Title: Orthogonal Non-negative Tensor Factorization based Multi-view Clustering
Venue: NeurIPS 2023
Accepted: yes
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

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 nonnegative 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.

1 Introduction

As one of the most typical methods in unsupervised learning, clustering has a wide scope of application [26; 4; 1] to assign data to different clusters according to the information describing the objects and their relationships. Non-negative matrix factorization (NMF) [19] is one of the representative methods of clustering, which is proved to be equivalent to K-means clustering [7]. Despite the widespread use of NMF, there are some drawbacks that have prompted some variants of NMF [8; 5; 27; 9; 3].

In particular, the one-side G-orthogonal NMF [8] can guarantee the uniqueness of the solution of matrix factorization and has excellent clustering interpretation. Also, Ding et al. proposed the semiNMF [9]. 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.

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 [13; 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-based 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 multi-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 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. This ensures that the NTF not only acknowledges the relationships between the views but also harnesses the complementary information they offer. Fig 1 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 betweenview 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.

Weprovide an 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 clustering-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 a 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 as in multiNMF. In addition, it sets a weight for a different view and introduces

view 1

…

…

view 2

view V

Multi-view Data

Low-dimensional

Representation 1

Low-dimensional

Representation 2

Low-dimensional

Representation V

…

Indicator Matrix

view 1

…

…

view 2

view V

Multi-view Data

Clustering

Fuse

Clustering

3rd-order tensor construction

Tensorized

Anchor Graph

NMF

Orth-NTF

NMF

NMF

Indicator Tensor

…

𝑯 (𝟏)

𝑯 (𝟐)

𝑯 (𝑽)

N

K

V

𝑯 (𝟏)

𝑯 (𝟐)

𝑯 (𝒗)

N

V

K

𝑯 (𝟏)

𝑯 (𝟐)

𝑯 (𝒗)

Construct

Rotate

Tensor

𝚫 (𝒌)

a 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 to improve clustering performance. Considering the above work, Wang et al. [29] proposed a graph regularization multi-view clustering method based on concept factorization (CF). CF is a variant of NMF and it is suitable for handling data containing negative.

As the size of data grows, lots of methods to accelerate matrix factorization are presented. Wang et al. [28] proposed a fast non-negative matrix triple factorization method. It 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 a 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. [37] introduced an anchor graph as the input of G-orthogonal NMF. The efficiency of matrix factorization is indeed improved due to the introduction of anchor graph.

Despite the fact that existing NMF-based multi-view clustering methods can perform the clustering tasks excellently, they 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.

3 Notations

We introduce the notations used throughout this paper. We use bold calligraphy letters for 3rd-order tensors, H ∈ R n 1 × n 2 × n 3 , bold upper case letters for matrices, H , bold lower case letters for vectors, h , and lower case letters such as h ijk for the entries of H . Moreover, the i -th frontal slice of H is H ( i ) . H is the discrete Fourier transform (DFT) of H along the third dimension, H = fft( H , [ ] , 3) . Thus, H = ifft( H , [ ] , 3) . The trace and transpose of matrix H are expressed as tr( H ) and H T . The F-norm of H is denoted by ∥ H ∥ F .

Definition 1 (t-product [18]) . Suppose A ∈ R n 1 × m × n 3 and B ∈ R m × n 2 × n 3 , the t-product A ∗ B ∈ R n 1 × n 2 × n 3 is given by

where A = bdiag( A ) and it denotes the block diagonal matrix. The blocks of A are frontal slices of A .

Definition 2. [12] Given H ∈ R n 1 × n 2 × n 3 , h = min( n 1 , n 2 ) , the tensor Schatten p -norm of H is defined as

where, 0 < p ⩽ 1 , σ j ( H ( i ) ) denotes the j-th singular value of H ( i ) .

It should be pointed out that for 0 < p ⩽ 1 when p is appropriately chosen, the Schatten p -norm provides quite effective improvements for a tighter approximation of the rank function [39; 36].

4 Methodology

4.1 Motivation and Objective

Non-negative matrix factorization (NMF) was initially presented as a dimensionality reduction method, and it is commonly employed as an efficient latent feature learning technique recently. Generally speaking, given a non-negative matrix X , the target of NMF is to decompose X into two non-negative matrices,

where X ∈ R n × p + , H ∈ R n × k + and G ∈ R p × k + . R n × p + means n -byp 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, ℓ 2 -norm and F-norm are frequently adopted as the objective function for the NMF. Considering that F-norm can make the model optimization easier, we 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 [8] and Semi-NMF [9]. By imposing an orthogonality constraint on one of the factor matrices in NMF, we obtain the objective function of the one-side orthogonal NMF,

If 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 we 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.

where S ∈ R n × m , H ∈ R n × k and G ∈ R m × k , m is 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 in the appendix.

As described in the previous section, the existing NMF-based multi-view clustering methods are essentially a matrix factorization on a single view combined with the integration of multiple views. It causes the loss of 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 non-negative matrix factorization is written in the following form:

The 3rd-order tensor construction process is illustrated in Fig 2.

In order to better exploit the complementary information and spatial structure between different views, we get inspiration from the excellent performance of the tensor Schatten p-norm [12; 38]. We introduce tensor Schatten p-norm regularization on the tensorial form of the cluster indicator matrix. Our objective function is formulated as follows:

where 0 < p ⩽ 1 , λ is the hyperparameter of the Schatten p -norm term.

Remark 1. The regularizer in the proposed objective (6) is used to explore the complementary information embedded in inter-views cluster assignment matrices H ( v ) ( v = 1 , 2 , · · · , V ). Fig. 2 shows the construction of tensor H , it can be seen that the k -th frontal slice ∆ ( k ) describes the similarity between N sample points and the k -th cluster in different views. The idea cluster assignment matrix H ( v ) 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, we impose tensor Schatten p-norm minimization [12] constraint on H , which can make sure each ∆ ( k ) has spatial low-rank structure. Thus ∆ ( k ) can well characterize the complementary information embedded in inter-views.

4.2 Optimization

Inspired by Augmented Lagrange Multiplier (ALM), we introduce two auxiliary variables Q and J and let H = Q , H = J , respectively, where Q ⩾ 0 . Then, we rewrite the model as the following unconstrained problem:

min L ( Q , H , G , J )

where Y 1 , Y 2 represent Lagrange multipliers and µ , ρ are the penalty parameters. The optimization process can therefore be separated into four steps:

· Solve G with fixed Q , H , J . (7) becomes:

After being implemented with discrete Fourier transform (DFT) along the third dimension. the equivalent representation of (8) in the frequency domain becomes:

where G = fft( G , [ ] , 3) , and the others in the same way.

Let Φ = ∥ ∥ ∥ S ( v ) -H ( v ) ( G ( v ) ) T ∥ ∥ ∥ 2 F , we can obviously get the following equation:

Setting the derivative ∂ Φ /∂ G ( v ) = 0 gives 2 G ( v ) -2( S ( v ) ) T H ( v ) = 0 . So the solution of (9) is:

· Solve H with fixed Q , G , J . (7) becomes:

And (12) is equivalent to the following in the frequency domain:

where H = fft( H , [ ] , 3) , and the others in the same way.

And (13) can be reduced to:

where W ( v ) 1 = Q ( v ) -Y ( v ) 1 µ and W ( v ) 2 = J ( v ) -Y ( v ) 2 ρ .

and it also can be reduced to:

is G ∗ = V [ I , 0 ]( Λ ) T .

Proof. From the SVD P = ΛS ( V ) T and together with (16), it is evident that

where H = ( V ) T GΛ , s ii and h ii are the ( i, i ) elements of S and H , respectively. It can be easily verified that H ( H ) T = I , where I is an identity matrix. Therefore -1 ⩽ h ii ⩽ 1 and s ii ⩾ 0 , Thus we have:

The equality holds when H is an identity matrix. tr( GP ) reaches the maximum when H = [ I , 0 ] .

So the solution of (15) is:

where B ( v ) = 2 S ( v ) G ( v ) + µ W ( v ) 1 + ρ W ( v ) 2 .

To solve (15), we introduce the following Theorem:

Theorem 1. Given G and P , where G ( G ) T = I and P has the singular value decomposition P = ΛS ( V ) T , then the optimal solution of

where Λ ( v ) and V ( v ) can be obtained by SVD B ( v ) = Λ ( v ) X ( V ( v ) )

· Solve Q with fixed H , G , J .

(20) is obviously equivalent to:

T (7) becomes:

According to [37], the solution of (21) is:

· Solve J with fixed Q , H , G . (7) becomes:

after completing the square regarding J , we can deduce

which has a closed-form solution as Lemma 2 [12]:

Lemma 2. Let Z ∈ R n 1 × n 2 × n 3 have a t-SVD Z = U ∗ S ∗ V T , then the optimal solution for

is X ∗ = Γ τ ( Z ) = U ∗ ifft( P τ ( Z )) ∗ V T , where P τ ( Z ) is an f-diagonal 3rd-order tensor, whose diagonal elements can be found by using the GST algorithm introduced in [12].

Now the solution of (24) is:

Finally, the optimization procedure for Multi-View Clustering via Orthogonal non-negative Tensor Factorization (Orth-NTF) is outlined in Algorithm 1.

Algorithm 1 Multi-View Clustering via Orthogonal non-negative Tensor Factorization (Orth-NTF)

return Clustering result (The position of the largest element in each row of the indicator matrix is the label of the corresponding sample).

4.3 Convergence Analysis

Theorem 2. [Convergence Analysis of Algorithm 1] Let P k = { Q k , H k , G k , J k , Y 2 ,k , Y 1 ,k } , 1 ≤ k < ∞ in (7) be a sequence generated by Algorithm 1 , then

P k is bounded with the assumption lim k → 0 max { µ k , ρ k } ( ¯ H ( v ) k +1 -¯ H ( v ) k ) = 0 ;

Any accumulation point of P k is a stationary KKT point of (7) .

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 1, which are ∥ Q k -H k ∥ ∞ ≤ ε , ∥ Q k -J k ∥ ∞ ≤ ε .

4.4 Complexity Analysis

For Orth-NTF, the storage requirements for G , H , Q , J , Y 1 and Y 2 have complexities of O ( V ( m + k ) n ) , O ( V ( n + k ) k ) , O ( V nk ) , O ( V nk ) , O ( V nk ) , O ( V nk ) and O ( V nk ) , respectively. Combining these, the total storage complexity for Orth-NTF is O ( V nm + vk 2 +6 V nk ) .

For the computational complexity, the process of constructing S has a computational complexity of O ( V nmd + V nm log( m )) . When updating the four variables, G , H , Q and J , their respective computational complexities are O ( V nmd + V nm log( m )) , O ( V m 2 k + V mk 2 ) , O ( V nk ) and O (2 V nk log( V k ) + 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 O ( V nkm + V m 2 k ) . Summing it all up, the overall computational complexity of our proposed method is O ( V nmd + V m 2 k ) .

(a) MSRC

(b) HandWritten4

(c) Mnist4

(d) AWA

5 Experiments

In this section, we demonstrate the performance of our proposed method through extensive experiments. It is compared with plenty of state-of-art multi-view clustering algorithms on some multi-view datasets. We evaluate the clustering performance by applying 7 metrics used widely, i.e. , 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-parameters 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 1. MSRC [34]; HandWritten4 [10]; Mnist4 [6]; AWA [11]; Reuters [2]; Noisy MNIST [32]; We choose the following 8 state-of-art multi-view clustering algorithms to compare with our proposed methods: AMGL [25]; MVGL [40]; CSMSC [24]; GMC [30]; LMVSC [17]; SMSC [16]; SFMC [21] FMCNOF [37]; FPMVS-CAG [31]; ETLMSC [35]; MSC-BG [38];

5.2 Experiments Result

The clustering performances are listed in Table 2 and Table 3. They contain four medium-scale datasets and two large-scale datasets. The corresponding experimental configurations and descriptions are included in the appendix. It is clear that our algorithm outperforms the other baseline algorithms on most of the datasets.

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

Dataset

MSRC

Metrics

ACC

NMI

Purity

PER

REC

F-score

ARI

AMGL [25] (IJCAI' 16)

0.751

±

0.000

0.704

±

0.000

0.789

±

0.000

0.621

±

0.000

0.744

±

0.000

0.674

±

0.000

0.615

±

0.000

MVGL [40] (TCYB' 17)

0.690

±

0.000

0.663

±

0.000

0.733

±

0.000

0.466

±

0.000

0.715

±

0.000

0.564

±

0.000

0.476

±

0.000

CSMSC [24] (AAAI' 18)

0.758

±

0.007

0.735

±

0.010

0.793

±

0.008

0.736

±

0.014

0.673

±

0.008

0.703

±

0.010

0.653

±

0.012

GMC[30] (TKDE' 19)

0.895

±

0.000

0.809

±

0.000

0.895

±

0.000

0.788

±

0.000

0.814

±

0.000

0.801

±

0.000

0.768

±

0.000

LMVSC [17] (AAAI' 20)

0.814

±

0.000

0.717

±

0.000

0.814

±

0.000

0.676

±

0.000

0.692

±

0.000

0.684

±

0.000

0.632

±

0.000

SMSC [16] (Inf Fusion' 20)

0.766

±

0.000

0.717

±

0.000

0.804

±

0.000

0.672

±

0.000

0.718

±

0.000

0.694

±

0.000

0.643

±

0.000

FMCNOF [37] (TIP' 21)

0.440

±

0.039

0.345

±

0.046

0.449

±

0.042

0.290

±

0.036

0.606

±

0.074

0.395

±

0.036

0.249

±

0.051

FPMVS-CAG [31] (TIP' 21)

0.786

±

0.000

0.686

±

0.000

0.786

±

0.000

0.684

±

0.000

0.642

±

0.000

0.731

±

0.000

0.629

±

0.000

SFMC [21] (TPAMI' 22)

0.810

±

0.000

0.721

±

0.000

0.810

±

0.000

0.657

±

0.000

0.782

±

0.000

0.714

±

0.000

0.663

±

0.000

ETLMSC [35] (TIP' 19)

0.962

±

0.000

0.937

±

0.000

0.962

±

0.000

0.926

±

0.000

0.931

±

0.000

0.928

±

0.000

0.917

±

0.000

MSC-BG [38] (TIP' 22)

0.981

±

0.000

0.960

±

0.000

0.981

±

0.000

0.961

±

0.000

0.963

±

0.000

0.962

±

0.000

0.956

±

0.000

ours

0.990

±

0.000

0.978

±

0.000

0.990

±

0.000

0.980

±

0.000

0.981

±

0.000

0.981

±

0.000

0.978

±

0.000

Dataset

HandWritten4

Metrics

ACC

NMI

Purity

PER

REC

F-score

ARI

AMGL [25] (IJCAI' 16)

0.704

±

0.000

0.762

±

0.000

0.732

±

0.000

0.591

±

0.000

0.781

±

0.000

0.670

±

0.000

0.628

±

0.000

MVGL [40] (TCYB' 17)

0.811

±

0.000

0.809

±

0.000

0.831

±

0.000

0.721

±

0.000

0.826

±

0.000

0.770

±

0.000

0.743

±

0.000

CSMSC [24] (AAAI' 18)

0.806

±

0.001

0.793

±

0.001

0.867

±

0.001

0.778

±

0.001

0.743

±

0.001

0.760

±

0.001

0.733

±

0.001

GMC[30] (TKDE' 19)

0.861

±

0.000

0.859

±

0.000

0.861

±

0.000

0.799

±

0.000

0.855

±

0.000

0.826

±

0.000

0.806

±

0.000

LMVSC [17] (AAAI' 20)

0.904

±

0.000

0.831

±

0.000

0.904

±

0.000

0.819

±

0.000

0.825

±

0.000

0.822

±

0.000

0.802

±

0.000

SMSC [16] (Inf Fusion' 20)

0.742

±

0.000

0.781

±

0.000

0.759

±

0.000

0.675

±

0.000

0.767

±

0.000

0.717

±

0.000

0.685

±

0.000

FMCNOF [37] (TIP' 21)

0.385

±

0.092

0.370

±

0.092

0.386

±

0.093

0.254

±

0.077

0.688

±

0.101

0.360

±

0.070

0.250

±

0.097

FPMVS-CAG [31] (TIP' 21)

0.744

±

0.000

0.753

±

0.000

0.744

±

0.000

0.681

±

0.000

0.636

±

0.000

0.762

±

0.000

0.642

±

0.000

SFMC [21] (TPAMI' 22)

0.853

±

0.000

0.871

±

0.000

0.873

±

0.000

0.775

±

0.000

0.910

±

0.000

0.837

±

0.000

0.817

±

0.000

ETLMSC [35] (TIP' 19)

0.938

±

0.001

0.893

±

0.001

0.938

±

0.001

0.886

±

0.001

0.890

±

0.001

0.888

±

0.001

0.876

±

0.001

MSC-BG [38] (TIP' 22)

0.889

±

0.000

0.922

±

0.000

0.889

±

0.000

0.871

±

0.000

0.893

±

0.000

0.882

±

0.000

0.869

±

0.000

ours

0.985

±

0.000

0.969

±

0.000

0.985

±

0.000

0.970

±

0.000

0.970

±

0.000

0.970

±

0.000

0.966

±

0.000

Dataset

Mnist4

Metrics

ACC

NMI

Purity

PER

REC

F-score

ARI

AMGL [25] (IJCAI' 16)

0.921

±

0.000

0.806

±

0.000

0.921

±

0.000

0.854

±

0.000

0.862

±

0.000

0.858

±

0.000

0.810

±

0.000

MVGL [40] (TCYB' 17)

0.919

±

0.000

0.803

±

0.000

0.919

±

0.000

0.851

±

0.000

0.860

±

0.000

0.856

±

0.000

0.807

±

0.000

CSMSC [24] (AAAI' 18)

0.641

±

0.000

0.601

±

0.010

0.728

±

0.008

0.607

±

0.014

0.767

±

0.008

0.677

±

0.010

0.553

±

0.012

GMC[30] (TKDE' 19)

0.920

±

0.000

0.807

±

0.000

0.920

±

0.000

0.853

±

0.000

0.861

±

0.000

0.857

±

0.000

0.809

±

0.000

LMVSC [17] (AAAI' 20)

0.892

±

0.000

0.726

±

0.000

0.892

±

0.000

0.808

±

0.000

0.812

±

0.000

0.810

±

0.000

0.747

±

0.000

SMSC [16] (Inf Fusion' 20)

0.909

±

0.000

0.774

±

0.000

0.909

±

0.000

0.834

±

0.000

0.841

±

0.000

0.837

±

0.000

0.783

±

0.000

FMCNOF [37] (TIP' 21)

0.697

±

0.119

0.490

±

0.102

0.711

±

0.096

0.558

±

0.118

0.683

±

0.073

0.611

±

0.095

0.460

±

0.145

FPMVS-CAG [31] (TIP' 21)

0.885

±

0.000

0.715

±

0.000

0.885

±

0.000

0.800

±

0.000

0.795

±

0.000

0.815

±

0.000

0.733

±

0.000

SFMC [21] (TPAMI' 22)

0.916

±

0.000

0.797

±

0.000

0.916

±

0.000

0.846

±

0.000

0.855

±

0.000

0.850

±

0.000

0.800

±

0.000

ETLMSC [35] (TIP' 19)

0.934

±

0.000

0.847

±

0.000

0.934

±

0.000

0.878

±

0.000

0.885

±

0.000

0.881

±

0.000

0.842

±

0.000

MSC-BG [38] (TIP' 22)

0.938

±

0.000

0.861

±

0.000

0.938

±

0.000

0.884

±

0.000

0.891

±

0.000

0.888

±

0.000

0.850

±

0.000

ours

0.977

±

0.000

0.926

±

0.000

0.977

±

0.000

0.955

±

0.000

0.956

±

0.000

0.955

±

0.000

0.941

±

0.000

Dataset

AWA

Metrics

ACC

NMI

Purity

PER

REC

F-score

ARI

MVGL [40] (TCYB' 17)

0.061

±

0.000

0.070

±

0.000

0.065

±

0.000

0.020

±

0.000

0.843

±

0.000

0.040

±

0.000

0.002

±

0.000

CSMSC [24] (AAAI' 18)

0.113

±

0.000

0.175

±

0.002

0.119

±

0.001

0.051

±

0.001

0.054

±

0.000

0.053

±

0.001

0.033

±

0.001

GMC[30] (TKDE' 19)

0.028

±

0.000

0.030

±

0.000

0.039

±

0.000

0.020

±

0.000

0.915

±

0.000

0.039

±

0.000

0.001

±

0.000

LMVSC [17] (AAAI' 20)

0.105

±

0.000

0.171

±

0.000

0.114

±

0.000

0.041

±

0.000

0.065

±

0.000

0.051

±

0.000

0.027

±

0.000

FMCNOF [37] (TIP' 21)

0.035

±

0.008

0.018

±

0.011

0.035

±

0.008

0.021

±

0.001

0.688

±

0.227

0.040

±

0.002

0.002

±

0.003

FPMVS-CAG [31] (TIP' 21)

0.106

±

0.000

0.183

±

0.000

0.111

±

0.000

0.069

±

0.000

0.047

±

0.000

0.142

±

0.000

0.042

±

0.000

SFMC [21] (TPAMI' 22)

0.042

±

0.000

0.044

±

0.000

0.049

±

0.000

0.023

±

0.000

0.592

±

0.000

0.044

±

0.000

0.006

±

0.000

ETLMSC [35] (TIP' 19)

0.631

±

0.000

0.783

±

0.000

0.656

±

0.000

0.498

±

0.000

0.580

±

0.000

0.536

±

0.000

0.526

±

0.000

MSC-BG [38] (TIP' 22)

0.493

±

0.000

0.550

±

0.000

0.511

±

0.000

0.059

±

0.000

0.494

±

0.000

0.105

±

0.000

0.073

±

0.000

ours

0.646

±

0.000

0.815

±

0.000

0.670

±

0.000

0.534

±

0.000

0.574

±

0.000

0.553

±

0.000

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 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.

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

variables. The result of the experiment is shown in Fig 7. It 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 factorization (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, we provide an optimization algorithm for the proposed method and validate the effectiveness of this approach in extensive experiments on different datasets.

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A Proof of Convergence

A.1 Proof of the 1st part

Lemma 3 (Proposition 6.2 of [20]) . Suppose F : R n 1 × n 2 → R is represented as F ( X ) = f ◦ σ ( X ) , where X ∈ R n 1 × n 2 with SVD X = U diag( σ 1 , . . . , σ n ) V T , n = min( n 1 , n 2 ) , and f is differentiable. The gradient of F ( X ) at X is

where θ = ∂f ( y ) ∂y | y = σ ( X ) .

To minimize ¯ Q ( v ) at step k + 1 in (21) in the main body, the optimal ¯ Q ( v ) k +1 needs to satisfy the first-order optimal condition

By using the updating rule ¯ Y ( v ) 1 ,k +1 = ¯ Y ( v ) 1 ,k + µ k ( ¯ H ( v ) k -¯ Q ( v ) k ) , we have

According to our assumption lim k → 0 µ k ( ¯ Q ( v ) k +1 -¯ Q ( v ) k ) = 0 , we know Y 1 ,k +1 is bounded.

To minimize J at step k +1 in (24) in the main body, the optimal J k +1 needs to satisfy the first-order optimal condition

Recall that when 0 < p < 1 , in order to overcome the singularity of ( | η | p ) ′ = pη/ | η | 2 -p near η = 0 , we consider for 0 < ϵ ≪ 1 the approximation

Letting J ( i ) = U ( i ) diag ( σ j ( J ( i ) ) ) V ( i )H , then it follows from Lemma 3 that

And then one can obtain

Let us denote ˜ F V = 1 √ V F V , F V is the discrete Fourier transform matrix of size V × V , F H V denotes its conjugate transpose. For J = J × 3 ˜ F V and using the chain rule in matrix calculus, one can obtain that is bounded.

And it follows that

thus Y 2 ,k +1 appears to be bounded.

Moreover, by using the updating rule Y 1 = Y 1 + µ ( H -Q ) , Y 2 = Y 2 + ρ ( H -J ) , we can deduce ( i = 1 , 2 )

Thus, summing two sides of (28) from k = 1 to n , we have

Observe that

we have the right-hand side of (29) is finite and thus L ( Q n +1 , G n +1 , H n +1 , J n +1 ; Y i,n ) is bounded. Notice from (7) in the main body

and each term of (30) is nonnegative, following from the boundedness of L ( Q n +1 , G n +1 , H n +1 , J n +1 ; Y i,n ) , we can deduce each term of (30) is bounded. And ∥ J n +1 ∥ p Sp ⃝ being bounded implies that all singular values of J n +1 are bounded and hence ∥ J n +1 ∥ 2 F (the sum of squares of singular values) is bounded. Therefore, the sequence { J k } is bounded.

Because

and in light of the boundedness of Q k , Y 1 ,k , it is clear that H k is also bounded.

And from (8) in the main body, it is evident that ∥ ¯ G ( v ) k ∥ 2 F ≤ ∥ ( ¯ S ( v ) ) T ∥ 2 F ∥ ¯ H ( v ) k ∥ 2 F , so ¯ G ( v ) k is also bounded. So G k is bounded.

A.2 Proof of the 2nd part

From Weierstrass-Bolzano theorem, there exists at least one accumulation point of the sequence P k . We denote one of the points P ∗ = { H ∗ , Q ∗ , G ∗ , J ∗ , Y ∗ 1 , Y ∗ 2 } . Without loss of generality, we assume {P k } + ∞ k =1 converge to P ∗ .

Note that from the updating rule for Y 1 , we have

Note that from the updating rule for Y 2 , we have

In the ¯ G ( v ) -subproblem (8) in the main body, we have

In the J -subproblem (24) in the main body, we have

Therefore, one can see that the sequences H ∗ , Q ∗ , G ∗ , J ∗ , Y ∗ 1 , Y ∗ 2 satisfy the KKT conditions of the Lagrange function (7) in the main body.

B Anchor Selection And Graph Construction

Inspired by [21], we adopt directly alternate sampling (DAS) to select anchors.

First of all, with the given data matrices { X ( v ) } V v =1 , we concatenate the data matrix of each view along the feature dimension. The connected feature matrix X ∈ R n × d can be represented as X = [ X (1) ; X (2) ; · · · ; X ( v ) ] , where d is the sum of the number of features in each view. Let θ i represent the i -th sample of the d -dimensional features, which can be calculated as

where dT = ∑ V v =1 d v , 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 θ = [ θ 1 , θ 2 , · · · , θ n ] ∈ R n . We choose the point where the maximum score is located as the anchor. The position of the largest score is

Then the 1st anchor of the v -th view is b ( v ) 1 = x ( v ) Index .

After that, let θ Index be the score of the anchor selected from the last round, then we normalize the score of each sample by:

Then the score θ i can be updated as

Finally, we repeat (32) - (34) m times to select m anchors. After selecting m anchors, we construct an anchor graph of each view S ( v ) , in the same way, as [21].

C More Details of the Experiments

C.1 Experimental Configurations

The Reuters and NoisyMNIST 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 , HandWritten 4 , Mnist 4 and AWA are implemented on a laptop computer with an Inter Core i5-8300H CPU and 16 GB RAM, using Matlab R2018b. Codes are available: https://github.com/xdjingli/Orth-NTF.

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 = 0.7, p = 0.5, λ = 100.

HandWritten4: anchor rate = 1.0, p = 0.1, λ = 1180.

Mnist4: anchor rate = 0.6, p = 0.1, λ = 5000.

AWA: anchor rate = 1.0, p = 0.5, λ = 1000.

Reuters: anchor rate = 0.005 (anchor number = 100), p = 0.4, λ = 1209800.

NoisyMnist: anchor rate = 0.03, p = 0.1, λ = 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 λ are variable parameters. In this section we 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 in Fig 4 and Fig 5. 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 p . We set the value of p to be 0.1 to 1.0 with a step of 0.1. We obtained the results of ACC, NMI, and Purity in experiments with different values of p as shown in Fig 6. The best clustering results are obtained on MSRC, HandWritten4, Mnist4 and AWA when the values of p are 0 . 5 , 0 . 2 , 0 . 1 , and 0 . 5 , respectively. It 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 λ . To determine the value of λ , 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 Fig 7. This figure highlights the clustering performance across different pairings of p and λ .

(a)MSRC

(b)HandWritten4

(c)Mnist4

(d)AWA

(a)MSRC

(b)HandWritten4

(c)Mnist4

(d)AWA

(a)MSRC

(b)HandWritten4

(c)Mnist4

(d)AWA

0.70.8

(a) MSRC

(b) HandWritten4

(c) Mnist4

(d) AWA

0.6

AC

0.5

ACC

0.5

ACC

0.5

ACC

0.4

0.2

0.01

0

（×10²)

0.1

1

10

100

0.1

0.2

0.3

P

0.6

0.9

0.02

0

入(x10)

0.2

2

20

200

0.1

0.20.3

0.4

P

0.5

0.6

0.9

0.05

0

入（x10)

0.5

5

50

500

0.1

0.20.3

0.4

P

0.5

0.6

0.7

0.80.9

0.01

0

入（×10)

0.1

1

10

100

0.1

0.2

0.3

0.40.5

P

1

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nExisting 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.\n\nSTRENGTHS:\n(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\nWEAKNESSES:\n(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\nQUESTIONS:\n(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\nLIMITATIONS:\nThe authors need to provide more abundant experimental results to further prove the effectiveness of the proposed method.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\n(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\nWEAKNESSES:\n(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\nQUESTIONS:\n(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\nLIMITATIONS:\nAuthors adequately addressed the limitations of the existing NMF-based multi-view clustering methods.", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-03"}

comments: 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.

STRENGTHS:
[a] The paper is well-written and easy to follow.
[b] The proposed model is concise and has good interpretability.
[c] The experimental results are substantial.

WEAKNESSES:
1. Some formulas are not strictly written, which variable to solve should be clearly written.
2. What does each letter of the matrix size represent?
3. In Algorithm 1, how to calculate the clustering label is not clear.
4. Authors are advised to report specific hyper-parameters on each dataset.
5. 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.
6. 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.’

QUESTIONS:
1.Some formulas are not strictly written, which variable to solve should be clearly written.
2.What does each letter of the matrix size represent?
3. In Algorithm 1, how to calculate clustering label is not clear.
4. Authors are advised to report specific hyper-parameters on each dataset.
5. 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.
2. It is encouraged that the author can discuss the practical application scenarios of multi-view clustering technology.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-04

comments: 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.

STRENGTHS:
1.The overall structure of the paper is clear and comprehensive.
2.The research problem and innovative aspects are well-defined, and they are supported by sufficient experimental evidence.
3.The proposed method demonstrates promising results for the task of multi-view clustering.
4.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.
2.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?
3.I think the author should provide the code to make the experimental results more convincing.

QUESTIONS:
Please refer to the weakness section.

LIMITATIONS:
N/A
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-06