Orthogonal Non-negative Tensor Factorization based Multi-view Clustering
Authors: Jing Li
Quanxue Gao
QIANQIAN WANG
Ming Yang
Wei Xia
Venue: NeurIPS 2023 poster (2023)
Keywords: Multi-view clustering
tensor Schatten p-norm
non-negative matrix factorization.

Abstract: Multi-view clustering (MVC) based on non-negative matrix factorization (NMF) and its variants have attracted much attention due to their advantages in clustering interpretability. However, existing NMF-based multi-view clustering methods perform NMF on each view respectively and ignore the impact of between-view. Thus, they can't well exploit the within-view spatial structure and between-view complementary information. To resolve this issue, we present orthogonal non-negative tensor factorization (Orth-NTF) and develop a novel multi-view clustering based on Orth-NTF with one-side orthogonal constraint. Our model directly performs Orth-NTF on the 3rd-order tensor which is composed of anchor graphs of views. Thus, our model directly considers the between-view relationship. Moreover, we use the tensor Schatten $p$-norm regularization as a rank approximation of the 3rd-order tensor which characterizes the cluster structure of multi-view data and exploits the between-view complementary information. In addition, we provide an optimization algorithm for the proposed method and prove mathematically that the algorithm always converges to the stationary KKT point. Extensive experiments on various benchmark datasets indicate that our proposed method is able to achieve satisfactory clustering performance.

Orthogonal Non-negative Tensor Factorization based Multi-view Clustering
Jing Li
Quanxue Gao Xidian University Xi' an, Shaanxi, China qxgao@xidian edw.. cn
Qianqian Wang
Xidian University
Xidian University
Xi' an, Shaanxi, China jinglxdssuu. xidian... edu. cn
Xi'an, Shaanxi, China qqwangaxidien ainnin .du cn
Ming Yang Harbin Engineering University Harbin, Heilongjiang, China yangmingmathh@mail.. com com
Wei Xia Xidian University
Xi' Xi'an, Shaanxi, China
xdweixia@gmail. com
Abstract
Multi-view clustering (MVC) based non-negative matrix factorization (NMF) and its variants have attracted much attention due to their advantages in clustering interpretability. However, existing NMF-based multi-view clustering methods perform NMF on each view respectively and ignore the impact between-view. Thus, they can't 't well exploit the within-view spatial structure and between-view complementary information. To resolve this issue, we present orthogonal non- negative tensor factorization (Orth-NTF) and develop novel multi-view clustering based on Orth-NTF with one-side orthogonal constraint. Our model directly performs Orth-NTF the 3rd-order tensor which is composed of anchor graphs of views. Thus, our model directly considers the between-vien relationship. Moreover, we use the tensor Schatten p-norm regularization as rank approximation of the 3rd-order tensor which characterizes the cluster structure of multi-view data and exploits the between-vien complementary information. In addition, provide optimization algorithm for the proposed method and prove mathematically that the algorithm always converges to the stationary KKT point. Extensive experiments on various benchmark datasets indicate that our proposed method is able to achieve satisfactory clustering performance.
Introduction
As of the most typical methods in unsupervised learning, clustering has wide scope of ap-
plication (26] 4 [ to assign data to different clusters according to the information describing the objects and their relationships. Non-negative matrix factorization (NMF) [99] one of the representative methods of clustering, which is proved to be equivalent to K-means clustering [ Despite the widespread use of NMF, there are some drawbacks that have prompted some variants of NMF 8 9
In particular, the one-side G-orthogonal NMF g can guarantee the uniqueness of the solution of matrix factorization and has excellent clustering interpretation. Also, Ding et al. proposed the semi- NMF 19]. The data matrix and one of the factor matrices are unconstrained, which allows semi-NMF to be more suitable for applications where the input data is mixed with positive and negative numbers.
Corresponding author.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Although the above methods can achieve outstanding clustering performance, they are all single-view clustering methods and cannot be adopted straightforwardly for multi-view clustering.
Multi-view clustering tends to achieve superior performance compared to traditional single-view clustering owing to the capability to leverage the complementary information embedded in the different views. Considering the superiority of MVC and NMF, lots of NMF-based multi-view clustering methods have been proposed [3 28 23 33 15 29 14 37). The NMF-based multi-view clustering methods can save time and space because it is unnecessary to construct affinity graphs while graph- bash- methods have to. However, usually, they decompose the original data matrix directly, which leads to a dramatic reduction in the efficiency of the algorithm when the dimension of the original data is huge.
Inspired by the idea of anchor graph. the above issues can be solve by carrying out NMF on the anchor graph (37]. Due to the fact that the dimension of the anchor graph is considerably smaller than the original affinity graph, it follows that the clustering efficiency can be improved. However, as is well-known, there exist two ways of NMF-based multi-view clustering methods. One is to integrate different views first and then implement the NMF on the integrated matrix; the other is to perform the NMF on different views separately and then integrate the result from each view. Both ways are essentially applications of NMF on a single view, and both need to reduce the multi-view data into matrices in the process, which causes the loss of the original spatial information.
To fix the aforesaid issues, we proposed a novel view clustering based on Orth-NTF with one-side orthogonal constraint. Specifically, Non-negative Matrix Factorization (NMF) is tailored primarily for second-order matrices. When processing third-order tensors, there's a need to first transform the tensor into matrix before applying NMF. This step can lead to loss of inherent spatial structural information from the third-order tensor. In contrast, Non-negative Tensor Factorization (NTF) sidesteps this issue. NTF directly decomposes third-order tensors. This ensures that the NTF not only acknowledges the relationships between the views but also harnesses the complementary information they offer. Fig delineates the distinction between traditional NMF-based clustering techniques and our NTF-based approach. Furthermore, by incorporating an orthogonal constraint, our model offers distinct physical interpretability for clustering. This suggests that each row of the indicator matrix contains a single non-zero element, and the position of this element directly corresponds to the label of the respective sample. A large number of experiments have shown that our methods have excellent clustering performance.
The main contributions are summarized below:
We introduce orthogonal non-negative tensor factorization, which considers the between- view relationship directly. Also, we use tensor Schatten p-norm regularization to characterize the cluster structure of multi-view data and can exploit the complementary information of between views.
We regard the anchor graph obtained from the original data as the input of the non-negative matrix factorization, which reduces the complexity of our proposed algorithm considerably.
We provide optimization algorithm for the proposed method and prove it always converges to the KKT stationary point mathematically. The effectiveness of its application on tensorial G-orthogonal non-negative matrix factorization is demonstrated by extensive experiments.
2
Related work
In recent years, multi-view clustering (MVC) has received increasing attention due to its excellent clustering performance. Also, non-negative matrix factorization (NMF) is an efficient technique in single-view clustering, which can generate excellent clustering results that are easy to interpret, and many NMF-based variants have been proposed. Therefore, multi-view mustering-based NMF and its variants have attracted tremendous interest recently.
As the first investigation of the multi-view clustering method based on joint NMF, multiNMF 23 implements NMF at each view and pushes the different clustering results of each view to a consensus. It provides new viewpoint for the subsequent NMF-based MVC methods. Influenced by multiNMF, He et al. proposed a multi-view clustering method combining NMF with similarity [15). It implements NMF on each view in multiNMF. In addition, it sets weight for a different view and introduces

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Teax x
2P14
M77
N
6111
Figure 1: Comparison between traditional NMF- multi-view clustering method (left) and Orth-NTF (right).
Figure 2: 3rd-order tensor construction process.
similarity matrix of data points to extract consistent information from different views. To better detect the geometric structure of inner-view space, Wang et al. [33 introduced graph regularization into the NMF-based multi-view clustering method improve clustering performance. Considering the above work, Wang et al. [29] proposed graph regularization multi-view clustering method based concept factorization (CF). CF is variant of NMF and it is suitable for handling data containing negative.
As the size of data grows, lots of methods accelerate matrix factorization are presented. Wang et al. (28) proposed fast non-negative matrix triple factorization method. constrains the factor matrix of NMF to a clustering indicator matrix, thereby avoiding the post-processing of the factor matrix. Inspired by the work of Wang, Han et al. [14] constrained the intermediate factor matrix in the triple factorization to diagonal matrix, reducing the number of matrix multiplications in the solution process. Another idea to deal with large-scale multi-view data is to introduce anchor graphs, since the number of anchor points is much smaller than the number of original data, multi-view clustering methods based on anchor graphs tend to reduce the computational complexity and thus are able to deal with large-scale data 22 17 21 Considering that previous NMF-based multi-view clustering methods are performed directly on the original data, Yang et al. 377 introduced an anchor graph as the input of G-orthogonal NMF. The efficiency of matrix factorization is indeed improved due the introduction of anchor graph.
Despite the fact that existing NMF-based multi-view clustering methods perform the clustering tasks excellently, they apply NMF to each view independently. Subsequently, they combine the low-dimensional representations from different perspectives to arrive unified shared representa- tion. This approach often overlooks the interrelationships between the views, which are crucial for clustering.
3 Notations
We introduce the notations used throughout this paper. We bold calligraphy letters for 3rd-order tensors, H E Rn Xmgxxs, bold upper case letters for matrices, H, bold lower case letters for vectors, h, and lower case letters such higk for the entries of H. Moreover, the i-th frontal slice of H. is
7 H. the discrete Fourier transform (DFT) H. along the third dimension, H fft( [].3] 3).. Thus, H. = ifft H., 3). The trace and transpose matrix H are expressed as (r(H) and H" The F-norm of H. is denoted by ||]
Definition (t-product 18 Suppose A E Re1xxxx and B E Bmxxggns the 1-product A B E T X is given
A B ifft(bdiag( (AB), (1,3),
where A = bdiag(A A) and it denotes the block diagonal matrix. The blocks of A are frontal slices of A.

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Definition 2. defined as
112
Given H E Ru X h min n2), the tensor Schatten p-norm of H is
o 7l )ー
| S3
(1)
where, < E 0, ( denotes the j-th singular value of H
<<ppL
It should be pointed out that for 0 < & 1 when is appropriately chosen, the Schatten p-norm provides quite effective improvements for a tighter approximation the rank function [39 36).
4 Methodology
4.1 Motivation and Objective
Non-negative matrix factorization (NMF) was initially presented as dimensionality reduction method, and it is commonly employed as an efficient latent feature learning technique recently. Generally speaking, given non-negative matrix X, the target of NMF is to decompose X into two non-negative matrices,
HG
where E =T H E R and G IR R"xP means n-by-p matrices with elements are all nonnegative. n and k means the number of samples and the number of clusters, respectively.
In order to approximate the matrix before and after factorization, f--norm and F-norm are frequently adopted as the objective function for the NMF. Considering that F-norm can make the model optimization easier, use F-norm to construct the objective function.
With the extensive use of NMF, more and more variants of NMF have emerged, among which are G-orthogonal NMF B and Semi-NMF [9]. By imposing an orthogonality constraint on one of the factor matrices NMF, we obtain the objective function of the one-side orthogonal NMF,
min Hg0.G
X - HGT -
s.t. H'H =I
we relax the nonnegative constraint on one of the factor matrices in the NMF and the input matrix X can also be mixed positive and negative, then can get Semi-NMF. Semi-NMF can be adapted to process input data that has mixed symbols. For G-orthogonal NMF and Semi-NMF, Ding et al. [8] presented the following lemma:
Lemma 1. G-orthogonal NMF and Semi-NMF are all relaxation of K-means clustering, and the main advantages of G-orthogonal NMF are (1) Uniqueness of the solution; (2) Excellent clustering interpretability.
Taking into account the one-side orthogonal NMF, we relax the nonnegative constraints on X and G. Moreover, inspired by FMCNOF (37], we construct the anchor graph S obtained from the original data X as the input of matrix factorization. Compared with the original data, the number of anchors is much smaller, therefore, by adopting the anchor graph constructed by anchors and original data points as the input of matrix factorization, we can reduce the computational complexity of the algorithm effectively.
min ||s H /. H20
s.t. H
(4)
where S 1 H gank and G E m m the number of anchors and we consider H as the cluster indicator matrix for clustering rows as described in [8] We will introduce the details of anchor selection and the construction of the anchor graph the appendix.
As described in the previous section, the existing NMF-based multi-view clustering methods are essentially matrix factorization causes the loss the original spatial structure of the multi-view data. We extend NMF to the 3rd-order tensor, which can process the multi-view data directly and can also take full advantage of the original spatial structure of the multi-view data. The objective function of tensorial one-side orthogonal neg-tege matrix factorization is written in the following form:
single view combined with the integration of multiple views.
min #20
S - 7 * g
s.t. H & H = I,
(5)
F

---

The 3rd-order tensor construction process is illustrated in Fig
In order better exploit the complementary information and spatial structure between different views, we get inspiration from the excellent performance the tensor Schatten p-norm 12 12:38] 38
We
introduce tensor Schatten p-norm regularization the tensorial form of the cluster indicator matrix. Our objective function is formulated as follows:
min $ - - 9t F + X I
S.I. H 020,7 0
7
(6)
where < & 1, is the hyperparameter of the Schatten p-norm term.
Remark 1. The regularizer in the proposed objective O used to explore the complementary
information embedded in inter-views cluster assignment matrices H(x) (v 1.2...., VV). Fig. shows the construction of tensor H. it can be seen that the k-th frontal slice A describes the similarity between sample points and the k-th cluster in different views. The idea cluster assignment matrix H()) should satisfy that the relationship between N data points and the k-th cluster is consistent in different views. Since different views usually show different cluster structures, impose tensor Schatten p-norm minimization [[2] constraint on H. which can make sure each (k) has spatial low-rank structure. Thus A (K) can well characterize the complementary information embedded in inter-views.
4.2 Optimization
Inspired by Augmented Lagrange Multiplier (ALM), introduce auxiliary variables and J
and let H = 2, H = J, respectively, where Q > 0. Then, we rewrite the model as the following unconstrained problem:
min (0.
min 2
s H
+
+
+
H
2
V1
2 /
+
0
H
J
+
where V1, 222 represent Lagrange multipliers and μ4, are the penalty parameters. The optimization process can therefore be separated into four steps:
Solve g with fixed 2, J. 7 becomes:
G 2
min S
(8)
After being implemented with discrete Fourier transform (DFT) along the third dimension. the equivalent representation of (8) in the frequency domain becomes:
min
S
H (2)
g
( F
(9)
where 9 [1, and the others in the same way.
Let ① = 5(() (t)) (g()) yT(
F we can obviously get the following equation:
① = tr s (0) (T
(1) T'
H (v)
TS (1) (n))
(g (1) "g"
2tr
+ tr
(10)
Setting the derivative 0b/ 00g
0 gives 2 (1) (0 _ 2 1ITH
0. So the solution of 9 is:
g () = = ) 1)
(11)
Solve H with fixed Q. J.G.J
becomes:
s H * gT
y1 2
P 2
D2
min H I
Q
+
H
J +
(12)
F 2
F
p
F
5

---

And 12 is equivalent to the following in the frequency domain:
)T
min
S H g
(13)
5
5
H
o
(t)
P
H
-
1)
J
+
2
p
where H = -ff((H,, [], 3), and the others in the same way.
And 13 can be reduced to:
2tr g H (1)
) T5 ptr
ptr Tw!? (14)
min
H
7
=I
where W 0)
) and W = -f(1) D.
and it also can be reduced to:
max (7L (( tr (H 1 TB²
(15)
where B (d))
=S 2 (1)
(1)
+ W PW (1)
To solve 15 we introduce the following Theorem:
Theorem Given G and P, where G(G)T I and P has the singular value decomposition P AS(V) then the optimal solution of
max tr(GP) G(G)
(16)
G (A))
Proof. From the SVD P AS(V)T and together with (16). is evident that
tr(GP) tr((AASV))) tr(S(V)
tr tr(SH)
(17)
where H = (V) 8ü and ha are the (i, i) elements of S and H, respectively. It can be easily verified that H(H) = where I is identity matrix. Therefore -1 & ha & 1 and 84 0, Thus we have:
tr tG()
Σ saha &
Ssi-
(18)
The equality holds when H is an identity matrix. tr (GP) reaches the maximum when H [ 0].
So the solution of 15p is:
H (v)
(v)
(19)
where A and can be obtained by SVD
B
X (0)
Solve Q with fixed H. J.J. 7 becomes:
V
min 2 2
H - 2
(20)
F
20
obviously equivalent to:
V
min
HH +
(21)
200
2

---

According to 37,, the solution of 21 is:
7 + 11
31
Q
(22)
Solve J with fixed Q H. g.
7 becomes:
min > A
P 2
H
J
22
(23)
F
after completing the square regarding J, we can deduce
y2
X + 1
T = arg min
H
I
(24)
2
F P
which has a closed-form solution as Lemma] 12
Lemma 2. Let Z IR X have t-SVD 1 S VT, then the optimal solution for
min ||
+ - |
21
(25)
is x - (if((((())((( Vr, where (Z is fddiggonal 3rd-order tensor, whose diagonal elements can be found by using the GST algorithm introduced [22].
Now the solution of 22 is:
J'=Ta V/2
(26)
Finally, the optimization procedure for Multi- View Clustering via Orthogonal non-negative Tensor Factorization (Orth-NTF) is outlined in Algorithm 1.
Algorithm Multi- Clustering via Orthogonal non-negative Tensor Factorization (Orth-NTF)
Input: matrices ( ) R Xife anchors mi; Output: Cluster labels Y of each data points.
Initialize: 10-3, 1079,
Q identity
1: Compute graph matrix ((*) of each views;
2: while not condition
3: Update 0 by solving
4: Update 7
solving 19
Update [v] by solving
6: Updltte by using 24
Update V 22. and 0:
V1
=
Vi
+
2). V3 V2 H(H J). min(ny, 10*3),
min(np, 1):
8: end while
Calculate clusters by using
Σe= H(') /V:
10: return Clustering (The position largest
indicator
comesponding sample).
4.3 Convergence Analysis
Theorem |Convergence Analysis of Algorithm Ty Let = (2x, & J2,, 1
l oc in (7) be sequence generated by Algorithm then
1. Pv is bounded with the assumption lima-~o max Ph: (H) k+l -H(()) 0:
2. Any accumulation point Ps a stationary KKT point of
The proof will be provided in the appendix and we need to mention that the KKT conditions can be used to determine the stop conditions for Algorithm which are | Ha S E,
|+ J xooooo S.

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Table 1: Multi-viee datasets used in our experiments
#Datasel
#Samples
#Veew
#Class
Mivature
MSRC
210
5
24, 576, $12, 256, 254
HandWritten4
2000
4
10
76, 216,47,6
Mnist4
4000
Reuters
18758
5
21531, 24892, 34251, 15506, 11547
Noisy
50000
2
784,
4.4 Complexity Analysis
For Orth-NTF, the storage requirements for 9. H. 2.J.V and 2/2 have complexities (VV + k)n), (n +))),, (Vn)), O(Vnk), O(Vnk), O(Vnk) and OVVnk), respectively. Combining these, the total storage complexity for Orth-NTF is O(Vnm + uk? + 6Vnk).
For the computational complexity, the process of constructing S has computational complexity of OV + V n' log(m)). When updating the four variables, g. H. Q and J. their respective computational complexities O(2V nk (Vk) + V2kn). Given that m, n, and V are relatively small constants, the primary computational cost associated with updating the variables stands at (Vnkm + V22k). Summing it all up, the overall computational complexity of our proposed method is O( + Vm²k).
D(Vmmd + V Vnmlog(m))). O(Vm²k k + Vmk²), O(Vnk) and
(a) MSRC
(b) HlandWiitend
(c) Minis44
(d) AWA
Figure Convergence experiments on MSRC, HandWritten4, Mnist4 and AWA.
5
Experiments
In this section, we demonstrate the performance of our proposed method through extensive experi- ments. It is compared with plenty of state-of-art multi-view clustering algorithms some multi-view datasets. We evaluate the clustering performance by applying metrics used widely, Le., 1) ACC; 2) NMI; 3) Purity; 4) PRE; 5) REC; 6) F-score; and 7) ARI. The higher the value the better the clustering results for all metrics mentioned above. Detailed experimental configurations and hyper-parameter on each dataset are in the appendix.
5.1 Datasets and Compared Baselines Methods
The following multi-view datasets are selected to examine our proposed method. The details of
the datasets are shown in Table I
MSRC [34]; HandWritten4 [10]; Mnist4 [6] AWA [I]]
Reuters (2] Noisy MNIST [32; We choose the following state-of-art multi-view clustering algo-
rithms compare with proposed methods: AMGL 25 MVGL 40): CSMSC (24: GMC
30):
LMVSC [7:: SMSC [66; SFMC [2]] FMCNOF 37
FPMVS-CAG B1: ETLMSC 35
MSC-BG 388;
5.2 Experiments Result
The clustering performances are listed in Table 2 and Table 3 They contain four medium-seatd datasets and two large-scale datasets. The corresponding experimental configurations and descriptions are included in the appendix. is clear that our algorithm outperforms the other baseline algorithms most of the datasets.
This advantage may stem from the fact that our model directly factorizes the tensorized anchor grapieco of anchor graphs from various views-into the product of two non-negative tensors, one being an index tensor. As a result, our model effectively captures both the spatial structural information and the complementary data present in the anchor graphs from different

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Table 2: Clustering performance MSRC, HandWritten4, Mnist4 and AWA. (The best result is in bold, and the second-bee best result is underlined.)
Dataset
MSRC
Metries
NMI
Purity
F-score
AMGL
(BCAT 16)
0.751-0.00
0.704-0.000
0.789-0.001
0.621 +0.00
0.744-0.000
0.674+0.00
0.615-00000
MYGL
40
TTCB 17)
0.600.0.000
0.66310.000
0.733_00000
0.666 10.00
0.715_00000
0.564110000
0:676_000000
CSMSC
24
(AAAT 18)
0.758-00007
0.735-0000
0.793-0.008
0.756-+0004
0.573-00008
0.703++0010
0.653-0012
GMC 20
(TKEE 19)
0.895_000000
0.809100000
0.895_00.000
0.788 10.000
0.814_00.00
0.001000000
0.268_00000
LMVSC
17
(AAAT
20)
0.81400.000
0.717=00000
0.81400.000
0.676==0.000
0.602-0.000
0.654=00000
0.632=0.000
SMSC 16
[nn Fuxion
20)
0.706-00.00
0,717:00000
0.804-0.000
0.672:10.00
0.718-0.000
0.694:0.000
0.643:0.000
FMCNOP
E7
[TIP 21)
0.440-00039
0.345:00046
0,469_0.042
0.25
0.605=000044
0.595
0.259=00051
0.66610.000
0.642_0.000
0.731100000
SPMC [11
22)
0.10-0000
0.721=00000
0.80000000
0065
0.752-00006
0.714=0000
0.663-00000
ETLMSC 33 (TIP' 19)
0.60-0000
0.0
0.962-0.00 000
0.926
00000
0
0.928:10.000
0.917:00000
BS 3 (TIP 22)
0,951
0,90000.000
0.981=0000
0,961 10.000
0.963-0000
0,962.00000
0.956-00000
OurS
0.990 0.000
0.978-0.001
1.990- 0.00N
0.900 0.000
0 0.000
0 +0,000
0 0.000
HandWriftend
Merries
NMI
Parily
PER
RFC
F-eere
AMGL (UCA)
0.762.00.000
0.591110.000
0.67010.000
0.628_00.00
MVGL 4M TTYB 17)
0.811_0.000
0.80910.000
0.831_0.000
0.721 10.000
0.825_00000
0.77010.000
0.243_00000
CSMSC 24 [AAA]
18)
0.80=00001
0.798-00001
0.807-0.001
0.778=20001
0.745=00001
0.760=00001
0.755=00001
GMC N0 (TDEE 19)
0.861-0.000
0.859-0.000
0.861-0.000
0.799-00000
0.855-00000
0.826+00000
0.806-00000
LMVSC 17 (AAAI
20)
0,001_00000
0.831200000
0.901_000000
0.81910.000
0.825_00.00
0,822100000
0.812-000000
SMSC 16 [Inf Pasion' 20)
0.742-0.000
0.751=0.00
0.759=0.000
00005=00000
0.767=0.000
0.717=00000
0.685-0.00 000
EMCNOF 17
[TTP' 21)
0.385_00092
0.370.00.02
0,386-00.93
0.254:0.077
0.688-00.00 101
0,360.10.070
0,250 0.097
FPMVS-CAGT
0.744-0.000
0.753-0.000
0.744-0.00
0.681-0000
0.636-0.000
0.762-+0.000
0.642-0.00
[1
22)
0.775
0.837-10.000
ETLMSC 55 (TIP 19)
0.80-00001
0:93--0000
0.886=00000
0.890-00001
0888=00000
0.876-00000
MSC-BG 18) (TIP' 22)
0.00000000
0.022-0.000
0.88-00000
0.87 100000
0 0.80000000
0:000000000
0.86000000
0.000
0.969-0.000
0.985_ 0.000
0.970000.000
0.900
0.970-10.000
0.966
Ditaset
Miist4
Metries
ACC
NMI
Purty
PER
RBC
F-score
ARI
AMGL 23 (BCAT 16)
0.021_00000
0.806
0.921
0.004L0.000
0.862_.0.00
0,858 10.000
0.810_.0.00
MVGIL CD (TYY8 17)
0.919=00000
0000000000
0.919=0.000
00001=00000
0.860=00000
0836200000
0.807=000000
CSMSC 21 (AAAF 18)
0.641-0.000
0.601-0.010
0.728-0.008
0.607-00014
0.767-0.008
0.677=00010
0.555-0.112
GMC 30 TODEE
0,920_00000
0,807
0,920_0.000
0.853
0.861_0.000
0.857-10.000
0.809_00000
LMVSC (AAAI'
0.892-0.000
0.726=0.000
0.892-0.00 060
0.808-=00000
0.812=0.000
0.810=0.000
0.747=0.000
T6
20)
0.909-0.000
0.754:0.000
0.909-0.000
0.834:10.00
0.841=00000
000
0.763=00000
FMCNOF 17
(TPP 21)
0.697-0.119
0.400-0.102
0.711-0.096
0.558 118
0.683-0.073
1611++0.95
0.460-0 145
FPMVS CAG
MJ (TIP 21)
0.885_0.000
0.71510.000
0.885_0.000
0.000
0.80010.000
0.795_0.000
0.8151000000
0.733_.0000
SPMC (1 (TPAMI 22)
0.916000000
0.757=00000
0.916=0000 000
0.846:
0.85500000
0.850=000000
0:8000000000
ETL [SS
(TIP 19)
0.934-00000
0.847-00.000
0.934-0.00 0.00
0.878
00000
0.885-00000
0.881 00.000
0.842-0000
BSC-BG 29 (TIP' 22)
0.008_00000
0.001000000
0.908_00000
0.800-00000
0.891_00000
0.800-00000
0.850_00000
0,977_0.00
0,92610.000
0.977_0.000
0,955 0,,000
0.956__0.00 0.000
0,95010.000
0.941_0..00
Dataset
AWA
Metries
ACC
NMI
Purity
PER
REC
F-score
ARI
MVGIL 40 (TCYR' 17)
B.06-D-000
0.000-0.000
0.065-D.000
0.020-00.000
0.843-00000
0.820++6000
0.802-00000
CSMSC 21 (AAA)' 18)
0.113_.0.00
0.105
0.119_00001
0,051
0.0.44_00000
0.053110.00
0.033_.0001
GMC 5U (TKDE 191
D.0E=00000
0,000000000
0.099=0.000
0.020=000000
0.915:00.00
00095000000
0:001=00000
LMVSC (AAAT 20)
0.005-00000
0,171=00000
0.114-0.000
0.041=0.000
0:065-0.000
0.051=000000
0.027-00000
EMCNOF D7 (TP 21)
0,005_00008
0.018100011
0,035_0.008
0,021 10.011
0.668_.0227
0.03010.002
0.002-.0003
FPMVS-
0.18310.000
0,069
0,142.10.000
SEMC [T (UPAMI 22)
0054
0.949-0.000
0.00
0.592=00000
0:05:20000
0.00K=00000 000
FIT MSC (30 (TIP 19)
0.631-0000
0.78-00000
0.656 000
0.498 000
0.580-0.00
0.536+00000
0. 0:526-0000 000
MSC-BG BGC-BBGPA 18 (TIP 22)
0,693_00000
0550
0.511 0,00
0R050 10000
0,894_0.000
01105001000
01015-0000
OMES
0.646-0.000 0.000
0.815-00000
0.670 0.000
0.534=0.000
0.574=00000
0.553=40.00
0.543=0.000
perspectives. Additionally, with orthogonal and non-negative constraints in place, our model offers clear interpretability for clustering. This means that each row of the indicator matrix for every view contains a single non-zero element, with its position indicating the label of the associated sample. Consequently, our model can immediately provide the label without necessitating any post-processing, a step which other methods still require.
5.3 Ablation Experiments
We do some ablation experiments on orthogonal constraint and Schatten p-norm on four datasets as shown in Table 4 It can be found that, tensor Schatten p-norm regularization is overall superior to orthogonal constraint. The reason is that tensor Schatten p-norm regularization effectively charac- terizes both the complementary information and spatial structure information of index matrices of different views. Compared to orthogonal and tensor Schatten p-norm constraints, Joint constraints have great contribution for clustering.
5.4 Experiments of convergence
We optimize the objective function iteratively by introducing two auxiliary variables Q and J. We test the convergence of our algorithm by checking the difference between H and the two auxiliary

---

Table 3: Clustering results and running time (sec.) Reuters and NoiSyMNIST. ("OM" means out of memory and "_ means the algorithm takes more than three hours to calculate.)
Basect Meriic
Rralen
Datret
Nukey MNNST
Purty
Time
Minios
NMI
Prrty
Tiar
MVGIL
23 CiCraa 07)
CM
OM
QM
MVGL 40
(CNB 171
GM
DM
OM
CM
USMSE (AAA) 180
OM
OM
OM
CM
CSMSC
(AAA 180
OM
0M
OM
CM
CMC (TDEE
0.56200000
08i5=0000
0.383:0000
643410000
SMSC
11 Fasion
OM
OM
OM
GM
SMBS [1]]
Piskee
GM
DM
OM
OM
IMENOF (TTP 11)
01333-00177
0175-410077
0.506-20052
FMONNR ID
[TTP 211
0.317-2616
025740617
0.30444057
10
MIIV--WA (mP 30
11716-010000
110-000011
0667--5000
PERIVSICATT (TIP 20
0.94--0000
85134000
0.90740100
FITT MOC L1 (TDP
OM
OM
0M
GM
ITFI MSC
(TPP
FM
DM
0M
CM
MSC-IG 16 (TIP)32
R660100000
0-854-11000
0.695 0:000
465 13
MISC-BER 15 (TEP 27)
GM
DM
OM
OM
557.09
CEP
0,701=8:000
0.729:90000
0,747200000
503 A6
Table 4: Ablation experiments on MSRC, HandWritten4, Mnist4 and AWA. (Corth and Csp represent the orthogonal constraint and the tensor Schatten p-norm regularization, respectively.)
Situations
MSRC
HandWrittend
Carth
Csp
ACC
NMI
Purity
ACC
NMI
Purity
0.776
0.653
0.776
0.559
0.572
0.581
0.785
0.659
0.785
0.594
0.621
0.615
0.886
0.819
0.886
0.725
0.741
0.725
0.990
0.987
0.990
0.985
0.969
0.985
Situations
Mnist4
AWA
Corth
Csp
ACC
NMI
Purity
ACC
NMI
Purity
x
x
0.898
0.750
0.898
0.020
0.020
0.020
V
x
0.905
0.759
0.905
0.092
0.147
0.097
0.912
0.789
0.912
0.212
0.125
0.213
0.977
0.926
0.977
0.646
0.815
0.670
variables. The result of the experiment is shown in Fig is evident that when the iteration reaches around 80, the difference decreases significantly until it is about zero finally. When the number of iteration increases, the clustering metric (such as ACC) overall improves gradually and tends to be constant with the convergence of the algorithm. It also indicates that our method has good clustering performances.
More experiments can be found in the appendix.
6
Conclusion
We are concerned in this paper with multi-view clustering based on semi-non-negative tensor fac- torization (Orth-NTF) with one-side orthogonal constraint. Our proposed model extends NMF to Orth-NTF so that the spatial structure information of the multi-view data can be utilized to improve the clustering performance. In addition, the complementary information embedded in different views is fully leveraged by imposing the tensor Schatten p-norm composed of cluster indicator matrices. To diminish the computational complexity, we adopt anchor graphs instead of the original multi-view data. Also, provide an optimization algorithm for the proposed method and validate the effectiveness of this approach in extensive experiments on different datasets.
Acknowledgements
The authors would like to thank the anonymous reviewers, AC and PC for their constructive comments and suggestions. This work was supported part by National NSFC under Grants 62176203, in part by the Natural Science Foundation of Shandong Province under Grant ZR2021021 020800988 in part by the State Key Laboratory of Multimodal Artificial Intelligence Systems under Grant 202200035, in part by Guangxi Key Laboratory of Digital Infrastructure under Grant GXDIOP2023010, in part by the Fundamental Research Funds for the Central Universities, the Innovation Fund of Xidian University.
10

---

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12

---

A Proof of Convergence
A.1 Proof of the 1st part
Lemma 3 (Proposition 6.2 of (20)) Suppose F R Knz - R is represented as = where X IR Xna with SVD X
on) n min(n1,12)) and f differentiable.
The gradient of F at X is
(F(X) Udia
(27)
0x
where af(y)
(X)(
To minimize Ø ( step k + 1 in (21) in the main body, the optimal 2 needs satisfy the first-order optimal condition
VI.e
2 = 76(.) +
14
By using the updating rule ((()
5 1,k + uu (71 - we have
1,6+1
5 1,
-2 (241) = O. (5)
According to our assumption lim Mk
2
2 = 0, we know V 11,++1 1 is bounded.
To minimize I at step k + in (24) in the main body, the optimal Th+1 needs to satisfy the first-order optimal condition
1 y2,,) = 0.
A k+1 I
PR(T k++1 Hir+1
Pk
Recall that when 0 < 1, in order to overcome the singularity of (1yl") = pn/ln1? near 17 0.
<
we consider for ( < the approximation
Pd
max 2-p, 2-)
Letting J = lt diag ())
(0)
HH
then it follows from Lemma that
0 T (3)
u
(1)
diag
po
D (()
max 2 P, P,,o, [o,(T" ( 12 12-p}
And then one can obtain
pa,
P
max 2- 10, (3) ||22p]
cl
12
p
OJ
(1-p)
1=1
11 aJ
So
bounded.
Let denote Fv - Fv. Fr the discrete Fourier transform matrix of size V V. F# denotes
its conjugate transpose. For J J X3 Fy and using the chain rule in matrix calculus, one can obtain that
aT P
X F
aJ
is bounded.
13

---

And it follows that
I1,++ = 22 + + - J JR+1)
AV k+11|88 = 2,,++1,
thus V2A4+1 appears to be bounded.
Moreover, by using the updating rule V = 1 + H(H 2), = 32 p(H J), we can deduce 2,
C(2 1 H I U+tii.,,.
(28)
- L gk. His J vik,)
L(2,; 9k: Hks I k: i,)
Pk Ph-1
1) 2,1 -22,k-111
1|
11 117
2pp--1
2p%
2pk-1
Pb + Mk-1
||
13 11
1131, Vi, +
2
2/1x
2/4k-1
Thus, summing two sides (28) from
n., we have
1(Qu+1,9nuc Hnt1: J 11 1=1: s.n))
<L(2), 91, H1, J1: (;:)))))
|02,,01
Pk: Pk-1 11 22226-211
2pn
2p0
2
(29)
|2'1,l1e
M4 + 1k-1
11 12,1--113))
2
2/10
2
Observe that
Pk + Pk-1
fb + Hk-1 < 00,
20
0C,
2
k=1
we have the right-hand side (29h) finite and thus L(Qn+1, 9 Hat1, J 1411:a.,) bounded. Notice from (7) in the main body
C( 9n, 1. H.+1..7 (+t1:Din))
Σ ((0)
H (1) (g(())
n+1 an11) |
+ A||J
n+1
+
Pn
1a+t1 - I +
22 1|?
2
Pr.
[5()) 1,m+1
+ Pn
2
n+1
2
2011
11?
(30)
and C(
each term of 30 is nonnegative, following from the boundedness of
I(@n+1c
n+. H.+11, n+ 1 v.,), we can deduce each term of (30) is bounded. And
[ m+11
being bounded implies that all singular values of J n+1 are bounded and hence
IJ n bounded.
F (the sum of squares of singular values) is bounded. Therefore, the sequence [Jx} is
Because
yI.+1 VI,A + (2x He:) Hi 2i VLA+1 V1.k
μt: and in light of the boundedness of Q V1.k- it is clear that H. is also bounded.
And from (8) in the main body, is evident that || 13, s II(S (1(5))) y (1) 12. so g(.e) is also bounded. So Gb is bounded.
14

---

A.2 Proof of the 2nd part
From Weierstrass-Bolee Bolzano theorem, there exists at least one accumulation point of the sequence Without loss of generality,
Pk. We denote one of the points p {H 2 , J Vi.V2 assume
(Pk - converge to Pa
Note that from the updating rule for 1 we have
II = VI. + Na (P. Q1) - Q" =
Note that from the updating rule for 2/2 we have
V 2 = 22 (HH Jk) = T = H.
In the g() -subproblem (8) the main body, we have
g (5 "H = -
In the J-subproblem (24) in the main body, we have
XV J J k+1
22 = V = AV |
Therefore, one can see that the sequences H'" g', , , y/ satisfy the KKT conditions of the Lagrange function (7) in the main body.
B Anchor Selection And Graph Construction
Inspired by [21]. adopt directly alternate sampling (DAS) to select anchors.
First of all, with the given data matrices (x(")) Hy= we concatenate the data matrix of each view along the feature dimension. The connected feature matrix X E RRad can represented as X = (X);; ( x( where d is the sum of the number of features each view. Let 0, represent the i-th sample of the d-dimensional features, which can be calculated as
df
0, Tra(Xij),
(31)
1=1
1 de, and Tra(-)) represents the transformation of the raw features. Specifically, if the features are negative, we process the features of each dimension by subtracting the minimum value in each dimension. Then we obtain the score vector 0 = [01 02, ... 0,] E R". We choose the point where the maximum score is located as the anchor. The position of the largest score
where d'' EV
Index = arg max 0,
(32)
Then the 1st anchor of the v-th view is 8(0) = x
(v)
Indexx
After that, let Ornden be the score of the anchor selected from the last round, then we normalize the score of each sample by:
0,
0: max 0'
(i = 1,2,.... ,.)
(33)
Then the score 0g can be updated as
8, +- 8 x (1 0.).
(34)
times to select anchors. After selecting anchors, construct an anchor graph of each view S(v), in the same way, as (21].
Finally, repeat (32)
(34
15

---

C More Details of the Experiments
C.1 Experimental Configurations
The Reuters and VoisyMNIST are implemented on a standard Windows 10 Server with two Intel (R) Xeon (R) Gold 6230 CPUs 2.1 GHz and 128 GB RAM, MATLAB R2020a. The MSRC, HandWritfend Mnist4 and AWA A are implemented on a laptop computer with an Inter Core i5- 15-8300H CPU and 16 GB RAM, using Matlab R2018b. Codes are available:
We repeated the all methods 20 times independently and showed the averages with the corresponding standard deviations. The specific hype-parameters on each dataset are as follows:
MSRC: anchor rate p 0.5, A= 100.
HandWritten4: anchor rate 1.0, p 0.1, À 1180.
Mnist4: anchor rate 0.6, p 0. = 5000.
AWA: anchor rate = p= À 1000.
0.5,
Reuters: anchor rate = 0.005 (anchor number = 100), p 0.4, 1209800.
NoisyMnist: anchor rate 0.03, p 0.1, À 200000.
C.2 Impact for Parameters
In our proposed algorithm, the number of anchors, the value of p from the tensor Schatten p-norm, and the value of are variable parameters. In this section take 4 datasets: MSRC, HandWritten4, Mnist4, and AWA as examples to analyze the effect of these variable parameters.
Effect of the number of anchors. We changed the anchor rate from 0.1 to 1.0 with step size 0.1. The changes of clustering results and algorithm running time along with the anchor rate were tested on MSRC, HandWritten4, Mnist4 and AWA, as shown Fig 4 and Fig When the anchor rate were 0.7, 1.0, 0.6 and 1.0, the best clustering results were obtained on MSRC, HandWritten4, Mnist4 and AWA, respectively. The time required for clustering is approximately linearly related to the increase of anchor rate.
Effect of the value of We set the value of to be 0.1 to 1.0 with a step of 0.1. We obtained the results of ACC, NMI, and Purity experiments with different values of p as shown Fig The best clustering results obtained on MSRC, HlandWrittend' Mnist4 and AWA when the values of are 0.5, 0.2, 0.1, and 0.5, respectively. indicates that tensor Schatten pnorm can take advantage of the low-rank of views which helps mine the complementary information of different views. This helps get better clustering results.
Effect of the value of X. To determine the value of X, we initially approximate its range using the magnitude of the tensor Schatten p-norm regularization, followed by more detailed fine-tuning within that range. The impact of varying parameter combinations on the method's performance can be seen in Fig This figure highlights the clustering performance across different pairings of p and A.
K8 (c)Maist4
Rarder
(WMSRC
(DHHndWWrilttdd
(dIAWA
Figure 4: Clustering results with different anchor rate on MSRC, HandWritten4, Mnist4 and AWA.
16

---

[c]MaisM
Figure 5: Time (sec.) with different number of anchors on MSRC, HandWritten4, Mnist4 and AWA.
(b)ttadWrrttteed
CelMnist4
GBAWA
Figure 6: The influence of p on clustering results on MSRC, HandWritten4, Mnist4 and AWA.
g
8
(a) MSRC
HandWaattedd
fcl Muistt
AW
Figure 7: The influence of À and on clustering results MSRC, HandWritten4, Mnist4 and AWA.
17


Reviews:
[{"review_id": "y0dGbJHSuH", "reviewer": "Reviewer_a6FX", "summary": "In this paper, the authors focus on the problem of multi-view clustering using semi-non-negative tensor factorization (Orth-NTF) with a one-side orthogonal constraint. The proposed model extends Non-negative Matrix Factorization (NMF) to Orth-NTF, allowing for the utilization of spatial structure information from multi-view data to enhance clustering performance. Moreover, the authors fully leverage the complementary information embedded in different views by incorporating a tensor Schatten p-norm composed of cluster indicator matrices. To reduce computational complexity, anchor graphs are adopted instead of the original multi-view data. The authors provide an optimization algorithm for the proposed method and demonstrate its effectiveness through extensive experiments conducted on various datasets.", "questions": "Please refer to the weakness section.", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "1.The overall structure of the paper is clear and comprehensive.\n2.The research problem and innovative aspects are well-defined, and they are supported by sufficient experimental evidence.\n3.The proposed method demonstrates promising results for the task of multi-view clustering.\n4.The provided optimization method includes detailed formula derivation, enhancing the understanding of the proposed approach.", "weaknesses": "1.The paper seems to lack a detailed explanation of the advantages of extending NMF to 3rd-order tensor NMF.\n2.The proposed method applies NMF to the anchor graph S. Couldn't other dimensionality reduction methods be used to learn low-dimensional representations of high-dimensional data and achieve acceleration in clustering? Why was the choice made to utilize the anchor graph?\n3.I think the author should provide the code to make the experimental results more convincing.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "__Q1__: The paper seems to lack a detailed explanation of the advantages of extending NMF to 3rd-order tensor NMF.\n\n__A1__: Thank you for your attention. Non-negative Matrix Factorization (NMF) is designed primarily for second-order matrices. When handling third-order tensors, one must first transform the tensor into a matrix to apply NMF. This transformation leads to a loss of spatial structural information inherent in the third-order tensor. Conversely, Non-negative Tensor Factorization (NTF) is adept at directly decomposing third-order tensors, effectively preserving the spatial information they contain.\n\nFor multi-view clustering, traditional NMF-based methods execute NMF separately on each view. They then amalgamate the low-dimensional representations from different perspectives to derive a common shared representation. However, these methods tend to overlook the interrelationships between the views, which play a crucial role in clustering. Our model, in contrast, applies NTF directly to the third-order tensor formed by anchor graphs from different views. As a result, it not only leverages the relationships between the views but also taps into the complementary information they offer.\n\nTo visually understand the distinction between prevailing NMF-based clustering techniques and our NTF-based model, __please refer to Figure 1 in the provided PDF__.\n\n__Q2__: The proposed method applies NMF to the anchor graph S. Couldn't other dimensionality reduction methods be used to learn low-dimensional representations of high-dimensional data and achieve acceleration in clustering? Why was the choice made to utilize the anchor graph?\n\n__A2__: Thank you for highlighting that. While other dimensionality reduction methods like PCA and LPP can indeed be utilized to derive low-dimensional representations and handle large-scale data effectively, NMF offers certain unique advantages. Compared to PCA and LPP, NMF boasts superior interpretability. When augmented with an orthogonal constraint, NMF gains even clearer physical interpretability in the realm of clustering. This means that each row of the indicator matrix possesses just one non-zero element, with the position of that element directly signifying the label of the corresponding sample. Additionally, the anchor graph excels in clustering due to its ability to adeptly encapsulate relationships between data points of any shape. Inspired by these strengths, we opted for anchor graph in our approach to large-scale multi-view clustering.\n\n__Q3__: I think the author should provide the code to make the experimental results more convincing.\n\n__A3__: Thanks very much. We're sorry we can't use any links in our reply. __We've sent an anonymized link to the codes to the AC as required__.\n"}, {"review_id": "plae19wQ25", "reviewer": "Reviewer_PAbp", "summary": "This article proposed a novel orthogonal non-negative tensor factorization strategy for multi-view clustering, which well takes into account within-view spatial structure and between-view complementary information. Meanwhile, the optimization step has good convergency.", "questions": "1.Some formulas are not strictly written, which variable to solve should be clearly written.\n2.What does each letter of the matrix size represent?\n3. In Algorithm 1, how to calculate clustering label is not clear.\n4. Authors are advised to report specific hyper-parameters on each dataset.\n5. In Section 5.3, the author draws a conclusion that the clustering time increases linearly with the increase of anchor rate. This description seems imprecise.", "limitations": "1. The source code and datasets are encouraged to release.\n2. It is encouraged that the author can discuss the practical application scenarios of multi-view clustering technology.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "[a] The paper is well-written and easy to follow.\n[b] The proposed model is concise and has good interpretability.\n[c] The experimental results are substantial.", "weaknesses": "1. Some formulas are not strictly written, which variable to solve should be clearly written.\n2. What does each letter of the matrix size represent?\n3. In Algorithm 1, how to calculate the clustering label is not clear.\n4. Authors are advised to report specific hyper-parameters on each dataset.\n5. In Section 5.3, the author draws a conclusion that the clustering time increases linearly with the increase of anchor rate. This description seems imprecise.\n6. Some typos, for example: \n  In Section 5.3, the Schatten p norm should be ‘Schatten p norm.’\n  In Section 5.2, ‘according to Tables 2 and 3’ should be ‘according to Tables 2 and 3.’", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "__Q1__: Some formulas are not strictly written, which variable to solve should be clearly written.\n\n__A1__: Thanks very much. We double checked our manuscript and corrected formulas to explicitly indicator which variables need to be solved.\n\n__Q2__: What does each letter of the matrix size represent?\n\n__A2__: Thanks very much. In our article, $n$, $m$ and $k$ represent the number of samples, the number of anchors and the number of clusters, respectively. We have explicitly pointed out the meaning of each letter of the matrix size in the revised paper.\n\n__Q3__: In Algorithm 1, how to calculate the clustering label is not clear.\n\n__A3__: Thanks very much. The position of the largest element in each row of the indicator matrix is the label of the corresponding sample. We have explicitly pointed out this in our revised paper.\n\n__Q4__: Authors are advised to report specific hyper-parameters on each dataset.\n\n__A4__: Thanks very much. The corresponding hyper-parameters for each dataset are as follows: __MSRC__: anchor rate=0.7, $p$=0.5, $\\lambda$=100; __HandWritten4__: anchor rate=1.0, $p$=0.1, $\\lambda$=1180; __Mnist4__: anchor rate=0.6, $p$=0.1, $\\lambda$=5000; __AWA__: anchor rate=1.0, $p$=0.5, $\\lambda$=1000; __Reuters__: anchor rate=0.005(anchor number=100), $p$=0.4, $\\lambda$=1209800; __NoisyMNIST__: anchor rate=0.03, $p$=0.1, $\\lambda$=200000. We have explicitly pointed out this in our revised paper.\n\n__Q5__: In Section 5.3, the author draws a conclusion that the clustering time increases linearly with the increase of anchor rate. This description seems imprecise.\n\n__A5__: Thanks very much. I corrected these inaccurate representations. It should be that the time required for clustering is approximately linearly related to the increase of anchor rate.\n\n__Q6__: Some typos, for example: In Section 5.3, the Schatten p norm should be ‘Schatten p norm.’ In Section 5.2, ‘according to Tables 2 and 3’ should be ‘according to Tables 2 and 3.’\n\n__A6__: Thanks very much. We double checked the manuscript and corrected them.\n\n__Q7__: The source code and datasets are encouraged to release.\n\n__A7__: Thanks. We're sorry we can't use any links in our reply. __We've sent an anonymized link to the AC as required__. The datasets we used is open source.\n\n__Q8__: It is encouraged that the author can discuss the practical application scenarios of multi-view clustering technology.\n\n__A8__: Thank you for pointing that out. Multi-view clustering techniques have been applied in a myriad of practical situations across diverse fields. To highlight a few applications: \n\n1. In social media analysis, multi-view clustering allows for grouping based on textual content, visual features, and user network structures, thereby aiding in community identification or anomaly detection. \n2. Recommender systems leverage this clustering technique across various user and item views, which can enhance the personalization, accuracy, and diversity of their recommendations."}, {"review_id": "hghgDuKCNI", "reviewer": "Reviewer_J3E2", "summary": "This paper presented an orthogonal semi-nonnegative tensor factorization and proposed a novel tensorized anchor graph factorization model for Multiview clustering. Compared with existing NMF-based multi-view clustering methods, the proposed model has the following advantages: First, the proposed model directly factorizes 3-order tensorized anchor graph for clustering, while existing methods perform NMF on each view respectively. Thus the proposed method well exploits the within-view and between-views spatial structure information. Second, the proposed model has good interpretability for clustering with orthogonal non-negative constraint on tensorized soft labels of views. Third, authors use the tensor Schatten p-norm regularization as a rank approximation of the 3rd-order tensor which characterizes the cluster structure of multi-view data and exploits the between-view complementary information. Fourth, the paper presented a convergence analysis in theory. Experimental results indicate the efficiency of the proposed model on some databases.", "questions": "(1) In the paper, how to select anchor points or construct anchor graph?\n\n(2) In algorithm 1, mu and \\pho will be infinity. This will make the algorithm be unstable.\n\n  (3) What is the storage complexity and computational complexity for the model?\n\n (4) It is unclear for the variables in (19).\n", "limitations": "   Authors adequately addressed the limitations of the existing NMF-based multi-view clustering methods. ", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(1)\tThe paper presented an orthogonal semi-nonnegative tensor factorization.\n\n(2)\tThe paper proposed a novel tensorized anchor graph factorization model for Multiview clustering with good interpretability for clustering by orthogonal non-negative constraint on tensorized soft labels of views, this avoids post-processing for clustering.\n\n(3)\tThe proposed model directly factorized 3-order tensorized anchor graph for clustering, while existing methods perform NMF does not.\n\n(4)\tThe paper mathematically proved the convergence of the proposed algorithm for clustering.\n", "weaknesses": "(1) The paper does not provide the storage complexity and computational complexity.\n\n(2) It is unclear how to select anchor points or construct anchor graph.\n\n(3) It is unclear for the variables in (19).\n\n(4) In algorithm 1, mu and \\pho will be infinity. This will make the algorithm be unstable.\n\n(5) In the experiments, how to select \\ eta and \\ lambda?\n\n(6) Figure 6 indicates that the proposed model has good convergence, how does the performance (such as ACC) of the proposed model change?\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "__Q1__: The paper does not provide the storage complexity and computational complexity.\n\n__A1__: Thank you for the clarification. For Orth-NTF, the storage requirements for $\\mathcal{G}$, $\\mathcal{Q}$, $\\mathcal{H}$, $\\mathcal{J}$, $\\mathcal{Y}_1$, and $\\mathcal{Y}_2$ have complexities of $\\mathcal{O}(V(m+k)n)$, $\\mathcal{O}(V(n+k)k)$, $\\mathcal{O}(Vnk)$, $\\mathcal{O}(Vnk)$, $\\mathcal{O}(Vnk)$, and $\\mathcal{O}(V(n+k)k)$, respectively. Combining these, the total storage complexity for Orth-NTF is $\\mathcal{O}(Vnm+6Vnk+2vk^2)$.\n\nThe process of constructing $\\mathcal{S}$ has a computational complexity of $\\mathcal{O}(Vnmd+Vnm\\log(m))$. When updating the four variables—G, Q, H, and J—their respective computational complexities are $\\mathcal{O}(Vnkm+Vnk\\log(k))$, $\\mathcal{O}(Vm^2 k+Vmk^2)$, $\\mathcal{O}(Vnk)$, and $\\mathcal{O}(2Vnk\\log(Vk)+V^2 kn)$. Given that $m$, $n$, $k$, and $V$ are relatively small constants, the primary computational cost associated with updating the variables stands at $\\mathcal{O}(Vnkm+Vm^2 k)$. Summing it all up, the overall computational complexity of our proposed method is $\\mathcal{O}(Vnmd+Vm^2 k)$.\n\n__Q2__: It is unclear how to select anchor points or construct anchor graph.\n\n__A2__: Thanks very much. We adopt directly alternate sampling (DAS) to select anchors inspired by [1], and we construct anchor graph in the same way as [1]. We also explain about anchor selection and anchor graph construction in our Supplementary Material.\n\n[1] Li, X., Zhang, H., Wang, R., and Nie, F. Multiview clustering: A scalable and parameter-free bipartite graph fusion method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(1):330–344, 2022.\n\n__Q3__: It is unclear for the variables in (19).\n\n__A3__: Thanks very much. Sorry for the confusion. $\\overline{\\Lambda}^{(v)}$ and $\\overline{V}^{(v)}$ can be obtained by SVD($\\overline{\\mathcal{B}}^{(v)}$) = $\\overline{\\Lambda}^{(v)} X {\\overline{V}^{(v)}}^T$, where $\\overline{\\mathcal{B}}^{(v)}$=$ 2\\overline{\\mathcal{S}}^{(v)}$ $\\overline{\\mathcal{G}}^{(v)}$+$\\mu \\overline{\\mathcal{H}}^{(v)}$ + $\\overline{\\mathcal{Y}}_1^{(v)}$.\n\n__Q4__: In algorithm 1, mu and \\pho will be infinity. This will make the algorithm be unstable.\n\n__A4__: Thanks very much. In the experiments, we set the maximum of mu and $\\rho$ to $10^{13}$. We have explicitly pointed out this in our revised paper.\n\n__Q5__: In the experiments, how to select \\ eta and \\ lambda?\n\n__A5__: Thank you for noting that. The value of $\\eta$ influences the algorithm's convergence speed, and based on our empirical observations, we set $\\eta$ = 1.6. As for $\\lambda$, we initially estimate its range by considering the magnitude of the tensor Schatten $p$-norm regularization and subsequently fine-tune within that established range.\n\n__Q6__: Figure 6 indicates that the proposed model has good convergence, how does the performance (such as ACC) of the proposed model change?\n\n__A6__: Thanks very much. \nWhen the number of iteration increases, the clustering metric (such as ACC) overall improves gradually and tends to be constant with the convergence of the algorithm (__See Figure 2 in provided PDF__). It also indicates that our method has good clustering performances.\n"}, {"review_id": "tUTKxJaeC1", "reviewer": "Reviewer_xoPB", "summary": "Existing NMF-based multi-view clustering methods perform NMF on each view respectively and ignore the impact of between-view. To solve these problems, this paper proposes an orthogonal non-negative tensor factorization method with one-side orthogonal constraint. This method can process the multi-view data directly and can also take full advantage of the original spatial structure of the multi-view data. Extensive experiments on various benchmark datasets indicate that the proposed method can obtain satisfactory clustering performance.", "questions": "(1)\tThe authors mainly introduce the NMF and its variants in Introduction and Related work. However, existing published work on multi-view clustering based on anchor graphs has not been discussed in detail.\n(2)\tThe authors should design more reasonable and complete experiments to prove the effectiveness of the proposed method. In addition, the quality of figures and tables should be improved.\n", "limitations": "The authors need to provide more abundant experimental results to further prove the effectiveness of the proposed method.", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "(1)\tThe proposed method can directly consider the between-view relationship and perform Orth-NTF on the 3rd-order tensor which is composed of anchor graphs of views.\n(2)\tThe construction of anchor graph reduces the complexity of the proposed algorithm, while the tensor Schatten p-norm regularization explores the cluster structure of multi-view data.\n", "weaknesses": "(1)\tThe Introduction and Related work mainly introduce the NMF and its advantages and disadvantages. However, the authors did not adequately explain the motivation for designing the tensor factorization and orthogonal constraint.\n(2)\tIn section 5.2, the author only briefly introduces that their method has achieved good results without any in-depth analysis, which cannot provide any valuable insights for the readers.\n(3)\tThe authors should add more ablation experiments to clearly point out which basic design (or technical idea) of the whole method makes the largest contribution.\n(4)\tIt is not clear how to determine the optimal parameter \\lambda in Eq. (6). The authors also failed to analyze in detail the impact of different parameter combinations on the performance of the method, which leads to unconvincing results.\n", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "__Q1__: The Introduction and Related work mainly introduce the NMF and its advantages and disadvantages. However, the authors did not adequately explain the motivation for designing the tensor factorization and orthogonal constraint.\n\n__A1__: Thank you for highlighting that. Non-negative Matrix Factorization (NMF) is tailored primarily for second-order matrices. When processing third-order tensors, there's a need to first transform the tensor into a matrix before applying NMF. This step can lead to a loss of inherent spatial structural information from the third-order tensor. In contrast, Non-negative Tensor Factorization (NTF) sidesteps this issue. NTF directly decomposes third-order tensors, effectively capturing the spatial information they contain.\n\nIn the realm of multi-view clustering, conventional NMF-based methods apply NMF to each view independently. Subsequently, they combine the low-dimensional representations from different perspectives to arrive at a unified shared representation. This approach often overlooks the interrelationships between the views, which are crucial for clustering. Our model, however, directly implements NTF on the third-order tensor made up of anchor graphs from the various views. This ensures that the NTF not only acknowledges the relationships between the views but also harnesses the complementary information they offer. __Figure 1 in the provided PDF__ delineates the distinction between traditional NMF-based clustering techniques and our NTF-based approach. Furthermore, by incorporating an orthogonal constraint, our model offers distinct physical interpretability for clustering. This suggests that each row of the indicator matrix contains a single non-zero element, and the position of this element directly corresponds to the label of the respective sample.\n\n__Q2__: In section 5.2, the author only briefly introduces that their method has achieved good results without any in-depth analysis.\n\n__A2__: Thank you for pointing that out. Based on the experimental results presented in our paper, our method significantly outperforms other clustering approaches. This advantage may stem from the fact that our model directly factorizes the tensorized anchor graph—comprised of anchor graphs from various views—into the product of two non-negative tensors, one being an index tensor. As a result, our model effectively captures both the spatial structural information and the complementary data present in the anchor graphs from different perspectives. Additionally, with orthogonal and non-negative constraints in place, our model offers clear interpretability for clustering. This means that each row of the indicator matrix for every view contains a single non-zero element, with its position indicating the label of the associated sample. Consequently, our model can immediately provide the label without necessitating any post-processing, a step which other methods still require.\n\n__Q3__: The authors should add more ablation experiments to clearly point out which basic design (or technical idea) of the whole method makes the largest contribution.\n\n__A3__: Thanks very much for this constructive advice. In the experiments, we added some ablation experiments on orthogonal constraint and Schatten p-norm on four datasets. __Please see Table 1 in provided PDF__.\n\nIt can be found that, tensor Schatten p-norm regularization is overall superior to orthogonal constraint. The reason is that tensor Schatten p-norm regularization effectively characterizes both the complementary information and spatial structure information of index matrices of different views. Compared to orthogonal and tensor Schatten p-norm constraints, Joint constraints have great contribution for clustering. \n\n__Q4__: It is not clear how to determine the optimal parameter \\lambda in Eq. (6). The authors also failed to analyze in detail the impact of different parameter combinations on the performance of the method.\n\n__A4__: Thank you for bringing that to our attention. To determine the value of $\\lambda$, we initially approximate its range using the magnitude of the tensor Schatten p-norm regularization, followed by a more detailed fine-tuning within that range. The impact of varying parameter combinations on the method's performance can be seen __in Figure 3 of the provided PDF__. This figure highlights the clustering performance across different pairings of $p$ and $\\lambda$.\n\n__Q5__: The authors introduce the NMF and its variants in Introduction and Related work. However, existing published work on multi-view clustering based on anchor graphs has not been discussed in detail.\n\n__A5__: Thanks very much. Our main contribution of the paper is to propose a non-negative tensor factorization (NTF) model with orthogonal constraint, and then apply it to tensorized graph for large-scale multi-view clustering. Considering the length limitation of paper, we did not discuss the anchor-based clustering methods in detail. It is true that, there have many anchor-based multi-view clustering methods, but all of them separately process the anchor graph of each view respectively, and then fuse them to obtain a common-shared anchor graph. In contrast, our model applies NTF directly to the third-order tensor formed by anchor graphs from different views. As a result, it not only leverages the relationships between the views but also taps into the complementary information they offer. According to your constructive advice, we will add a detail description for anchor-based clustering methods in the revised paper.\n\n__Q6__: The authors should design more experiments. In addition, the quality of figures and tables should be improved.\n\n__A6__: Thank you for your advice. We have added ablation experiments as well as experiments on the impact of different parameter combinations on clustering performance (__See Table 1 and Figure 3 in the provided PDF__). We have adjusted the figures and tables to improve their readability and presentation."}]

Author rebuttal:
Supplementary PDF is uploaded here as required.

Meta-review: After the author-review discussions, all reviewers agree accpeting the paper. I would ecournage the authors to improve their paper according to the reviews.
Decision: Accept (poster)