plant| Espgame |Human|\n|-|-|-|-|-|-|-|-|\n|TMvML|AP| **0.811±0.020** | **0.771±0.008** | **0.440±0.008** | **0.608±0.007** | **0.306±0.001** | **0.631±0.010**|\n|TMvML-TNN|AP|0.738±0.014|0.747±0.014|0.382±0.015|0.601±0.017|0.270±0.020|0.621±0.007|  \n|TMvML|Cov| **0.300±0.070** | **0.460±0.002** | **0.266±0.006** | **0.169±0.013** | **0.409±0.008** | **0.150±0.003** |          \n|TMvML-TNN|Cov|0.344±0.032|0.467±0.009|0.279±0.003|0.171±0.012|0.452±0.009|0.162±0.005|\n\nExperimental results demonstrate that TMvML consistently outperforms TMvML-TNN across all datasets. This compelling evidence proves that our proposed LTR is more effective than traditional TNN in modeling the complex high-order correlations in multi-view multi-label learning tasks, particularly in preserving discriminative singular values while suppressing noise-corrupted ones.\n\n\n\n\n**W2**: We agree that ablation study to validate the rotation operation in the tensor classifier construction would provide concrete evidence of the rotation’s contribution to capturing label consistency and view correlations. We compared TMvML with a variant that removes the rotation operation (denoted as TMvML-NoRot), and the results are summarized below:\n\n|  |    | Emotions | Yeast | Corel5k | Plant| Espgame |Human|\n|-|-|-|-|-|-|-|-|\n|TMvML|AP| **0.811±0.020** | **0.771±0.008** | **0.440±0.008** | **0.608±0.007** | **0.306±0.001** | **0.631±0.010** |\n|TMvML-NoRot|AP|0.628±0.021|0.733±0.021|0.231±0.002|0.511±0.015|0.191±0.001|0.532±0.005|\n|TMvML |Cov| **0.300±0.070** | **0.470±0.002** | **0.266±0.006** | **0.169±0.013** | **0.409±0.008** | **0.150±0.003** |\n|TMvML-NoRot|Cov|0.473±0.005|0.485±0.004|0.350±0.002|0.210±0.007|0.498±0.000 |0.185±0.002|\n\nThe results show that TMvML consistently outperforms TMvML-NoRot across all datasets. This significant performance gap highlights the critical role of rotation in extraction of both cross-view consistent correlations and multi-label semantic relationships simultaneously.\n\n**W3**: Thank you for pointing this out. We will enlarge the font of the coordinate axes to improve readability and ensure better clarity in the visualization.", "claims_and_evidence": "The paper's main claims are supported by convincing evidence. Extensive experiments on six datasets demonstrate TMvML’s superiority over state-of-the-art methods across multiple metrics.", "methods_and_evaluation_criteria": "The proposed methods make sense for the problem. TMvML innovatively leverages tensor classifier to encode high-order correlations across both multi-view and multi-label, while the Laplace Tensor Rank (LTR) constraint effectively balances the preservation of critical semantic relationships and the suppression of noise. The experimental evaluation utilizes widely recognized MVML benchmark datasets and state-of-the-art baseline methods.", "theoretical_claims": "I checked the correctness of the proofs for theoretical claims, including the theorems and proofs related to the effectiveness of LTR and closed-form solution in optimization.", "experimental_designs_or_analyses": "I checked the validity of the experimental designs and analyses. Extensive experiments are conducted on six widely-used MVML benchmark datasets, with results averaged over multiple runs to ensure statistical reliability. The issues are listed behind in the Weaknesses.", "relation_to_broader_scientific_literature": "The method TMvML is the first attempt to utilize tensorized low-rank MVML classifier in MVML problem.", "essential_references_not_discussed": "There are no related works that are not currently discussed in the paper.", "other_strengths_and_weaknesses": "The paper proposes a Tensorized Multi-View MultiLabel Classification method (TMvML), which is the first attempt to utilize tensorized low-rank MVML classifier to achieve the high-order feature correlations extraction and multi-label semantic correlations characterization simultaneously. Meanwhile, the paper designs a new Laplace Tensor Rank (LTR), which serves as a tighter surrogate of tensor rank to effectively capturing high-order fiber correlations.\n\nThere are also some weaknesses：\n\n1. In tensor-based methods, the Tensor Nuclear Norm (TNN) is commonly used to capture the low-rank structure of tensors [1,2]. The proposed Laplace Tensor Rank (LTR) should be compared with TNN in the experiments to better highlight its effectiveness and potential advantages.\n\n2. The proposed tensor classifier construction involves merging view-specific mapping matrices and rotating the resulting tensor to align label-view interactions. While this rotating design is theoretically motivated, ablation studies would strengthen the claim that rotation is essential for capturing label consistency and view correlations. Such experiments would provide concrete evidence of the rotation operation’s contribution to the method’s overall performance.\n\n3. In Fig 6, the font of the coordinate axis should be enlarged.\n\n[1] Zhao S, Wen J, Fei L, et al. Tensorized incomplete multi-view clustering with intrinsic graph completion[C] // Proceedings of the AAAI Conference on Artificial Intelligence. 2023, 37(9): 11327-11335.\n\n[2] Zhang C, Li H, Lv W, et al. Enhanced tensor low-rank and sparse representation recovery for incomplete multi-view clustering[C] // Proceedings of the AAAI conference on artificial intelligence. 2023, 37(9): 11174-11182.", "other_comments_or_suggestions": "I would like to learn about the authors' response to the weaknesses listed above, which may give me a clearer perspective on the paper's contribution.", "questions_for_authors": "I would like to learn about the authors' response to the weaknesses listed above, which may give me a clearer perspective on the paper's contribution.", "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": "OP5bAAl7j9", "reviewer": "Reviewer_DSYf", "summary": "In this paper, the authors propose a novel approach that introduces a low-rank tensor classifier combined with the innovative Laplace Tensor Rank (LTR), which jointly captures high-order feature correlations and label dependencies. Extensive experiments across six benchmark datasets demonstrate TMvML’s superior performance.", "questions": "", "limitations": "", "rating": 5, "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": "Thank you for the feedback on our paper. We appreciate the time and effort you have put into reviewing our work. In this rebuttal, we respond to the concerns raised in the reviews.\n\n**W1:** In this figure, we tested the ability of multiple low-rank tensor norms (including TNN, ETR, LTSpN and LTR) to approximate the true rank of three-dimensional tensors. Specifically, we constructed **a series of three-dimensional tensors with a fixed true rank of 3, while varying singular values that progressively increase from 0 to 9.** The horizontal axis represents the singular values, while the vertical axis represents the approximation value of the rank function. Such setup allows us to evaluate how well each method approximates the true rank under different singular value distributions. \n\nWe will revise the figure to include a detailed caption explaining the experimental setup. We hope these changes address the reviewer’s concerns.\n\n**W2:** To clarify, our learning process follows a **transductive learning paradigm**, where the model leverages both labeled and unlabeled data during training but only uses the labels from the training set for supervision. The matrix $\\bf S$ is a filtering matrix designed to ensure that the optimization process only utilizes the label information from the training set, while excluding any label information from the test set. Specifically, $\\bf S$ is defined as a diagonal matrix:\n\n$\\mathbf{S}_{ii}=\\begin{cases}1&\\text{if the }i\\text{-th sample belongs to the training set,}\\\\\\\\0&\\text{otherwise (samples belongs to the test set).}\\end{cases}$", "claims_and_evidence": "Yes, the central claims are supported by evidence. Extensive experiments across six benchmark datasets demonstrate TMvML’s superior performance. The superiority of LTR over existing tensor rank approximations is validated through ablation studies.", "methods_and_evaluation_criteria": "Yes, the methodological design is reasonable. The rotation operation on the tensor classifier is a clever design choice, enabling the exploration of interactions between different views and labels through frontal slice comparisons.", "theoretical_claims": "Yes, the proofs for Theorem 3.1 is correct.", "experimental_designs_or_analyses": "Yes, the experimental designs is reasonable. The use of six widely adopted MVML datasets and five standard metrics ensures a comprehensive and fair comparison. The inclusion of statistical tests further strengthens the reliability of the results.", "relation_to_broader_scientific_literature": "TMvML builds on prior work in tensor-based method and multi-label classification. The paper extends the principles of low-rank representation to the MVML setting, leveraging a novel tensorized classifier to capture high-order correlations across views and labels.", "essential_references_not_discussed": "All relevant works critical to understanding the main contribution of the method are cited in the paper.", "other_strengths_and_weaknesses": "Strengths: \n\n(1) This work innovatively leverages a concise low-rank MVML tensor classifier to excavate cross-view feature correlations and characterize multi-label semantic relationships simultaneously. The whole paper is well organized and easy to understand.\n\n(2) This paper designs a new Laplace Tensor Rank, which preserves larger singular values and discards smaller ones to obtain an accurate low-rank tensor representation. Such new component is welcome to multi-view community.\n\nWeaknesses：\n\n(1) The Figure 2 is ambiguous. Unclear labeling of vertical and horizontal coordinates, for example, why the true rank is 3 and the singular value gets progressively larger from 0 to 9.\n\n(2) Some expressions are not accurate enough. For example, The matrix S is designed to ensure the multi-view representation is predictive corresponding to the known labels, and it is not clear.", "other_comments_or_suggestions": "See the above weaknesses.", "questions_for_authors": "See the above 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": {}}]

Author rebuttal:

Meta-review: This paper proposes Tensorized Multi-View Multi-Label Classification (TMvML), introducing a novel Laplace Tensor Rank (LTR) to jointly model high-order feature correlations and multi-label dependencies. The method is evaluated on six benchmark datasets, demonstrating superior performance over existing approaches.

TR provides a tighter low-rank surrogate than traditional tensor norms (e.g., TNN). Extensive experiments (ablation studies, sensitivity tests, statistical significance) support claims. The paper makes a significant contribution to MVML by introducing a unified tensor framework with rigorous theory and empirical validation. All reviewers were satisfied with the rebuttal, and the revised manuscript (with minor clarifications) will strengthen the final version.
Decision: Accept (poster)