# High-dimensional multi-view clustering methods

K. Jbilou
Approximation and Numerical Analysis department
ULCO University
Dunkerque, France 62228
<EMAIL_ADDRESS>
& A. Ratnani
Khwarizmi department
Mohammed VI Polytechnic University
Benguerir, Morocco 43150
<EMAIL_ADDRESS>
A. Zahir
Khwarizmi department
Mohammed VI Polytechnic University
Benguerir, Morocco 43150
<EMAIL_ADDRESS>

###### Abstract

Multi-view clustering has been widely used in recent years in comparison to
single-view clustering, for clear reasons, as it offers more insights into the
data, which has brought with it some challenges, such as how to combine these
views or features. Most of recent work in this field focuses mainly on tensor
representation instead of treating the data as simple matrices. This permits
to deal with the high-order correlation between the data which the based
matrix approach struggles to capture. Accordingly, we will examine and compare
these approaches, particularly in two categories, namely graph-based
clustering and subspace-based clustering. We will conduct and report
experiments of the main clustering methods over a benchmark datasets.

_Keywords_ Multi-view clustering $\cdot$ Graph based Multi-view clustering
$\cdot$ Subspace based Multi-view clustering $\cdot$ Tensor learning $\cdot$
Representation learning

##  1 Introduction

Clustering (Li et al., 2019; Yang and Wang, 2018; Zhao et al., 2017) is a
fundamental topic of data mining as well as machine learning, especially when
there are no labels for data objects. Clustering results are often used in
subsequent applications, such as community detection, recommendation, and
information retrieval using features. A feature is any individual measurable
property or characteristic of an object, for example, an image is represented
by different kind of features as colors, edges or texture, a document can be
represented by different language. A video may be encoded in different amount
of images and sound.

The combine of these features are refereed as multi-view data, which it gave
birth to a new paradigm. Although each view may have enough information by
it’s own, it likely provides complementary information to the other views,
this leaning paradigm is humanly nature, as we also tend to learn in a similar
fashion, as in many real-life problems multi-view data arise naturally. The
challenge in this new paradigm, is how to combine these features for a better
result than the single view. In this paper, we focus on multi-view
unsupervised learning and particularly, multi-view clustering. Note that we
will focus on a more particular subject, that will be mentioned after giving
the appropriate definition at the right time.

One naive MVC technique is to use a single-view clustering algorithm on
concatenated information acquired from several views. This strategy however,
may fail if greater emphasis is placed on particular specific view than on
others. As a result, multi view subspace clustering that learns common
coefficient matrices, multi view k-means (Cai et al., 2013) and multi-kernel
based MVC (Guo et al., 2014) have received increased attention.

The remainder of the paper is structured as follows: In Section 2, we present
the notations throughout the paper which are listed as follows according to
Kilmer and Martin (2011), and state some definitions that we will use
throughout this review, then in section 3, we will talk about the single view
clustering, and in section 4, we will introduce the different categories in
the multi view clustering, both the matrix based as a first part, and the
tensor based in the second part, experimental results are given in Section 5.
Finally, we give concluding remarks and future possible work in the last
section 6.

##  2 Preliminaries

This section will present some preliminaries and notations and related to the
subject. For convenience, we summarize the frequently used notations in table
1.
Throughout this review, we will denote the matrices are denoted by boldface
capital letters (e.g $X$), an element of a matrix is denoted by $x_{ij}$.
Vectors will be denoted by a boldface lowercase letters (i.e $\mathbf{x}$),
and an element of a vector is denoted by $x_{i}$. The Frobenius norm of $X$ is
denoted by $\|X\|_{F}$. The $\ell_{2,1}$ norm of a matrix that is not
necessary $X$, is defined as sum of norm-2 of its columns, which encourages
the sparsity property of these columns. $\mathbf{I}_{n}$ denotes an identity
matrix with size n.

We denote an indirect graph with $M$ views as
$G=\left(V,E_{(1)},E_{(2)},\cdots,E_{(V)}\right)$, where $V$ is the set of
nodes and $E_{(v)}\subset V\times V$ is the set of edges from view $v$ of $G$.
We denote $S_{(v)}\in\mathbb{R}^{n\times n}$ the weighted affinity matrix in
view $v$, and it’s entries denote the pairwise affinity between nodes of $G$
in view $v$.
We denote by the diagonal matrix $D^{(v)}$ the degree matrix of $G$ in view
$v$ where $D_{i}^{(v)}=\sum_{j=1}^{n}S_{i,j}^{(v)}$. We also denote by
$L^{(v)}=D^{(v)}-A^{(v)}$ the Laplacian matrix of $G$ in view $v$, if not
specified otherwise.

We will denote tensors calligraphy letters (i.e. $\mathcal{X}$). For third
order tensors $\mathcal{X}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}$, the
i-th horizontal slice, j-th lateral slice and k-th frontal slice are defined
by $\mathcal{X}(i,:,:,),\mathcal{X}(:,j,:)$ and $\mathcal{X}(:,:,k)$
respectively.
We can transform a tensor to a matrix in mode k, named the mode-k
Matricization of a tensor $\mathcal{X}$ denoted,
$\mathcal{X}_{(k)}\in\mathbb{R}^{N_{k}\times J}$ where $J=\Pi_{q=1,q\neq
k}^{3}N_{q}$
We can also define the product of a tensor by a matrix $U$ in mode k, named
the k-mode (matrix) of a tensor denoted $\times{}_{k}$:

$\mathcal{Y}=\mathcal{X}\times{}_{k}U\Leftrightarrow Y_{(k)}=UX_{(k)}$

. We define some tensor decomposition, namely the CP, tucker and T-svd
decomposition:
a) The Tucker decomposition is a form of higher-order PCA. It decomposes a
tensor into a core tensor multiplied (or transformed) by a matrix along each
mode:

$\mathcal{Z}\approx\mathcal{G}\times_{1}U_{1}\times_{2}U_{2}\times_{3}U_{3},$
(1)

where the tensor $\mathcal{G}$ is called, the core tensor. The matrices
$U_{k}$, $k=1,\ldots,3$ are the factor matrices (which are usually orthogonal)
and can be thought of as the principal components in each mode.
b) The CANDECOMP / PARAFAC (CP) factorization by:

$\mathcal{Z}\approx\llbracket
Z^{(1)},Z^{(2)},Z^{(3)}\rrbracket=\sum_{r=1}^{R}z_{r}^{(1)}\circ
z_{r}^{(2)}\circ z_{r}^{(3)},$ (2)

with $Z^{(k)}=\left[z_{1}^{(k)},\cdots,z_{R}^{(k)}\right]$ are factor
matrices, $R$ is the number of factors, and $\llbracket\cdot\rrbracket$ is
used for shorthand, and $\circ$ is the outer product.
These two decomposition can be defined for higher order tensors, the next
decomposition (T-SVD) is reserved for the 3 order tensors.
c) The T-SVD decomposition: Before defining it, or even the T-product. We
start by defining the Discrete Fourier Transformation for vectors before
showing The DFT on tensors.

The Discrete Fourier Transformation (DFT) is essential in the tensor-tensor
product. We provide some background information and notations here. The DFT
matrix $F_{n}$ has the following structure:

$F_{n}=[w^{(i-1)(j-1)}],\;i,j=1,\ldots,n,$

where $w=e^{\frac{2\pi i}{n}}$ is the primitive n-root of 1.
The matrix $F_{n}$ verifies $F_{n}F_{n}^{*}=nI_{n}$, which means that
$F_{n}^{-1}=\frac{1}{n}F_{n}^{*}$. The symbol ∗ represents the conjugate
transpose. The DFT of a vector $v$ denoted by $\widetilde{v}=fft(v)$ is given
by

$\widetilde{v}=F_{n}v\in\mathbb{C}^{n}.$ (3)

The Fast Fourier Transform (FFT) is a less expensive method for computing
$\widetilde{v}$, which costs only $O(nlogn)$, instead of $O(n^{2})$. Denote
the circulant matrix of $v$ as

$\operatorname{circ}(v)=\left[\begin{array}[]{cccc}v_{1}&v_{n}&\cdots&v_{2}\\\
v_{2}&v_{1}&\cdots&v_{3}\\\ \vdots&\vdots&\ddots&\vdots\\\
v_{n}&v_{n-1}&\cdots&v_{1}\end{array}\right]\in\mathbb{R}^{n\times n}$

It can be diagonalized by the DFT matrix, i.e.

$F_{n}\operatorname{circ}(v)F_{n}^{-1}=diag(\widetilde{v}).$ (4)

For any vector $w\in\mathbb{R}^{n}$, the associated $\widetilde{w}=F_{n}w$
satisfies :

$\widetilde{w}_{1}\in\mathbb{R},\quad
conj(\widetilde{w_{i}})=\widetilde{w}_{n-i+2},\text{for}\;i=2,3\dots\lfloor\frac{n+1}{2}\rfloor$
(5)

Now we consider the DFT for tensors, We denote $\widetilde{\mathcal{X}}$ the
discrete Fast Fourier transform (FFT) of $\mathcal{X}$ along the third
dimension i.e. $\widetilde{\mathcal{X}}=fft(\mathcal{X},[\,],3)$. Similarly,
the tensor can be obtained back by the inverse FFT (IFFT) along the third
dimension $\mathcal{X}=ifft(\widetilde{\mathcal{X}},[\,],3)$. For a tensor
$\mathcal{X}$ whose frontal slices are
$X^{(1)},\ldots,X^{\left(N_{3}\right)}$, we define the block diagonal matrix,
the block curculant matrix, an unfolding operator and it’s inverse, unfold as:

$\operatorname{bcirc}(\mathcal{X})=\left[\begin{array}[]{cccc}X^{(1)}&X^{\left(N_{3}\right)}&\cdots&X^{(2)}\\\
X^{(2)}&X^{(1)}&\cdots&X^{(3)}\\\ \vdots&\vdots&\ddots&\vdots\\\
X^{\left(N_{3}\right)}&X^{\left(N_{3}-1\right)}&\cdots&X^{(1)}\end{array}\right].$
$\operatorname{bdiag}(\mathcal{X})=\left[\begin{array}[]{llll}X^{(1)}&&&\\\
&X^{(2)}&&\\\ &&\ddots&\\\ &&&X^{\left(N_{3}\right)}\end{array}\right].$
$\operatorname{unfold}(\mathcal{X})=\left[\begin{array}[]{c}X^{(1)}\\\
X^{(2)}\\\ \vdots\\\ X^{\left(N_{3}\right)}\end{array}\right],\text{ fold
}(\operatorname{unfold}(\mathcal{X}))=\mathcal{X}.$

The block circulant matrix can be block-diagonalized as follows:

$\left(F_{N_{3}}\otimes
I_{N_{1}}\right)\cdot\operatorname{bcirc}(\mathcal{X})\cdot\left(F_{n_{3}}^{-1}\otimes
I_{N_{2}}\right)=\operatorname{bdiag}(\mathcal{\widetilde{X})},$ (6)

where $\otimes$ denotes the Kronecker product. By using the same lemma as
before we have :

$\left\\{\begin{array}[]{l}\widetilde{X}^{(1)}\in\mathbb{R}^{N_{1}\times
N_{2}}\\\
\operatorname{conj}\left(\widetilde{X}^{(i)}\right)=\widetilde{X}^{\left(N_{3}-i+2\right)},i=2,\cdots,\left\lfloor\frac{N_{3}+1}{2}\right\rfloor.\end{array}\right.$
(7)

In contrast, for any given $\widetilde{\mathcal{X}}\in\mathbb{C}^{N_{1}\times
N_{2}\times N_{3}}$ satisfying (7), there exists a real tensor
$\mathcal{X}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}$ such that (6) is
satisfied.
The T-product $\mathcal{X}*\mathcal{B}$, with
$\mathcal{B}\in\mathbb{R}^{N_{2}\times L\times N_{3}}$ is a tensor of size
$N_{1}\times L\times N_{3}$ defined by:

$\mathcal{X}*\mathcal{B}=\text{ fold
}(\operatorname{bcirc}(\mathcal{X})\cdot\text{ unfold }(\mathcal{B})).$ (8)

The T-product is also called the circular convolution operation. It is shown
that

$\mathcal{C}=\mathcal{X}*\mathcal{B}\iff\widetilde{C}=\widetilde{X}\widetilde{B},$

which suggests an efficient way to calculate the T-product based on the FFT.
The transpose of $\mathcal{X}$ denoted by
$\mathcal{X}^{T}\in\mathbb{R}^{N_{2}\times N_{1}\times N_{3}}$ is obtained by
transposing each of the frontal slices and then reversing the order of
transposed frontal slices 2 through $N_{3}$.
The identity tensor $\mathcal{I}$ is a tensor whose first frontal slice is the
identity matrix and the remaining frontal slices has zero entries.
A tensor is orthogonal if it satisfies the relationship:

$\mathcal{X}^{T}*\mathcal{X}=\mathcal{X}*\mathcal{X}^{T}=\mathcal{I}.$

A tensor is called, f-diagonal if each of its frontal slices is a diagonal
matrix.
The T-SVD decomposition theorem (Kilmer and Martin, 2011; Kilmer et al., 2013;
Hao et al., 2013) states that $\mathcal{X}$ can be factorized as:

$\mathcal{X}=\mathcal{U}*\mathcal{S}*\mathcal{V}^{*},$ (9)

where $\mathcal{U}\in\mathbb{R}^{N_{1}\times N_{1}\times
N_{3}},\mathcal{V}\in\mathbb{R}^{N_{2}\times N_{2}\times N_{3}}$ are
orthogonal, and $\mathcal{S}\in\mathbb{R}^{N_{1}\times N_{2}\times N_{3}}$ is
an f-diagonal tensor. It can also be written as

$\mathcal{X}=\sum_{i=1}^{\min\left(N_{1},N_{2}\right)}\mathcal{U}(:,i,:)*\mathcal{S}(i,i,:)*\mathcal{V}(:,i,:)^{\mathrm{T}}.$

We also have a truncated T-SVD representation that provide an optimal
approximation in the same way as the truncated matrix SVD: for
$k<\min\left(N_{1},N_{2}\right)$, we define
$\mathcal{X}_{k}=\sum_{i=1}^{k}\mathcal{U}(:,i,:)*\mathcal{S}(i,i,:)*\mathcal{V}(:,i,:)^{\mathrm{T}}$,
then we have the following result:

$\mathcal{X}_{k}=\underset{\widetilde{\mathcal{X}}\in\mathbb{M}}{\operatorname{argmin}}\|\mathcal{X}-\widetilde{\mathcal{X}}\|_{F},$

where
$\mathbb{M}=\left\\{\mathcal{C}=\mathcal{X}*\mathcal{Y}\mid\mathcal{X}\in\mathbb{R}^{N_{1}\times
k\times N_{3}},\mathcal{Y}\in\mathbb{R}^{k\times N_{2}\times N_{3}}\right\\}$
which gives a best low rank approximation to the matrix in terms of the
Frobenius norm under rank k constraint.
We define the T-SVD based tensor nuclear norm (T-TNN) (Zhang et al., 2014):

$\|\mathcal{X}\|_{\circledast}=\sum_{i=1}^{N_{3}}\left\|\widetilde{X}^{(i)}\right\|_{*},$

where $\left\|.\right\|_{*}$ is the nuclear norm, which is exactly the sum of
all its singular values. It is proven to be a valid norm and the tightest
convex relaxation to $l_{1}$ norm of the tensor multi-rank.
We can further define a more generalization of this norm, called the weighted
tensor nuclear norm:

$\|\mathcal{X}\|_{\omega,\circledast}=\sum_{i=1}^{N_{3}}\left\|\widetilde{X}^{(i)}\right\|_{\omega,*}=\sum_{i=1}^{N_{3}}\sum_{j=1}^{\min\left(N_{1},N_{2}\right)}\omega_{j}*\sigma_{j}\left(\widetilde{X}^{(i)}\right)\text{,
}$

where, $\sigma_{j}\left(\widetilde{X}^{(i)}\right)$ denotes the $j$ largest
singular value of $\widetilde{X}^{(i)}$.
We also define the Shatter p-norm, for $0<p\leqslant 1$:

$\|\mathcal{X}\|_{Sp}^{p}=\sum_{i=1}^{N_{3}}\left\|\widetilde{\mathcal{X}}^{(i)}\right\|_{Sp}^{p}=\sum_{i=1}^{N_{3}}\sum_{j=1}^{\min\left(N_{1},N_{2}\right)}\sigma_{j}\left(\widetilde{\mathcal{X}}^{(i)}\right)^{p}.$

Another norm that will be presented is the t-Gamma tensor quasi-norms
presented by (Wu et al., 2020) defined as:

$\|\mathcal{X}\|_{t-\gamma}=\frac{1}{N_{3}}\sum_{n_{3}=1}^{N_{3}}\left\|\widetilde{X}^{(n_{3})}\right\|_{\gamma},$

where, $\|.\|_{\gamma}$ represents the Gamma quasi-norm for matrices (Wu et
al., 2020) , for $Z\in\mathbb{R}^{N_{1}\times N_{2}}$

$\|Z\|_{\gamma}=\sum_{i=1}^{\min(N_{1},N_{2})}\frac{(1+\gamma)\sigma_{i}(Z)}{\gamma+\sigma_{i}(Z)},\gamma>0.$

The table 1 summarizes the different important notations that will be used in
this work, for convenience, we add a dashline that separates the notations
related to tensors.

Table 1: Summary of notations in this paper $x$,$\mathbf{x}$ | Scalar, vector. |  $X$, $\mathcal{X}$ | Matrix, tensor
---|---|---|---
$\|X\|_{F}$ | $\|X\|_{F}=\sqrt{\sum_{ij}X_{ij}^{2}}.$ | $\|X\|_{*}$ | Nuclear norm.
$\operatorname{tr}(X)$ | The trace function. | $\|X\|_{2,1}$ | $\|X\|_{2,1}=\sum_{j}\|X(:,j)\|_{2}.$
$\operatorname{Diag}(X)$ | The diagonal entries. | $\|X\|_{\gamma}$ | Gamma quasi-norm.
$X^{*}$ | The conjugate transpose. |  |
$\mathcal{X}_{ijk}$ | The $(i,j,k)$-th entry of $\mathcal{X}.$ | $\|\mathcal{X}\|_{F}$ | $\|\mathcal{X}\|_{F}=\sqrt{\sum_{ijk}\mathcal{X}_{ijk}^{2}}$
$\mathcal{X}(i,:,:)$ | The $i$-th horizontal slice of $\mathcal{X}.$ | $\|\mathcal{X}\|_{1}$ | $\|\mathcal{X}\|_{1}=\sum_{ijk}\mid\mathcal{X}_{ijk}\mid.$
$\mathcal{X}(:,i,:)$ | The $i$-th lateral slice of $\mathcal{X}.$ | $\|\mathcal{X}\|_{2,1}$ | $\|\mathcal{X}\|_{2,1}=\sum_{i,j}\|\mathcal{X}(i,j,:)\|_{2}.$
$X^{(i)}$ | The $i$-th frontal slice of $\mathcal{X}.$ | $\|\mathcal{X}\|_{\omega,\circledast}$ | Weighted based tensor nuclear norm.
$\widetilde{\mathcal{X}}$ | $\widetilde{\mathcal{X}}=\operatorname{fft}(\mathcal{X},[],3).$ | $\|\mathcal{X}\|_{\circledast}$ | T-SVD based tensor nuclear norm.
$\mathcal{X}_{(i)}$ | Mode-i matricization of $\mathcal{X}.$ | $\|\mathcal{X}\|_{Sp}^{p}$ | Shatten p-norm.
$\mathcal{X}^{T}$ | The transpose of $\mathcal{X}.$ | $\|\mathcal{X}\|_{FF1}$ |  $\|\mathcal{X}\|_{FF1}=\sum_{i}\|\mathcal{X}(i,:,:)\|_{2}$.
$\widetilde{\mathcal{X}}$ | The Fast Fourier Transform of $\mathcal{X}$. | $\|\mathcal{X}\|_{t-\gamma}$ | T-Gamma tensor quasi-norms.

Next, we will show some clustering methods. There are roughly two categories
of clustering that depend on the number of views: Single view clustering, and
the more general framework, the multi-view clustering.

##  3 Single view clustering

Single view clustering is performing the clustering task using only single
view feature. It fails to explore the correlations among the other features
from different sources.
Given a feature matrix $X\in\mathbb{R}^{d\times n}$, where $n$ represents the
number of samples, and $d$ is the dimension feature. Two categories that are
going to be presented: Graph based and subspace.

###  3.1 Single view graph based clustering:

In single view graph based clustering, each node or a vertex of a graph
corresponds to an object and each edge represents a relationship between two
objects. The general methods that are in this category, follow the following
steps to partitions the $n$ data points into $c$ clusters:

  1. 1.

Construct a data graph matrix $S\in\mathbb{R}^{n\times n}$, where each entry
in $S$ represents the similarity between the points.

  2. 2.

Compute the graph Laplacian matrix $L_{S}=D_{S}-\frac{S^{T}+.S}{2}$, where
$D_{\mathrm{S}}$ is a diagonal matrix:
$d_{i,i}=\sum_{j}\dfrac{s_{ij}+s_{ji}}{2}$.

  3. 3.

Compute the embedding matrix $F\in\mathbb{R}^{n\times c}$ by solving the
following problem
$\min_{F\in\mathbb{R}^{n\times
c}}\operatorname{Tr}\left(F^{T}L_{S}F\right)=\min_{F\in\mathbb{R}^{n\times
c}}\frac{1}{2}\sum_{i,j}s_{ij}||f_{i}-f_{j}||_{2}$.

  4. 4.

Clustering F into $c$ groups with an additional clustering algorithm (e.g.,
K-means).

###  3.2 Single view subspace clustering

The subspace learning approach is built on the assumption that all the views
are generated from a latent subspace. Without loss of generality, we can write
the data matrix $X$ in the matrix form $X=XZ+E$, where
$Z\in\mathbb{R}^{n\times n}$ is the representation matrix,
$E\in\mathbb{R}^{d\times n}$ is the error matrix. The general method that
falls in this category aims to solve the following optimization problem

$\min_{Z,E}\Psi(Z)+\lambda\varphi(E)\text{ s.t }X=XZ+E,$ (10)

where $\Psi(Z)$ is the regularization which imposes desired property on
$Z\in\mathbb{R}^{n\times n},\varphi(E)$ is the loss function to remove noise
and $\lambda>0$ is a balanced parameter that is controlled by the user. It can
also be added other constraints, for example $Diag(Z)=0$, which enforces that
each data point can only be represented as a combination of the remaining
points except itself. After getting the representation matrix $Z$, we compute
the affinity matrix and further employ it as the input of the spectral
clustering algorithm for achieving the final clustering result.
As such, Low-Rank Representation method (LRR) (Liu et al., 2010) introduces
the low rank regularization to subspace clustering by putting the
regularization as the nuclear norm, and the loss function as the $l_{2-1}$
norm. On the other hand, Sparse Subspace Clustering (SSC) (Elhamifar and
Vidal, 2013), represents each data point as a sparse linear combination of
other points and captures the data local structure. SSC is based on self
representation property i.e. data matrix stands for a dictionary. According to
the experimental results in (Wang et al., 2013), LRR misclassifies different
data points than SSC. As a result, in order to capture both the global and
local structure of the data, low-rank and sparsity requirements must be
combined (Wang et al., 2013).

##  4 Multi-view clustering

Multi-view clustering tries to benefit from all these views, to perform the
clustering task. The success of the multi-view leanings is based on two
principles: the complementary principal and the consensus principal.
Specifically, the complementary principle, which means that a view can contain
information that the others do not have. The consensus principle aims to
maximize the agreement on the different views, which makes the MVC has better
robustness partition and accuracy than the single view clustering. The core
challenge of MVC is how to integrate all the different features to obtain a
reliable affinity matrix.

MVC can also be categorized into roughly 5 categories, co-training style
algorithms, multi-kernel learning, multi-view graph clustering, multi-view
subspace clustering and multi-task multi-view clustering. Co-training based
methods search for the results that agree across different views using a co-
training strategy, by bootstraps the partitions of different views by using
the prior or learned knowledge from one another. The method relies on
conditional independence. Co-regularization based methods jointly regularize
the hypotheses to explore the complementary information. Multiple Kernel
Learning (MKL) methods usually combine different kernels by adding them
equally or learning the combination weights either linearly or non-linearly.
The method have high computational complexity.
We will first discuss the data construction:
Given a multi-view data $\\{X^{(v)}\\}_{v=1,\dots,V}$, where
$X^{(v)}=[x_{1}^{(v)},\ldots,x_{n}^{(v)}]\in\mathbb{R}^{d_{v}\times n}$ is the
data matrix of the $v$-th view, $d_{v}$ is the dimension of the $v$-th view,
$n$ is the number of data points and $V$ is number of views. Although the
feature dimension is variable, we can always construct a third order tensor
$\mathcal{X}$, first, we create block diagonal matrix
$B_{i}\in\mathbb{R}^{D\times V}$ with $D=\sum_{v}d_{v}$ as such:

$B_{i}=\left[\begin{array}[]{llll}X^{(1)}(:,i)&&&\\\ &X^{(2)}(:,i)&&\\\
&&\ddots&\\\ &&&X^{(V)}(:,i)\end{array}\right]$

then twisting it to obtain $\mathcal{B}_{i}\in\mathbb{R}^{D\times 1\times V}$,
and finally, construct the tensor $\mathcal{X}\in\mathbb{R}^{D\times n\times
V}$ by concatenating the n tensors $\mathcal{B}_{i}$ along the second mode as
in figure 1.

Figure 1: Constructing a third order tensor from the matrices in
$\mathbb{R}^{d_{v}\times n}$

By following these steps, we can rid of the variable dimension, while
preserving the structure of tha data.

###  4.1 Multi-view graph clustering

Multi-view graph clustering aims to find a fusion graph of all views, then
apply the classical clustering methods on the fusion graph, as spectral
clustering or graph-cut algorithms to produces the final clustering.

Figure 2: The general method of multi-view graph-based clustering Yang and
Wang (2018)

Spectral clustering relies on computing the eigenvectors of the graph
Laplacian matrix.
One of the most representative spectral clustering methods is the co-
regularized spectral clustering. It performs standard spectral clustering on
each view to get the corresponding indicator matrix, and then gets the common
indicator matrix by minimizing disagreement between indicator matrices.
However, it treats each indicator matrix equally, which is not ideal in real
applications.

As it is said, the challenge is how to assign an appropriate weight to each
view, even though we find some method Chaudhuri et al. (2009) that don’t
bother themselves of these different views, this strategy neglects the
importance of different graphs and may suffer when an unreliable graph is
added to, other methods like in Xia et al. (2010) adds new hyper parameters,
but it’s effect on the clustering is cannot be underestimated, and other
methods try, to learns the weight automatically, which eliminates the need of
any additional hyper parameter, as we present some.

The method used in Auto-weighted Multiple Graph Learning (AMGL) Nie et al.
(2016), supposes that the normalized Laplacian matrices $L_{S}^{(v)}$ are
constructed, and then solve the following problem:

$\min_{F\in\mathbb{R}^{n\times
c}F^{T}F=\mathbf{I}}\sum_{v=1}^{V}\sqrt{\operatorname{Tr}\left(F^{T}L_{S}^{(v)}F\right)}.$
(11)

which is solved by fixing the parameters $\alpha$ then solves:

$\min_{F\in\mathbb{R}^{n\times
c}F^{T}F=\mathbf{I}}\sum_{v=1}^{V}\alpha^{(v)}\operatorname{Tr}\left(F^{T}L_{S}^{(v)}F\right).$

Then, the parameters is updated using the formula:

$\alpha^{(v)}=\dfrac{1}{\operatorname{Tr}\left(F^{T}L_{S}^{(v)}F\right)}.$

The method can also be used for semi supervised clustering, but it is not in
the scope of this review. In AMGL method, there are no hyper parameters, it
can learns both the weight of each view and also the embedding matrix $F$, and
then we treat each row of $F$ as a new representation of each data point and
compute the clustering labels by using k-means algorithm.

After merging the view, some methods modify the traditional spectral
clustering framework, they don’t need to search the uniform similarity matrix,
but instead, try to learn an indicator matrix, that is used directly for
clustering. One can also impose a constraint Laplacian rank, which guarantees
that we have c clusters (components in the language of graphs), as stated by
the following theorem.

###### Theorem 1

(Marsden, 2013) The multiplicity c of the eigenvalue 0 of the Laplacian matrix
$L_{S}$ is equal to the number of connected components in the graph with the
similarity matrix $S$.

In (Nie et al., 2017a), a Parameter-weighted Multi view Clustering (PwMC)
method has been proposed. This method is based on the generalization of the
CLR method: It learns a new similarity matrix from the old similarities of the
v-th view $S^{(v)}$, computed from the original data:

$\displaystyle\min_{w_{(v)},U}$
$\displaystyle\sum_{v=1}^{V}w^{(v)}\left\|U-S^{(v)}\right\|_{F}^{2}+\gamma\|w\|_{2}^{2}$
(12) s.t. $\displaystyle w_{(v)}\geq
0,\mathbf{1}_{v}^{T}\mathbf{w}=1,u_{ij}\geq 0,\mathbf{1}^{T}\mathbf{u}=1$
$\displaystyle\operatorname{rank}\left(L_{U}\right)=n-c.$

The rank constraint is hard to tackle, since the diagonal matrix $D_{S}$
depends also on $S$. As the Laplacian is positive semi-definite, it’s
eigenvalues are positive, which makes the constraint rank:
$\operatorname{rank}\left(L_{U}\right)=n-c$ can be ensured by
$\sum_{i=1}^{c}\sigma_{i}\left(L_{U}\right)=0$, where $\sigma_{i}(V)$
represents the i-th smallest eigenvalue of a matrix $M$. Thus we can replace
the constraint, by adding $\lambda\sum_{i=1}^{c}\sigma_{i}\left(L_{U}\right)$
to the objective function, with $\lambda$ large enough. In addition, we can
further make the objective function easier to solve, since we have the
following theorem:

###### Theorem 2

(Fan et al., 1949) We have the following result

$\sum_{i=1}^{c}\sigma_{i}\left(L_{S}\right)=\min_{F\in\mathbb{R}^{n\times
c},F^{T}F=\mathbf{I}}\operatorname{Tr}\left(F^{T}L_{S}F\right).$

The method depends on the hyper parameter $\gamma$, which affects the
clustering accuracy since it is confirmed to be dataset-related. A new method
proposed named Self-weighted Multi-view Clustering (SwMC) (Nie et al., 2017b),
deals with this problem by removing completely $\gamma\|w\|_{2}^{2}$, from the
equation (12).

As SwMc, supposes that the similarity matrix of each view is given, or
calculated separately by k-means for example, Some approaches bypass this
stage since the performance is significantly dependent on the view graphs that
are generated. Due to the complex distribution of data in real-world
applications, it is challenging to manually create a suitable similarity
graph. When the graph quality is inadequate, the performance of these
approaches drops dramatically. This decreases the algorithm flexibility. One
solution is to calculate the similarity matrix directly from the data points,
which is found more reliable and noise reducing, in addition, it calculates
the weights automatically and eliminates the post processing step as well, the
method named Multi-View Clustering and Semi-Supervised Classification with
Adaptive Neighbours (MLAN) (Nie et al., 2017a) solves the following problem:

$\displaystyle\min_{S}\sum_{v=1}^{V}\sqrt{\operatorname{tr}\left(X^{(v)}L_{S}X^{(v)}\right)}+\alpha\|S\|_{F}^{2}$
(13) $\displaystyle\text{ s.t. }\mathbf{s}_{i}^{T}\mathbf{1}=1,0\leq
s_{ij}\leq 1,\quad\operatorname{rank}\left(L_{S}\right)=n-c,$

Where the following relationship is used instead in (Nie et al., 2017a) :

$\sum_{v=1}^{V}\operatorname{tr}\left(X^{(v)}L_{S}X^{(v)}\right)=\sum_{v=1}^{V}\sum_{i,j}\left\|\mathbf{x}_{i}^{(v)}-\mathbf{x}_{j}^{(v)}\right\|_{2}^{2}s_{ij}.$
(14)

This part (14) is usually added to give an affinity matrix the following
property: High intra-cluster similarity and low inter-cluster similarity: For
each data point $x_{i}$, it belongs to one of the c classes, and can be
connected by any other data point $x_{j}$ with a probability $s_{ij}$, this
probability can be seen as the similarity between the points, thus, a large
probability should be given to the closer sampler, in other words, a small
distance $\left\|\mathbf{x}_{i}^{(v)}-\mathbf{x}_{j}^{(v)}\right\|_{2}^{2}$,
which justifies the first part of the equation. In other words, the multi-view
graph embedding tries to preserve the local structure of the graphs, and
graphs with similar local structure tend to be close to each other in the
original multi-view feature space.
MLAN only learns a global graph for all views (without building a graph for
each view), thus a method named Graph-based Multi-view Clustering (GMC) Wang
et al. (2020) tries to find both while guarding the same objective:

$\displaystyle\min_{S^{(v)},U}$
$\displaystyle\sum_{v=1}^{V}\left(\operatorname{tr}\left(X^{(v)}L_{S}X^{(v)}\right)+\beta\left\|S^{(v)}\right\|_{2,1}^{2}\right)$
(15)
$\displaystyle+\sum_{v=1}^{V}w_{v}\left\|U-S^{(v)}\right\|_{F}^{2}+2\lambda\operatorname{Tr}\left(F^{T}L_{U}F\right)$
$\displaystyle\text{ s.t. }s_{ii}^{(v)}=0,s_{ij}^{(v)}\geq
0,\mathbf{1}^{T}\mathbf{s}_{i}^{(v)}=1,$ $\displaystyle u_{ij}\geq
0,\mathbf{1}^{T}\mathbf{u}_{i}=1,F^{T}F=\mathbf{I}.$

As most methods used only one similarity graph. A recent work, (Rong et al.,
2021) created a multi-metric similarity graph refinement and fusion strategy
for MVC in which several similarity graphs were built using different metrics
and an informative unified graph was obtained by adaptively fusing these
similarity graphs.

###  4.2 Multi-view subspace clustering

Multi-view subspace clustering attempts to discover a latent space from all
the different views in order to deal with high-dimensional data, and then
applies a traditional clustering approach to this subspace.

Figure 3: The general method of multi-view subspace clustering Yang and Wang
(2018)

It normally consists of the following steps:

  1. 1.

Learning the latent matrices or tensors using various learning methods.

  2. 2.

Computing the affinity (similarity) matrix by averaging latent matrices or
tensors.

  3. 3.

Applying spectral clustering on the similarity matrix.

Existing multi-view clustering algorithms may be divided into two types:
Matrix-based and tensor-based representation learning.
Matrix theory, which organizes data in matrix form, is central to matrix-
oriented approaches. It is discovered that this stream has a common flaw. It
solves the problem by simply merging the views without taking into
consideration the high-order correlations between these views at the same
time, which is why the second stream (tensor-based representation), which has
already received a lot of attention, will be the subject of this discussion.
We can generalize certain single-view methods to multi-view ways by addressing
the following problem:

$\displaystyle\min_{Z^{(v)},E^{(v)}}$
$\displaystyle\sum_{v=1}^{V}\left(\Psi(Z^{(v)})+\lambda\varphi(E^{(v)})\right)$
(16) s.t $\displaystyle X^{(v)}=X^{(v)}Z^{(v)}+E^{(v)}.$

After obtaining these multiple learned subspace representation
$\left\\{Z^{(v)}\right\\}_{v=1}^{V}$, the final affinity matrix is calculated
by combining all subspace representation of each view:

$S=\dfrac{1}{2V}\sum_{v=1}^{V}\left(\mid{Z}^{(v)}\mid+\mid{Z}^{(v)^{\mathrm{T}}}\mid\right).$
(17)

Finally, using the combined affinity matrix $S$, a spectral clustering
approach or k-means is performed; however, this formulation ignores the link
between subspaces and treats each subspace representation individually.

We get multiple subspace clustering methods by using alternative norms for the
loss function and regularization term, to mention a few.

In the method named consistency-specificity multi-view subspace clustering
(CSMSC) by Luo et al. (2018), they divided the learned subspace representation
$Z^{(v)}$ into a learned consistent $C$ that is shared across the views, and
specific self-representation matrices $D^{(v)}$ with data under different
views, as for the regularization function, the nuclear norm is chosen for $C$,
to guarantee the low rank property in order to excavate more shared
information among different views, and the $l_{2}$ norm for $D^{(v)}$ for the
connectedness property, and the $l_{21}$ norm for the loss function:

$\displaystyle\min_{C,D^{(v)},E^{(v)}}$
$\displaystyle\lambda_{1}\|C\|_{*}+\lambda_{2}\sum_{v=1}^{V}\left(\left\|D^{(v)}\right\|_{2}^{2}+\left\|E^{(v)}\right\|_{2,1}\right)$
(18) s.t. $\displaystyle X^{(v)}=X^{(v)}\left(C+D^{(v)}\right)+E^{(v)}.$

In this case, the learned affinity matrix is calculated differently:

$S=\frac{\mid C\mid+\mid C^{T}}{2}\mid+\frac{1}{2V}\sum_{v=1}^{V}\mid
D^{(v)}\mid+\mid D^{(v)}\mid^{T}.$ (19)

Instead of the classical independence tests such as Spearmans rho and Kendalls
tau that can only discover linear relationships. In the paper diversity-
induced Multi-view Subspace Clustering (DiMSC), Cao et al. (2015) used The
Hilbert Schmidt Independence Criterion (HSIC) (the empirical version) as a
diversity term in to recover the links between distinct subspace
representations, and to enhance the complementary between the multi-view
representations.

###  4.3 Tensorized MVC

####  4.3.1 Tensorized subspace based MVC

As previously stated, treating data as matrices rather than as a full tensor
implies that the relationship between subspace is neglected, and the tensor-
based problem often solves the following problem:

$\displaystyle\min_{\mathcal{Z},\mathcal{E}}$
$\displaystyle\psi(\mathcal{Z})+\lambda\varphi(\mathcal{E})$ (20) S.t.
$\displaystyle X^{(v)}=X^{(v)}Z^{(v)}+E^{(v)}$
$\displaystyle\mathcal{Z}=\Phi_{1}\left(Z^{(1)},Z^{(2)},\ldots,Z^{(V)}\right)$
$\displaystyle\mathcal{E}=\Phi_{2}\left(E^{(1)},E^{(2)},\ldots,E^{()}\right),$

where $\Phi$ is a suitable regularized tensor norm, $\varphi$ is a valid
tensor norm for noise cancellation or sparsity, and the functions $\Phi_{1}$
and $\Phi_{2}$ produce the tensor $\mathcal{Z}$ and $\mathcal{E}$ by merging
various matrices $Z(v)$ and $E(v)$ respectively, to a three-mode tensor. It
was intended to define two functions rather than one since various techniques
may utilize them differently; hence, $\Phi_{1}$ rotates the tensor as shown in
the figure 4.

Figure 4: Tensor construction and rotation it

Tensor rotation can be advantageous in a variety of ways: Because it performs
the transformation along the third dimension, the coefficients of the
representation matrices are kept in the Fourier domain, reducing computing
complexity greatly.

For example, inspired by the essential connection between spectral clustering
and the Markov chain, a method called essential tensor learning for multi-view
spectral clustering (TLMSC) Wu et al. (2018) builds a tensor based on multi-
view transition probability matrices of the Markov chain; It first creates a
similarity matrix of each view, then generates the transition probability
$P^{(v)}=D^{(v)^{-1}}S^{(v)}$, and feeds it to the proposed model:

$\displaystyle\min_{\mathcal{Z},\mathcal{E}}\|\mathcal{Z}\|_{\circledast}+\lambda\|\mathcal{E}\|_{2,1}$
(21) $\displaystyle\text{ s.t. }\mathcal{P}=\mathcal{Z}+\mathcal{E}$
$\displaystyle\mathcal{P}=\Phi_{1}\left(P^{(1)},P^{(2)},\ldots,P^{(V)}\right),$

where $\phi_{1}$ constructs and rotates the tensor to $n\times V\times n$.
TLMSC gives impressive result, however it employs the soft thresholding
function to shrink each singular values with the same parameter. As a result,
the prior knowledge of matrix singular values is not used even if it is
important for the clustering, thus a method known as weighted tensor-nuclear
norm minimization (WTNNM) Xie et al. (2021) is introduced to deal with this
issue; It shrinks singular values with different parameters and captures the
high order correlation underlying multi-view data by using $\psi$ as the
weighted nuclear norm of tensor rather than the tensor nuclear norm.
The present tensor-nuclear norm minimization keeps some big singular values
while rejecting others. This suggests that the bigger single values are
significant, whilst the lower singular values are insignificant. However, in
practice, such an assumption is not always correct, and higher single values
may include unwanted information.
In Cheng et al. (2019), a method named tensor based Representation Learning
method for Multi-view Clustering (tRLMvC) Cheng et al. (2019), adopts a
different idea than the previous ones presented, it searches the
representation tensor using T- product, then performs a factorization via the
Tucker decomposition, the problem that is solved is:

$\displaystyle\min_{\mathcal{Z},\mathcal{C},U_{1},U_{3},U_{3}}$
$\displaystyle\|\mathcal{X}-\mathcal{X}*\mathcal{Z}\|_{F}^{2}+\alpha\|\mathcal{Z}\|_{FF1}$
(22)
$\displaystyle+\beta\left\|\mathcal{Z}-\mathcal{C}\times_{1}U_{1}\times_{2}U_{2}\times_{3}U_{3}\right\|_{F}^{2},$

Then, any single view clustering algorithm such as k-means may be employed on
the matrix $H=[U_{1},U_{2}]$, with the column vector $U_{3}$ characterizing
the contribution of each view.

####  4.3.2 Tensorized graph based MVC

To acquire clustering results, current matrix-based algorithms execute
traditional spectral clustering after collecting the similarity matrix. None
of them explore how to increase the stability of spectral clustering or how to
utilize higher order correlation in data, hence the tensor-based technique is
introduced to address this issue.

To obtain the clustering results, the approach in Xie et al. (2018) first
pursues the subspace representation tensor using the tensor nuclear norm
(different tensor rank approximations used in similar methods) before
generating the final affinity matrix. Of this way, the two primary components
of spectral clustering; The representation tensor and the affinity matrix, are
learnt separately.

To address this issue, a method named Graph regularized Low-rank Tensor
representation and the Affinity matrix (GLTA) Chen et al. (2019) solves the
problem by simultaneously learning the representation tensor and the affinity
matrix, with the following optimization problem:

$\displaystyle\min_{\mathcal{Z},\mathcal{E},S,\omega}$
$\displaystyle\sum_{v=1}^{V}\Big{(}\alpha\operatorname{tr}\left(Z^{(v)}L^{(v)}Z^{(v)^{T}}\right)+\beta\omega_{v}\left\|Z^{(v)}-S\right\|_{F}^{2}\Big{)}+\|\mathcal{E}\|_{2,1}$
(23) $\displaystyle\text{ s.t. }X^{(v)}=X^{(v)}Z^{(v)}+E^{(v)}$
$\displaystyle\mathcal{Z}=\mathcal{C}\times{}_{1}U_{1}\times{}_{2}U_{2}\times{}_{3}U_{3},U_{i}^{\prime}U_{i}=I$
$\displaystyle\mathcal{E}=\Phi_{2}\left(E^{(1)},E^{(2)},\ldots,E^{(V)}\right),\quad\omega_{v}\geq
0,\mathbf{1}^{T}\mathbf{\omega}=1,$

where $L^{(v)}$ is the normalized Laplacian matrix, GLTA learns also the local
view-specific geometrical structures and the diverse contributions of distinct
attributes. Specifically, when creating the final affinity matrix, it uses the
Tucker decomposition to encapsulate the low-rank attribute, the manifold
regularization to show the local view-specific geometrical structures, and
gives appropriately varying weights to distinct characteristics.

In the method Task-driving affinity matrix for accurate Multi-view clustering
(TAMMC), Wang et al. (2021) tries to preserve the local affinities of all
views via a graph regularization on self-expressive tensor, by penalizing a
Laplacian rank on a learned common subspace by solving the following problem:

$\displaystyle\min_{S,\mathcal{Z}}$
$\displaystyle\sum_{v=1}^{V}\left(\alpha\operatorname{Tr}\left(Z^{(v)}L^{(v)}Z^{(v)^{T}}\right)+\lambda\|S-Z^{(v)}\|_{F}^{2}\right)$
(24)
$\displaystyle+\frac{1}{2}\|\mathcal{X}-\mathcal{X}*\mathcal{Z}\|_{F}^{2}+\theta\|\mathcal{Z}\|_{TNN}$
$\displaystyle\text{ s.t. }\quad\mathbf{1}^{T}\mathbf{s}_{i}=1,\quad
s_{ij}\geqslant 0,\operatorname{rank}\left(L_{S}\right)=n-c,$

where, the Laplacian matrix of each view, is calculated using Gaussian
distance based on t-nearest neighbors. The first term represents the
approximation error using the Tensor Frobenius norm, the second and third
terms penalize the tensor representation, thereby making it low-rank and
imposing the estimated subspace to preserve the local affinity (Samples that
are near in raw space are likely to remain close in the mapped subspace). The
last term is used to make a consensus representation. With the rank Laplacian
constraint condition, the affinity matrix is used directly for clustering.

A more recent work, a method named Multi-view Spectral Clustering with
Adaptive Graph Learning and Tensor Schatten p-norm Zhao et al. (2022), which
the suggested problem is:

$\displaystyle\min_{F,\mathcal{Z},\mathbf{w},\mathbf{\alpha}}$
$\displaystyle\sum_{v=1}^{V}\left(\frac{1}{w_{v}}\operatorname{tr}\left(X^{(v)}L^{(v)}X^{(v)}\right)+\lambda\frac{1}{\alpha_{v}}\operatorname{Tr}\left(F^{T}L^{(v)}F\right)\right)+\beta\|\mathcal{Z}\|_{Sp}^{p}$
(25) s.t. $\displaystyle
F^{T}F=\mathbf{I},\mathbf{1}^{T}\mathbf{z_{i}^{(v)}}=1,z_{ij}^{(v)}\geqslant
0$ $\displaystyle\mathbf{1}^{T}\mathbf{w}=1,w_{v}\geqslant
0,\mathbf{1}^{T}\mathbf{\alpha}=1,\alpha_{v}\geqslant 0,$

where $L^{(v)}$ is the Laplacian matrix of the corresponding similarity matrix
$Z_{(v)}$. The Schatten p-norm regularizer is used to minimize the divergence
between graphs. It guarantees also that the rank of the learned graph
approximates the target rank, i.e. cluster number.
In practice, noises and redundancy information are frequently mixed in with
the original features. As a result, the consensus similarity graph that was
learned m may be erroneous. Li et al. (2022) overcomes this issue by training
the adaptive neighbor graph in a new low dimensional embedding space rather
than the original feature space. First, it employs the view-specific
similarity graph $W^{(v)}$ using an adaptive neighbor graph learning:

$\displaystyle\forall i=1,...,n$
$\displaystyle\min_{W^{(v)}_{i}}\sum_{j=1}^{n}\left\|x_{i}^{(v)}-x_{j}^{(v)}\right\|_{2}^{2}w^{(v)}_{ij}+\gamma
w_{ij}$ (26) $\displaystyle w^{(v)}_{ij}\geq
0,\mathbf{1}^{T}\mathbf{w}^{(v)}_{i}=1.$

The new low-embedding space is learnt:

$\displaystyle\min_{F^{(v)},\mathcal{T}}$
$\displaystyle-\lambda\sum_{v=1}^{V}\operatorname{tr}\left(F^{(v)}A^{(v)}F^{(v)^{T}}\right)+\frac{1}{2}\|\mathcal{F}-\mathcal{T}\|_{F}^{2}+\|\mathcal{T}\|_{w,*}$
(27) $\displaystyle\text{ s.t. }F^{(v)^{T}}F^{(v)}=\mathbf{I}_{c}$
$\displaystyle\mathcal{F}=[\overline{F}^{(1)}\overline{F}^{(1)^{T}};\cdots;\overline{F}^{(n)}\overline{F}^{(n)^{T}}],$

where the normalized spectral embedding matrix $\overline{F}^{(v)}$ is
obtained by normalizing the rows of $F^{(v)}$, and the normalized affinity
matrix is

$A^{(v)}=D^{(v)^{-0.5}}W^{(v)}D^{(v)^{0.5}},$ (28)

and the tensors $\mathcal{F},\mathcal{T}\in\mathbf{R}^{n\times V\times n}$.
Lastrly, the consensus graph $S$ can be learned using:

$\displaystyle\min_{S}$
$\displaystyle\sum_{v=1}^{V}\sum_{i,j=1}^{n}\left\|\overline{f}_{i}^{(v)}-\overline{f}_{j}^{(v)}\right\|_{2}^{2}s_{ij}+\gamma
s_{ij}$ (29) $\displaystyle s_{ij}\geq 0,\mathbf{1}^{T}\mathbf{s}_{i}=1.$

##  5 Experiments

In this section, we do some experiments on various data sets, with varying
amounts of samples, views, and clusters. This allows for a thorough evaluation
of the performance of the various algorithms.
For the algorithms that has $\lambda\sum_{i=1}^{c}\sigma_{i}(L)$ minimization
part. We set $\lambda=1$, as most of the papers in their codes, in each
iteration, we double $\lambda$ if the number of components is less than the
number of clusters wanted, and we divide it by 2, if the number of components
is less than c. In practice, we can know the comparison between the number of
components and the number of clusters mentioned in Marsden (2013) using the
fact that if $\sum_{i=1}^{c+1}\sigma_{i}(L)\leq 0$ implies to the fact that
number of connected components of the graph is larger than c, and
$\sum_{i=1}^{c}\sigma_{i}(L)\geq 0$ implies that number of connected
components of the graph is smaller than c.

###  5.1 Data sets

We used multiple different real world data sets used, that are commonly used
in the literature, Which can be categorized into 3 Types: document, image and
graph. The majority of image multi-view data sets are derived from initial
single-view image data sets. The popular pre-computed multi-view picture data
sets are shown. Readers can also create such multi-view data sets on their own
for future research.
The vast majority of image multi-view data sets are created from single-view
image data sets. Popular pre-computed multi-view image data sets are
displayed. The document data sets are either obtained from a single view data
set or naturally described in numerous views. Readers can also generate their
own multi-view data sets for future investigation.

  * •

One-hundred plant species leaves data set leafs (100Leaves)
111https://archive.ics.uci.edu/ml/datasets/One-
hundred+plant+species+leaves+data+set: It consists of 1600 samples, of 100
different species, 4 views are given: shape descriptor, fine scale margin and
texture histogram.

  * •

Handwritten digit data set.
222http://archive.ics.uci.edu/ml/datasets/Multiple+Features: is from the UCI
repository. It consists of 2000 samples of handwrittend digits, with a 6
views, It has 10 clusters (0-9).

  * •

3 sources data set (3Sources)
333http://elki.dbs.ifi.lmu.de/wiki/datasets/MultiView: It consists of 169
news, provided by three news companies, including the BBC, Reuters, and The
Guardian. Each piece of news was manually labeled with one of six relevant
designations.

  * •

WebKB data set (WebKB) 444https://linqs.soe.ucsc.edu/data: It consists of 203
web pages, gathered from computer science departments of university, with 4
classes, each single web page is described by the title, the content of the
page and the anchor text of the hyperlink.

  * •

BBC data set(BBC) 555http://mlg.ucd.ie/datasets/segment.html: It consists of
685 documents collected from the BBC news website. Each document has 4
segments with 5 manually annotated labels.

While these are provided as raw features, some graph-based approaches employ
the similarity matrix as the input of each view, giving the user several
options for how to generate the graph( as complete or k-nn…) and the metrics(
as binary,cosine, or gaussian kernel…) from the raw input features. We
generate exhaustive graphs from the same data set as previously, and we
display the best performance on the best graph. We also included two pre-
computed graph data sets:

  * •

MSRC-v1 666http://mldta.com/dataset/msrc-v1/ is a class-based picture
collection from Microsoft Research in Cambridge containing 240 samples, with 7
classes. Each picture is represented by 6 distinct features: the HOG, the
Color moment, the CENT, the LBP, the CENTRIST, and the SIFT feature.

  * •

Yale Face Database(Yale) 777https://www.robots.ox.ac.uk/~vgg/data/flowers/17/
The database contains 165 GIF images with 15 classes. There are 11 ²s per
subject.

  * •

17 Category Flower Dataset(Flo17) 888www.robots.ox.ac.uk/vgg/data/flowers/17
is a class-based picture of 17 type of flowers, with 80 images of each class,
which means, a total of 1360 samples. It has 17 distinct features.

The characteristics of these raw feature data sets are given in the table
below 2, which displays the number of samples, views, clusters, and $d_{v}$,
which represents the dimension of features in view $v$. More information on
the data set may be obtained by following the hyperlinks in the footnotes.

Table 2: Summary of the benchmark data sets used Data set | # samples | # views | # clusters | $\mathrm{d}_{1}$ | $d_{2}$ | $d_{3}$ | $d_{4}$ | $d_{5}$ | $d_{6}$
---|---|---|---|---|---|---|---|---|---
3Sources | 169 | 3 | 6 | 3560 | 3631 | 3068 | - | - | -
WebKB | 203 | 3 | 4 | 1703 | 230 | 230 | - | - | -
100Leaves | 1600 | 3 | 100 | 64 | 64 | 64 | - | - | -
BBC | 685 | 4 | 5 | 4659 | 4633 | 4665 | 4684 | - | -
HW | 2000 | 6 | 10 | 216 | 76 | 64 | 6 | 240 | 47

###  5.2 Compared methods

Following that, we will compare the state of the art algorithms, for which
public code, A couple modification were done on these data set in order to
work with the same data set, and to work with the same clustering measures.
The k-means algorithm adopted is also available. Note that the code of some
algorithms were created from scratch, since they were not available to the
public, but we majorly focused on the methods that the code exist in any
programming language. The different methods are summarized in the table below,
where we’ve devised the methods on two columns: Methods using the matrix
related theory, the ones used the tensor related theory. We distinguish by
emphasizing the methods that require the similarity matrices instead as input.

Table 3: Benchmark methods Matricized | Tensorial
---|---
GBS 999https://github.com/cswanghao/gbs | CSMSC 101010https://github.com/XIAOCHUN-CAS/Consistent-and-Specific-Multi-View-Subspace-Clustering | t-SVD-MSC 111111https://www.researchgate.net/publication/324151918_the_source_of_paper_On_Unifying_Multi-View_Self-Representations_for_Clustering_by_Tensor_Multi-Rank_Minimization
GMC 121212https://github.com/cshaowang/gmc | MVLRSCC 131313https://github.com/mbrbic/Multi-view-LRSSC | MVSC-TPR 141414http://www.scholat.com/portaldownloadFile.html?fileId=4623
MCGC 151515https://github.com/kunzhan/MCGC | MVGL 161616https://github.com/kunzhan/MVGL | CGL 171717https://github.com/guanyuezhen/CGL
AMGL 181818https://github.com/kylejingli/AMGL-IJCAI16 | MLAN191919https://github.com/jxqhhh/MLAN |  MVSC-TLRR 202020https://github.com/jyh-learning/MVSC-TLRR
SwMC 212121https://github.com/kylejingli/SwMC-IJCAI17 |  RG-MVC 222222https://github.com/wx-liang/RG-MVC |  MCSCS p_norm 232323Created from algorithm in the paper Zhao et al. (2022)

###  5.3 Evaluation metrics

Each algorithm is run ten times to randomize the experiments. Different
measures are utilized to evaluate the performance of these multi view
clustering algorithms based using well-known benchmark data sets. In the
clustering task, several measures favor various traits.
The accuracy (ACC), the normalized mutual information (NMI), the F1 measure
(F-measure), and the adjusted rand index (ARI) are the main ones used:

\- NMI (Strehl and Ghosh, 2002) is an information theoretic metric that
measures the mutual information between the cluster assignments and the ground
truth labels. It is normalized by the average of entropy of both ground labels
and the cluster assignments, it even allows the comparison of two partitions
even with a different number of clusters:

$NMI=\dfrac{2I(\Omega,\Theta)}{H(\Omega)H(\Theta)},$ (30)

where $I$ represents the mutual information, and $H$ the information entropy.
$\Omega=\\{\omega_{1},\omega_{2},\ldots,\omega_{K}\\}$ is the set of clusters
and $\Theta=\\{c_{1},c_{2},\ldots,c_{J}\\}$ is the set of classes.
-ARI (Hubert and Arabie, 1985) computes a similarity measure between two clustering by considering all pairs of samples and counting pairs that are assigned in the same or different clusters in the predicted and true clustering. The adjusted Rand index is the corrected-for-chance version of the Rand index. It also allows the comparison of two partitions with different number of clusters:

$\displaystyle ARI$ $\displaystyle=\dfrac{RI-
Expected(RI)}{max(RI)-Expected(RI)}$ (31)
$\displaystyle=\dfrac{\sum_{ij}{\binom{n_{ij}}{2}}-\frac{1}{\binom{n}{2}}\sum_{i}{\binom{n_{i}}{2}}\sum_{j}{\binom{n_{j}}{2}}}{{\frac{1}{2}}\left[\sum_{i}{\binom{n_{i}}{2}}+\sum_{j}{\binom{n_{j}}{2}}\right]-\frac{1}{\binom{n}{2}}\sum_{i}{\binom{n_{i}}{2}}\sum_{j}{\binom{n_{j}}{2}}},$

where $n$ represents the number of samples, $n_{i}=\\#c_{i}$ is the number of
samples in the cluster $c_{i}$, while $n_{j}=\\#w_{j}$ is the number of
samples in the class $w_{j}$ and $n_{i}^{j}=\\#\lvert w_{i}\cap c_{j}\rvert$
is the number of data points of cluster $w_{i}$ assigned to the class $c_{j}$.
In general, an ARI value lies between 0 and 1. The index value is equal to 1
only if a partition is completely identical to the intrinsic structure and
close to 0 for a random partition.
-Purity(Manning et al., 2008) is a simple and transparent evaluation of the degree of each cluster containing a class of data points. It’s value can be calculated via the following formula:

$Purity=\frac{1}{n}\sum_{i}\max_{j}n_{i}^{j}.$ (32)

-ACC (Lovasz and Plummer, 1986) is the unsupervised counterpart of classification accuracy. It differs from usual accuracy metrics in that it employs a mapping function $map$ to determine the optimal mapping between the algorithm’s cluster assignment output $Y$ and the ground truth $Y^{0}$. This mapping is essential because an unsupervised algorithm may use a label other than the actual ground truth label to describe the same cluster, The best mapping can be found by using the Kuhn–Munkres algorithm in Lovasz and Plummer (1986):

$ACC=\max_{map}\dfrac{\sum_{i=1}^{n}\delta(Y_{i}^{0},map(Y_{i}))}{n},$ (33)

where $\delta(a,b)$ is a judgment function, it gives 1 if $a=b$ and 0
otherwise.
-$F_{\beta}$ measure is another metric that will be used, that can be calculated from the recall $R$ and the precision $P$ of a test, there is another parameter $\beta$( where it will be used equals to 1 in the numerical tests), which gives us the control of penalizing the false negatives more strongly than false positives by selecting a value greater than 1, as in ACC, we also used the permutation mapping that maximizes the matching of ground truth and predictive labels:

$\displaystyle P=\frac{TP}{TP+FP}\qquad R=\frac{TP}{TP+FN}\qquad
F_{\beta}=\frac{(\beta^{2}+1)PR}{\beta^{2}P+R},$

where $TP,TN,FP$, and $FN$ denote the number of similar pairs (same class)
assigned to the correct cluster, number of dissimilar pairs (different class)
assigned to the wrong correct cluster, number of dissimilar pairs assigned to
the wrong cluster, and number of similar pairs assigned to the wrong cluster,
respectively.
For all metrics, a higher value indicates the better performance of the
clustering.

###  5.4 Comparison result and analysis

For some methods, a post clustering (k -means) is needed on the obtaining
embedding data, as it is sensitive to initial values, we repeat the k-means
clustering processing 20 trials to avoid the randomness perturbation and
report the result with the lowest value for the objective function of k-means
clustering among the 20 results.
The comparison results are shown in Tables 8, 7, 5, 4, and 6. The best value
of each measure is shown in bold. We note that the standard deviation is
avoided to be show for the k-means clustering trials since simply, not all the
methods use it, we also mention that the values are rounded to the ten-
thousandths:

Table 4: Clustering measure on 3sources data set

Method | Fscore | Precision | Recall | NMI | ARI | ACC | Purity
---|---|---|---|---|---|---|---
CGL | 0.6224 | 0.7097 | 0.5555 | 0.6800 | 0.5263 | 0.6604 | 0.7976
GBS | 0.7159 | 0.6052 | 0.8760 | 0.7096 | 0.6082 | 0.7692 | 0.8047
GMC | 0.6047 | 0.4844 | 0.8045 | 0.6216 | 0.4431 | 0.6923 | 0.7456
AMGL | 0.2765 | 0.1711 | 0.7604 | 0.8490 | 0.2652 | 0.6352 | 0.7028
T-SVD-MS | 0.7029 | 0.7100 | 0.6961 | 0.6362 | 0.6098 | 0.7547 | 0.7967
MVSC-TPR | 0.5420 | 0.5722 | 0.5172 | 0.4561 | 0.4117 | 0.5979 | 0.6757
CSMSC | 0.3800 | 1.0000 | 0.2346 | 0 | 0 | 0.3299 | 1.0000
MVGL | 0.3455 | 0.2255 | 0.7381 | 0.1522 | 0.0163 | 0.3846 | 0.4320
Pairwise MLRSSC | 0.6443 | 0.6045 | 0.6969 | 0.6167 | 0.5460 | 0.6840 | 0.7491
Centroid MLRSSC | 0.5662 | 0.4629 | 0.7492 | 0.6470 | 0.4725 | 0.6482 | 0.6683
PairwiseKMLRSSC | 0.4675 | 0.3798 | 0.6118 | 0.4837 | 0.3525 | 0.5098 | 0.5240
Centroid KMLRSSC | 0.4745 | 0.3847 | 0.6214 | 0.4824 | 0.3620 | 0.5169 | 0.5272
MLAN | 0.9112 | 0.9112 | 0.9112 | 0.7881 | 0.8171 | 0.9112 | 0.9112

Table 5: Clustering measure on WebKB data set

Method | Fscore | Precision | Recall | NMI | ARI | ACC | Purity
---|---|---|---|---|---|---|---
CGL | 0,5025 | 0,6249 | 0,4202 | 0,3178 | 0,2725 | 0,4729 | 0,7783
GBS | 0,7227 | 0,6148 | 0,8766 | 0,465 | 0,4844 | 0,7833 | 0,798
GMC | 0,6878 | 0,5821 | 0,8404 | 0,4133 | 0,4169 | 0,7586 | 0,7734
AMGL | 0,5489 | 0,3925 | 0,9128 | 0,037 | -0,002 | 0,5099 | 0,5308
T-SVD-MS | 0,562 | 0,5642 | 0,5609 | 0,2536 | 0,2772 | 0,6182 | 0,6805
MVSC-TPR | 0,4751 | 0,5664 | 0,4092 | 0,2042 | 0,217 | 0,5539 | 0,6941
CSMSC | 0,4533 | 0,2435 | 0,509 | 0,4755 | 0,2983 | 0,4102 | 0,4835
MVGL | 0,6144 | 0,5761 | 0,6581 | 0,2805 | 0,3358 | 0,6995 | 0,7389
Pairwise MLRSSC | 0,4453 | 0,3339 | 0,6746 | 0,3675 | 0,2504 | 0,4204 | 0,4493
Centroid MLRSSC | 0,3816 | 0,2629 | 0,7036 | 0,3539 | 0,214 | 0,3963 | 0,4032
Pairwise KMLRSSC | 0,4655 | 0,3818 | 0,6021 | 0,2438 | 0,2338 | 0,5234 | 0,57
Centroid KMLRSSC | 0.4528 | 0.3649 | 0.6043 | 0.237 | 0.2264 | 0.5015 | 0.5436
MLAN | 0.3448 | 0.7881 | 0.2226 | 0.4921 | 0.5119 | 0.7881 | 0.8029

Table 6: Clustering measure on 100Leaves data set

Method | Fscore | Precision | Recall | NMI | ARI | ACC | Purity
---|---|---|---|---|---|---|---
CGL | 0,6245 | 0,7097 | 0,5654 | 0,6871 | 0,5263 | 0,6604 | 0,8024
GBS | 0,3867 | 0,2555 | 0,7951 | 0,8711 | 0,3779 | 0,6694 | 0,7131
GMC | 0,5042 | 0,3521 | 0,8874 | 0,9292 | 0,4974 | 0,8237 | 0,8506
AMGL | 0,2765 | 0,1711 | 0,7604 | 0,849 | 0,2652 | 0,6352 | 0,7028
T-SVD-MS | 0,8549 | 0,9428 | 0,7825 | 0,9708 | 0,8534 | 0,8568 | 0.9606
MVSC-TPR | 0,5787 | 0,5209 | 0,6515 | 0,8532 | 0,5743 | 0,6607 | 0,7028
CSMSC | 0,3455 | 0,5543 | 0,4521 | 0.6433 | 0,3894 | 0,3467 | 0,4870
MVGL | 0,2299 | 0,1545 | 0,4491 | 0,732 | 0,219 | 0,5056 | 0,5463
Pairwise MLRSSC | 0,0186 | 0,9994 | 0,0094 | 0,0006 | 0 | 0,0103 | 0,9997
Centroid MLRSSC | 0,0186 | 0,9991 | 0,0094 | 0,0009 | 0,000 | 0,0104 | 0,9995
Pairwise KMLRSSC | 0,0824 | 0,6515 | 0,044 | 0,4501 | 0,066 | 0,0781 | 0,7601
Centroid KMLRSSC | 0.4528 | 0.0821 | 0.663 | 0.0438 | 0.4523 | 0.0656 | 0.0784
MLAN | 0.0706 | 0.0706 | 0.0706 | 0.9494 | 0.8242 | 0.87375 | 0.8956

Table 7: Clustering measure on BBC data set

Method | Fscore | Precision | Recall | NMI | ARI | ACC | Purity
---|---|---|---|---|---|---|---
CGL | 0,9061 | 0,8906 | 0,9269 | 0,9185 | 0,8761 | 0,8886 | 0,9296
GBS | 0,5451 | 0,4031 | 0,8414 | 0,4839 | 0,3337 | 0,6044 | 0,6102
GMC | 0,6333 | 0,5012 | 0,86 | 0,5628 | 0,4789 | 0,6934 | 0,6934
AMGL | 0,3748 | 0,238 | 0,8891 | 0,02 | 0,0082 | 0,3431 | 0,3477
T-SVD-MS | 0,3516 | 0,7055 | 0,2361 | 0,0896 | 0,0034 | 0,3396 | 0,8132
MVSC-TPR | 0,574 | 0,6430 | 0,4039 | 0,3498 | 0,2344 | 0,4366 | 0,8132
CSMSC | 0,48374 | 0,5903 | 0,6345 | 0,243 | 0,245 | 0,245 | 0,674
MVGL | 0,3686 | 0,2329 | 0,8841 | 0,0806 | 0,0043 | 0,3489 | 0,3509
Pairwise MLRSSC | 0,7291 | 0,6643 | 0,8118 | 0,7213 | 0,6571 | 0,7302 | 0,7673
Centroid MLRSSC | 0,6498 | 0,5202 | 0,8669 | 0,6971 | 0,5751 | 0,6562 | 0,6562
Pairwise KMLRSSC | 0,408 | 0,3338 | 0,5274 | 0,4228 | 0,2761 | 0,4622 | 0,4725
Centroid KMLRSSC | 0.4051 | 0.3442 | 0.4948 | 0.4093 | 0.263 | 0.4661 | 0.4885
MLAN | 0.5591 | 0.5591 | 0.5591 | 0.7274 | 0.7460 | 0.8715 | 0.8715

Table 8: Clustering measure on HW data set

Method | Fscore | Precision | Recall | NMI | ARI | ACC | Purity
---|---|---|---|---|---|---|---
CGL | 0,8584 | 0,7941 | 0,963 | 0,9354 | 0,8405 | 0,8617 | 0,8741
GBS | 0,8327 | 0,7788 | 0,8945 | 0,8787 | 0,8128 | 0,847 | 0,8655
GMC | 0,8653 | 0,826 | 0,9085 | 0,9041 | 0,8496 | 0,882 | 0,882
AMGL | 0,4011 | 0,3081 | 0,5788 | 0,6162 | 0,3115 | 0,5536 | 0,5893
T-SVD-MS | 0,9042 | 0,9729 | 0,8464 | 0,9602 | 0,8926 | 0,8936 | 0,983
MVSC-TPR | 0,6745 | 0,6346 | 0,8045 | 0,8341 | 0,7563 | 0,7732 | 0,7422
CSMSC | 0,4354 | 0,6349 | 0,7409 | 0,753 | 0,65 | 0,632 | 0,6244
MVGL | 0,6367 | 0,557 | 0,7429 | 0,7231 | 0,59 | 0,6895 | 0,7245
Pairwise MLRSSC | 0,6498 | 0,5202 | 0,8669 | 0,6971 | 0,5751 | 0,6562 | 0,6562
Centroid MLRSSC | 0,7266 | 0,8678 | 0,625 | 0,8021 | 0,6907 | 0,7438 | 0,9244
Pairwise KMLRSSC | 0,5399 | 0,6179 | 0,4794 | 0,5958 | 0,4818 | 0,6361 | 0,7412
Centroid KMLRSSC | 0.5399 | 0.6179 | 0.4794 | 0.5958 | 0.4818 | 0.6361 | 0.7412
MLAN | 0.1965 | 0.1965 | 0.1965 | 0.9374 | 0.9384 | 0.972 | 0.972

As it shows from the tables, there is clearly, an advantage of using the
tonsorial theory, these type methods has a batter performances on all kind of
measures on most of the data.

Table 9: Clustering measure of graph methods on different data set

Method | Data set | Fscore | Precision | Recall | NMI | ARI | ACC | Purity
---|---|---|---|---|---|---|---|---
SwMC | 3sources | 0.5888 | 0.4637 | 0.8063 | 0.6036 | 0.4166 | 0.6746 | 0.7278
RG-MVC | 3sources | 0.4956 | 0.456 | 0.7963 | 0.6354 | 0.345 | 0.5244 | 0.6833
MCSCS p_norm | 3sources | 0.2768 | 0.3422 | 0.2324 | 0 | 0 | 0.3314 | 0.3442
MVSC-TLRR | 3sources | 0.5197 | 0.7299 | 0.4138 | 0.4498 | 0.3167 | 0.5707 | 0.8272
SwMC | BBC | 0.6167 | 0.4846 | 0.8476 | 0.5534 | 0.4536 | 0.6964 | 0.6964
RG-MVC | BBC | 0.3453 | 0.3948 | 0.7340 | 0.345 | 0.354 | 0.5464 | 0.345
MCSCS p_norm | BBC | 0.38 | 0.439 | 0.2346 | 0 | 0 | 0.3299 | 0.3847
MVSC-TLRR | BBC | 0.564 | 0.4566 | 0.4353 | 0.6309 | 0.3453 | 0.5635 | 0.7353
SwMC | Flower | 0.2804 | 0.181 | 0.6221 | 0.5566 | 0.1506 | 0.5391 | 0.53
RG-MVC | Flower | 0.1621 | 0.1834 | 0.1453 | 0.2342 | 0.1931 | 0.3453 | 0.2431
MCSCS p_norm | Flower | 0.1773 | 0.1917 | 0.1654 | 0.2886 | 0.1225 | 0.2709 | 0.319
MVSC-TLRR | Flower | 0.2666 | 0.2726 | 0.2609 | 0.4108 | 0.2203 | 0.3781 | 0.4082
SwMC | HW | 0.7668 | 0.6849 | 0.8711 | 0.862 | 0.7376 | 0.75 | 0.787
RG-MVC | HW | 0.6734 | 0.435 | 0.6543 | 0.739 | 0.945 | 0.6453 | 0.6953
MCSCS p_norm | HW | 0.5026 | 0.5733 | 0.4505 | 0.5886 | 0.44 | 0.5969 | 0.6924
MVSC-TLRR | HW | 0.7745 | 0.6934 | 0.8313 | 0.876 | 0.735 | 0.654 | 0.6941
SwMC | Leaves | 0.3644 | 0.2949 | 0.4766 | 0.7625 | 0.3569 | 0.5581 | 0.5844
RG-MVC | Leaves | 0.2409 | 0.2584 | 0.4209 | 0.7433 | 0.3583 | 0.5143 | 0.5243
MCSCS p_norm | Leaves | 0.4079 | 0.4743 | 0.3582 | 0.7605 | 0.4015 | 0.5502 | 0.6264
MVSC-TLRR | Leaves | 0.2138 | 0.3646 | 0.1517 | 0.6735 | 0.2032 | 0.4077 | 0.5272
SwMC | MSRCV1 | 0.8412 | 0.8304 | 0.8522 | 0.853 | 0.8152 | 0.919 | 0.919
RG-MVC | MSRCV1 | 0.4946 | 0.5314 | 0.4632 | 0.6326 | 0.4611 | 0.5855 | 0.6778
MCSCS p_norm | MSRCV1 | 0.3856 | 0.3314 | 0.4612 | 0.5936 | 0.3947 | 0.4634 | 0.4563
MVSC-TLRR | MSRCV1 | 0.7368 | 0.7988 | 0.6889 | 0.7824 | 0.6896 | 0.785 | 0.8667
SwMC | WebKB | 0.7004 | 0.5919 | 0.8576 | 0.4351 | 0.4397 | 0.7685 | 0.7833
RG-MVC | WebKB | 0.6433 | 0.5948 | 0.7003 | 0.4112 | 0.534 | 0.493 | 0.7103
MCSCS p_norm | WebKB | 0.4945 | 0.4347 | 0.5734 | 0 | 0 | 0.5271 | 0.593
MVSC-TLRR | WebKB | 0.6944 | 0.5903 | 0.8344 | 0.4230 | 0.4093 | 0.6734 | 0.7304
SwMC | Yale | 0.7734 | 0.6343 | 0.6301 | 0.6781 | 0.6433 | 0.823 | 0.8423
RG-MVC | Yale | 0.6434 | 0.6121 | 0.5834 | 0.6453 | 0.5498 | 0.7 | 0.7994
MCSCS p_norm | Yale | 0.5759 | 0.6012 | 0.5529 | 0.6207 | 0.5041 | 0.6852 | 0.7212
MVSC-TLRR | Yale | 0.7519 | 0.8386 | 0.6875 | 0.8278 | 0.7209 | 0.7841 | 0.9084

###  5.5 Further work

We can also, compare the time complexity, but since the methods are not all in
the same language (Matlab and Python), we decided not to stop in order to not
falsify the reel comparison between the methods.
For the graph based methods, we can further investigate the hyper graph, which
differs from the graph by letting the edges to be associated to more than two
nodes, as we can calculate the Laplacian by an appropriate formula, and then
continue the rest of the word as the normal graphs, as in Lu et al. (2020),
named HLR-MSCLRT, where the following objective is the same as in (20), by
simply replacing the Laplacian of a graph matrix of a view $L^{(v)}$ by its
Hyper graph Laplacian $L_{h}^{(v)}$.

##  6 Conclusion

As it is shown, for clustering multi view is the way to go, as it brings with
its multiple views, multiple data that can give more information, having also
new methods, a part from the ones that can be generalized from the single
view, without forgetting the new challenges of how to combine and use all the
views, the tensor representation of the data, has a clear advantages of just
considering data as matrices, by exploiting the higher order correlation in
the data, which has been used, different norms and ranks, The coherence and
complimentary nature of diverse views of view are therefore automatically
taken into account. Two main categories were mainly discussed are the graph
based, and the subspace based, we can even give the graph based method more
credits, as it gives insight on the local view specific geometrical
structures, and can also be combined in some methods that tries to find a low
rank representation tensor with a Laplacian term, while also some method tries
to learn the affinity matrices of each view and others only the global
affinity matrix.

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