# Multi-view MERA Subspace Clustering

Zhen Long∗, Ce Zhu∗, Jie Chen∗, Zihan Li∗, Yazhou Ren†,Yipeng Liu∗ Z. Long, C.
Zhu, J. Chen, Z. Li and Y. Liu are with the School of Communication and
Information Engineering, and University of Electronic Science and Technology
of China (UESTC), Chengdu 611731, China. (email<EMAIL_ADDRESS><EMAIL_ADDRESS>†Y. Ren is with the School of Computer Science and
Engineering, UESTC, Chengdu, 611731, China.

###### Abstract

Tensor-based multi-view subspace clustering (MSC) can capture high-order
correlation in the self-representation tensor. Current tensor decompositions
for MSC suffer from highly unbalanced unfolding matrices or rotation
sensitivity, failing to fully explore inter/intra-view information. Using the
advanced tensor network, namely, multi-scale entanglement renormalization
ansatz (MERA), we propose a low-rank MERA based MSC (MERA-MSC) algorithm,
where MERA factorizes a tensor into contractions of one top core factor and
the rest orthogonal/semi-orthogonal factors. Benefiting from multiple
interactions among orthogonal/semi-orthogonal (low-rank) factors, the low-rank
MERA has a strong representation power to capture the complex inter/intra-view
information in the self-representation tensor. The alternating direction
method of multipliers is adopted to solve the optimization model. Experimental
results on five multi-view datasets demonstrate MERA-MSC has superiority
against the compared algorithms on six evaluation metrics. Furthermore, we
extend MERA-MSC by incorporating anchor learning to develop a scalable low-
rank MERA based multi-view clustering method (sMREA-MVC). The effectiveness
and efficiency of sMERA-MVC have been validated on three large-scale multi-
view datasets. To our knowledge, this is the first work to introduce MERA to
the multi-view clustering topic. The codes of MERA-MSC and sMERA-MVC are
publicly available at https://github.com/longzhen520/MERA-MSC.

##  1 Introduction

Multi-view data are ubiquitous in the real world. For instance, certain news
can be reported in text, images, and video; the multi-view Yale database can
be represented by local binary patterns (LBP), Gabor, and intensity features.
Such multi-view data, which provide consensual and complementary information,
have given rise to a series of multi-view learning based tasks [56, 34, 50].
Among them, multi-view subspace clustering (MSC) separates multi-view data
into several groups by assuming it shares a common latent low-dimensional
subspace, driving numerous applications in image processing and computer
vision [46, 43, 31, 48, 29].

Inspired by the success of low-rank representation (LRR) [38] and sparse
subspace clustering (SSC) [8] on single-view data, where each sample can be
represented by a combination of other samples, many self-representation based
MSC methods have been proposed. Among them, tensor-based MSC ones show
promising performances by assuming that their self-representation tensors are
low-rank [53, 49, 10, 6]. For example, Zhang et al. [53] firstly stack all
subspace representations of each view to a 3rd-order tensor and consider
Tucker low-rank constraint [23] on it to explore the high-order correlations
among views. However, Tucker nuclear norm is denoted as the sum of the nuclear
norms of the unfolding matrices, which captures the low-rank information from
unbalanced unfolding matrices, resulting in inferior performance [49, 2]. To
alleviate it, Xie et al. [49] consider the tubal norm based on tensor singular
value decomposition (t-SVD) [18] for low-rank self-representation tensor,
where the tensor is rotated to effectively capture the consensus among
multiple views.

However, the self-representation tensor contains both inter- and intra-view
similarity information. For t-SVD, the rotation operation of the self-
representation tensor can well explore the correlations across different
views, but it also suffers from inadequate exploration of intra-view
information, because matrix singular value decompositions (SVDs) are only
performed in the first two modes, and linear transformations are considered in
the third mode [26, 24, 25]. On account of these, Jia et al. [15] introduce
low-rankness to the frontal and the horizontal slices of the self-
representation tensor to characterize the intra-view and inter-view
relationships, respectively. On exploring inter/intra-view information in the
self-representation tensor, one question naturally arises: whether the
inter/intra-view information can be well captured simultaneously with a new
tensor decomposition. In fact, as advanced tensor decompositions, tensor
networks show significant advantages in capturing the correlation of higher
order data [2, 42, 27].

In this paper, we consider a stable tensor network, namely the multi-scale
entanglement renormalization ansatz (MERA) decomposition [37, 1, 51], to
capture well both inter- and intra-view information of the self-representation
tensor. MERA factorizes a $D$-th order tensor into several orthogonal/semi-
orthogonal factors and one top core factor, providing a more expressive and
powerful representation of a high-order tensor. To make it clear, Fig. 1
provides the reconstructed images obtained by different tensor decompositions,
including t-SVD, Tucker decomposition, and MERA decomposition, under the same
compression ratio (CR), i.e., CR=5%. From Fig. 1(b-d), it is evident that the
low-rank MERA approximation exhibits better recovery performance in terms of
PSNR than the other approximation methods. This implies that MERA has a more
powerful representation capability in comparison to t-SVD and Tucker
decomposition.

Figure 1: Low-rank exploration performance of t-SVD, Tucker, and MERA on RGB
image at the same 5% compression ratio; (a) is the original image with size
$640\times 960\times 3$; The images in (b)-(d) correspond to the recovered
images obtained by low-rank t-SVD approximation, low-rank Tucker
approximation, and low-rank MERA approximation, respectively. Figure 2:
Framework of MERA-MSC. Each data $\mathbf{X}_{v}$ can be first expressed with
its linear combination, e.g., (a)
$\mathbf{X}_{v}=\mathbf{X}_{v}\mathbf{Z}_{v}$; (b) these self-representation
matrices $\mathbf{Z}_{v},v=1,\cdots,V$ are stacked into a self-representation
tensor $\mathcal{Z}$; (c) $\mathcal{Z}$ will be reshaped into a 5-th order
tensor, on which low-rank MERA approximation is performed to capture both
inter- and intra-view correlations. In each iteration, $\mathbf{Z}_{v}$ is
adaptively obtained from both $\mathbf{X}_{v}$ and the updated
$\hat{\mathbf{Z}}_{v}$ from low-rank MERA approximation. At last, the affinity
matrix can be constructed by
$\frac{1}{V}\sum_{v=1}^{V}|\mathbf{Z}_{v}|+|\mathbf{Z}_{v}^{\operatorname{T}}|$,
which will then be fed into the spectral clustering algorithm.

Based on it, we develop a low-rank MERA-based MSC (MERA-MSC) optimization
model, and the alternating direction method of multipliers (ADMM) framework
[3] is adopted to solve it. The processing of MERA-MSC is illustrated in Fig.
2, where the self-representation tensor will be adaptively updated from the
multi-view data and low-rank MERA approximation in each iteration.
Iteratively, the final self-representation tensor will be used to construct
the affinity matrix, which will then be fed into the spectral clustering
algorithm[39] for achieving MSC task. Furthermore, we extend the MERA-MSC
method using anchor learning and developed a scalable MERA based multi-view
clustering (sMERA-MVC) algorithm in the experimental part.

Our contributions are summarized as follows:

  * •

Motivated by the powerful representation ability of MERA decomposition, we
apply it to capture the inter/intra-view correlations within the self-
representation tensor for MSC. To the best of our knowledge, this is the first
work to introduce MERA to MVC tasks.

  * •

An effective algorithm named MERA-MSC is proposed and evaluated on five
benchmark datasets. Experimental results show MERA-MSC outperforms the state-
of-art methods, as evaluated using six commonly used metrics. Building upon
this, an sMERA-MVC is further developed, which clusters large-scale multi-view
data effectively and efficiently.

  * •

The outstanding clustering performance of MERA-MSC and sMERA-MVC highlights
the capability of low-rank MERA approximation to explore higher-order
correlation, which also provides some inspiration for applying it to other
tasks, like tensor classification, tensor regression.

Figure 3: The graphical illustration of tensor contraction.

##  2 NOTATIONS AND PROBLEM FORMULATION

###  2.1  Notations

A scalar, a vector, a matrix, and a tensor are written as $x$, $\mathbf{x}$,
$\mathbf{X}$, and $\mathcal{X}$, respectively. Indices typically range from 1
to their capital version, e.g., $i=1,\cdots,I$. For a $D$-th order tensor
$\mathcal{X}\in\mathbb{R}^{I_{1}\times\cdots\times I_{D}}$, its
mode-{$d_{1},d_{2}$} unfolding matrix is represented as
$\mathbf{X}_{\left(d_{1},d_{2}\right)}\in\mathbb{R}^{I_{d_{1}}I_{d_{2}}\times\prod_{e\neq
d_{1},d_{2}}I_{e}}$ by arranging the {$d_{1},d_{2}$}-th mode of $\mathcal{X}$
as the row while the rest modes as the column, and its inverse operator is
defined by
$\mathcal{X}$=$\operatorname{fold}_{d_{1},d_{2}}(\mathbf{X}_{\left(d_{1},d_{2}\right)})$.
As shown in Fig. 3, the tensor contraction of two tensors
$\mathcal{A}\in\mathbb{R}^{I_{1}\times I_{2}\times J_{1}\times J_{2}}$ and
$\mathcal{B}\in\mathbb{R}^{J_{1}\times J_{2}\times L_{1}\times L_{2}}$ is
achieved by contracting their common indices $\\{J_{1},J_{2}\\}$, which is
denoted as
$\mathcal{C}=\mathcal{A}\times_{\\{J_{1},J_{2}\\}}\mathcal{B}\in\mathbb{R}^{I_{1}\times
I_{2}\times L_{1}\times L_{2}}$ whose entries are calculated by
$c_{i_{1},i_{2},l_{1},l_{2}}=\sum_{j_{1},j_{2}}a_{i_{1},i_{2},j_{1},,j_{2}}b_{j_{1},j_{2},l_{1},l_{2}}.$

###  2.2  MERA Decomposition

Before introducing the MERA decomposition, it is necessary to introduce two
fundamental building blocks: isometry and disentangler.

######  Definition 1 (Isometry).

[37, 1] As shown in Fig. 4(a), an isometry
$\mathcal{W}\in\mathbb{R}^{I_{1}\times\cdots\times I_{K}\times R}$ is denoted
as a $(K+1)$-th order tensor with semi-orthogonal unfolding matrix
$\mathbf{W}\in\mathbb{R}^{I_{1}\cdots I_{K}\times R}$, e.g.,
$\mathbf{W}^{\operatorname{T}}\mathbf{W}=\mathbf{I}_{R}$, where $R$ is the
outgoing dimension of an isometry and $K$ is the number of bottom legs.

######  Definition 2 (Disentangler).

[37, 1] As shown in Fig. 4(b), a disentangler
$\mathcal{U}\in\mathbb{R}^{I_{1}\times I_{2}\times I_{1}\times I_{2}}$ is
denoted as a 4-way tensor with a square orthogonal unfolding matrix
$\mathbf{U}\in\mathbb{R}^{I_{1}I_{2}\times I_{1}I_{2}}$, namely,
$\mathbf{U}^{\operatorname{T}}\mathbf{U}=\mathbf{I}_{I_{1}I_{2}}$.

######  Definition 3 (Layer).

[37, 1] A layer $\mathcal{C}$ is denoted as a result of tensor contraction
between isometries and disentanglers in the same layer. For example, as shown
in Fig. 4(c), a 7-th order layer $\mathcal{C}$ can be constructed by two
isometries $\mathcal{W}_{1}\in\mathbb{R}^{{I}_{1}\times{I}_{2}\times R_{1}}$,
$\mathcal{W}_{2}\in\mathbb{R}^{{I}_{3}\times{I}_{4}\times{I}_{5}\times R_{2}}$
and one disentangler $\mathcal{U}\in\mathbb{R}^{I_{2}\times I_{3}\times
I_{2}\times I_{3}}$, as follows:

$\mathcal{C}=\mathcal{U}_{1}\times_{\\{I_{2}\\}}\mathcal{W}_{1}\times_{\\{I_{3}\\}}\mathcal{W}_{2}\in\mathbb{R}^{I_{1}\times
I_{2}\times I_{3}\times I_{4}\times I_{5}\times R_{1}\times R_{2}}.$

Figure 4:  The graphical representations of isometry, disentangler, and layer,
respectively.

######  Definition 4 (MERA Decomposition).

[37, 1] For a $D$-th order tensor $\mathcal{Z}\in\mathbb{R}^{I_{1}\times
I_{2}\times\cdots\times I_{D}}$, its MERA representation can be composed by
different layers and one top core, which can be recursively obtained by:

$\mathcal{G}^{l-1}=\mathcal{G}^{l}\times_{\\{R^{l}_{1},\cdots,R^{l}_{P_{l}}\\}}\mathcal{C}_{l},$
(1)

where $R^{l}_{p_{l}},p_{l}=1,\cdots,P_{l},l=1,\cdots,L$ are the MERA ranks;
$P_{l}$ is the number of isometries in $l$-th layer, and $L$ is the total
number of layers. $\mathcal{G}^{0}=\mathcal{Z}$, and
$\mathcal{G}^{L}=\mathbf{B}$, where $\mathbf{B}$ is the top core. In a nested
way, we can construct the MERA decomposition.

For simplification, we use $\mathcal{Z}=f(\mathbf{B},\mathcal{W},\mathcal{U})$
to represent an MERA decomposition, where
$\mathcal{U}=(\mathcal{U}_{1}^{l},\cdots,\mathcal{U}_{P_{l}-1}^{l})$ and
$\mathcal{W}=(\mathcal{W}_{1}^{l},\cdots,\mathcal{W}_{P_{l}}^{l})$,
$\mathcal{W}^{l}_{p_{l}}$ and $\mathcal{U}^{l}_{p_{l}}$ are the $p_{l}$-th
isometry and disentangler in the $l$-th layer, $l=1,\cdots,L$. Fig. 5 shows
the graphical representation of MERA decomposition for a 5-th order tensor.

Figure 5:  An graphical illustration of MERA decomposition for a 5-th order
tensor.

###  2.3 Related Works

According to different representation forms of self-representation matrices in
multi-view data, current MSC methods are mainly divided into two categories:
one is matrix-based MSC, which learns a shared self-representation matrix of
all views; the other is tensor-based MSC, which stacks self-representation
matrices of all views into a tensor to explore the correlations among views.

####  2.3.1 Matrix-based MSC

Current matrix-based MSC methods generally consider that multi-view data
possess a shared consistent feature while each view has its specific feature
that provides complementary information, namely consistency and diversity [43,
20]. Specifically, [43] considers a shared indicator matrix and a position-
aware exclusivity term as the consistent and the diversity representations,
respectively. [28] divides the self-representation matrix of each view into
two parts: the low-rank consensus shared by all views and the specific part of
every single view constrained by $\ell_{2}$ norm. Furthermore, instead of the
original space, [57] projects multi-view data into a latent space to exploit
shared-specific information. Besides, [32] considers the interaction between
the learned self-representation and the clustering calculation and proposes a
unified MSC framework to learn a better subspace self-representation for
clustering.

####  2.3.2 Tensor based MSC

Tensor-based MSC methods can well capture high-order correlations among
multiple views, showing improved clustering performance [53, 35]. The key idea
of tensor-based MSC methods is to find a good low-rank tensor approximation
for the self-representation tensor. Given multi-view dataset
$\mathbf{X}_{v}\in\mathbb{R}^{D_{v}\times N}$, $v=1,\cdots,V$, where $V$ and
$N$ are the numbers of views and samples, respectively, and $D_{v}$ represents
the feature dimension in the $v$-th view. The general tensor-based MSC
optimization model can be formulated as [53]:

$\displaystyle\min_{\\{\mathbf{E}_{v},\mathbf{Z}_{v}\\}_{v=1}^{V}}\Phi(\mathcal{Z})+\lambda\Psi(\mathbf{E})$
$\displaystyle\text{s.~{}t.}\quad\mathbf{X}_{v}=\mathbf{X}_{v}\mathbf{Z}_{v}+\mathbf{E}_{v},v=1,\cdots,V,$
(2)
$\displaystyle\quad\quad\mathcal{Z}=\Omega(\mathbf{Z}_{1},\mathbf{Z}_{2},\cdots,\mathbf{Z}_{V}),\mathbf{E}=[\mathbf{E}_{1};\mathbf{E}_{2};\cdots;\mathbf{E}_{V}],$

where the term $\Omega(\cdot)$ merges all self-representation matrices
$\mathbf{Z}_{v}(v=1,\cdots,V)$ to a third-order tensor
$\mathcal{Z}\in\mathbb{R}^{N\times N\times V}$, $\Phi$ depicts a tensor rank
term to explore the relationship of observed samples, $\Psi$ is another term
in accordance with corrupted noise type, and $\lambda$ is a trade-off
parameter to balance the effect of the $\Phi$ and $\Psi$ regulations. Usually,
sparse noise or outliers will be considered in subspace clustering. Hence, the
corresponding convex surrogates $\ell_{1}$ or $\ell_{2,1}$ will be used to
remove the sparse noise.

Tucker decomposition [19] and t-SVD [40] are two commonly used tensor
approximation methods for MSC. For example, [53] considers Tucker
decomposition and its corresponding tensor nuclear norm minimization to
extract the low-rank information. Following it, [7] considers simultaneously
learning the low-rank Tucker self-representation tensor and the affinity
matrix to preserve their correlations. In addition, there are many t-SVD based
works with various tensor norms, e.g., standard t-SVD based tensor nuclear
norm [49], weighted tensor nuclear norm [11], and weighted tensor Schatten
$p$-norm [47]. To further enhance the clustering performance, $\ell_{1,1,2}$
norm term can be added on the self-representation tensor to capture the local
correlation in the view dimension [52].

##  3 PROPOSED METHOD

The self-representation tensor obtained from the current MSC works is mainly
based on low-rank assumptions, which can explore the global relationship among
views. However, the existing works are mainly based on Tucker or t-SVD, which
can not fully explore the inter/intra-view information within the self-
representation tensor. In this work, we apply low-rank MERA decomposition for
MSC, which can adaptively attain a well-explored structure of the self-
representation tensor.

###  3.1 Low-Rank MERA approximation

Low-rank tensor networks have been shown to be more effective in approximating
high-order data [27, 42, 2], hence the self-representation tensor
$\mathcal{Z}\in\mathbb{R}^{N\times N\times V}$ is rearranged into a 5th-order
tensor $\mathcal{Y}\in\mathbb{R}^{A\times Q\times A\times Q\times V}$ with
$N=AQ$ for low-rank MERA approximation, while ensuring that $A$ and $Q$ are
probably close to equilibrium. In this part, we mainly give solutions of low-
rank MERA factors for a 5th-order tensor in detail.

Given $\mathcal{Y}\in\mathbb{R}^{I_{1}\times\cdots\times I_{5}}$, its low-rank
MERA approximation is to find MERA factors with predefined MERA ranks
$R_{1}=R_{2}=R$, which can be formulated as

$\min_{\mathcal{U},\mathcal{W},\mathbf{B}}\frac{1}{2}\|\mathcal{Y}-f(\mathbf{B},\mathcal{W},\mathcal{U})\|_{\operatorname{F}}^{2}.$
(3)

As shown in Fig. 5, there are four MERA factors including one disentangler
$\mathcal{U}_{1}$, two isometries $\mathcal{W}_{1}$, $\mathcal{W}_{2}$, and
one top core $\mathbf{B}$. Each variable can be alternately updated while the
others are fixed.
Update $\mathcal{U}_{1}$: With fixed $\mathcal{W}_{1}$,$\mathcal{W}_{2}$, and
$\mathbf{B}$, the subproblem of $\mathcal{U}_{1}$ can be rewritten into a
matrix form as:

$\min_{\mathbf{U}_{1}}\frac{1}{2}\|\mathbf{Y}_{(2,3)}-\mathbf{U}_{1}\mathbf{M}_{u}\|_{\operatorname{F}}^{2},\text{
s. t.
}\mathbf{U}_{1}^{\operatorname{T}}\mathbf{U}_{1}=\mathbf{I}_{I_{2}I_{3}},$ (4)

where $\mathbf{Y}_{(2,3)}$ and $\mathbf{M}_{u}$ are the mode-{2, 3} unfolding
matrices of $\mathcal{Y}$ and $\mathcal{M}_{u}$, respectively.
$\mathcal{M}_{u}$ is the tensor contraction result of other variables, i.e.,
$\mathcal{M}_{u}=\mathbf{B}\times_{\\{R_{1}\\}}\mathcal{W}_{1}\times_{\\{R_{2}\\}}\mathcal{W}_{2}\in\mathbb{R}^{I_{1}\times
I_{2}\times I_{3}\times I_{4}\times I_{5}}$; $\mathbf{U}_{1}$ is the
mode-{$1,2$} unfolding matrix of $\mathcal{U}_{1}\in\mathbb{R}^{I_{2}\times
I_{3}\times I_{2}\times I_{3}}$.

The optimization problem (4) is equivalent to the following well-known
orthogonal Procrustes problem:

$\max_{\mathbf{U}_{1}}\quad\operatorname{trace}((\mathbf{U}_{1})^{\operatorname{T}}\mathbf{Y}_{(2,3)}(\mathbf{M}_{u})^{\operatorname{T}}),$
(5)

where $\operatorname{trace}(\mathbf{A})$ calculates the sum of the diagonal
elements of $\mathbf{A}$, which can be achieved by the Matlab command “trace”.
In this way, the updating of $\mathcal{U}_{1}$ can be obtained by

$\mathcal{U}_{1}=\operatorname{fold}_{1,2}(\mathbf{S}_{u}\mathbf{D}_{u}^{\operatorname{T}}),$
(6)

where $\mathbf{S}_{u}$ and $\mathbf{D}_{u}$ are the left and right singular
vector matrices of the matrix
$\mathbf{Y}_{(2,3)}(\mathbf{M}_{u})^{\operatorname{T}}$.
Update $\mathcal{W}_{1}$, $\mathcal{W}_{2}$: Following the way of solving
$\mathcal{U}_{1}$, the updating of $\mathcal{W}_{p},~{}p=1,2$ can be obtained
by

$\mathcal{W}_{p}=\operatorname*{arg\,min}_{\mathcal{W}_{p}:\mathbf{W}_{p}^{\operatorname{T}}\mathbf{W}_{p}=\mathbf{I}_{R_{p}}}\frac{1}{2}\|\mathcal{Y}-f(\mathbf{B},\mathcal{W},\mathcal{U})\|_{\text{F}}^{2}.$
(7)

Update $\mathbf{B}$: The subproblem of $\mathbf{B}$ can be rewritten as:

$\min_{\mathbf{B}}\quad\frac{1}{2}\|\mathcal{Y}-\mathcal{C}\times_{\\{R_{1},R_{2}\\}}\mathbf{B}\|_{\operatorname{F}}^{2},$
(8)

where $\mathcal{C}$ is a 7-th order layer in Definition 3. According to the
orthogonal/semi-orthogonal property of disentangler $\mathcal{U}_{1}$ and
isometries $\mathcal{W}_{p},p=1,2$, the updating of $\mathbf{B}$ can be
obtained by

$\mathbf{B}=\mathcal{Y}\times_{\\{I_{1},I_{2},I_{3},I_{4},I_{5}\\}}\mathcal{C}.$
(9)

Repeatedly update equations (6), (7), and (9) for several iterations to
achieve low-rank MERA approximation, which is summarized in Algorithm 1.

Algorithm 1 Low-rank MERA approximation

1: Input: 5-th order data $\mathcal{Y}$, MERA ranks $R_{1}=R_{2}=R$

2: Initialize: $\mathbf{B}$, $\mathcal{W}$, $\mathcal{U}$, Maximum iterations
$S=10$

3: for $s=1:S$ do

4: Update $\mathcal{U}^{s}$ via equation (6)

5: Update $\mathcal{W}^{s}$ via equation (7)

6: Update $\mathbf{B}^{s}$ via equation (9)

7: end for

8: $\hat{\mathcal{Y}}$=$f(\mathbf{B}^{S}$, $\mathcal{W}^{S}$,
$\mathcal{U}^{S}$)

9: Output: Low-rank MERA approximation tensor $\hat{\mathcal{Y}}$

###  3.2 Low-Rank MERA for MSC

As shown in Fig. 2, applying low-rank MERA decomposition on learned self-
representation tensor $\mathcal{Z}$ for MSC can be formulated as the following
optimization problem:

$\displaystyle\min_{\\{\mathbf{E}_{v},\mathbf{Z}_{v}\\}_{v=1}^{V}}\quad\sum_{v=1}^{V}\lambda\|\mathbf{E}_{v}\|_{2,1}$
(10)
$\displaystyle\quad\text{s.~{}t.}\quad\mathbf{X}_{v}=\mathbf{X}_{v}\mathbf{Z}_{v}+\mathbf{E}_{v},v=1,\cdots,V,$
$\displaystyle\quad\qquad\hat{\mathcal{Z}}=f(\mathbf{B},\mathcal{W},\mathcal{U})$

where
$\mathcal{Z}=\Omega\left(\mathbf{Z}_{1},\mathbf{Z}_{2},\cdots,\mathbf{Z}_{V}\right)\in\mathbb{R}^{N\times
N\times V}$,
$\hat{\mathcal{Z}}=\operatorname{reshape}(\mathcal{Z},[A,Q,A,Q,V])$. We use
operator $\mathfrak{R}$ to represent these two steps. And $\mathfrak{R}^{-1}$
denotes its inverse operator.
$\hat{\mathcal{Z}}=f(\mathbf{B},\mathcal{W},\mathcal{U})$ means applying low-
rank MERA approximation to explore both inter-view and intra-view information.

###  3.3 Solutions

To make the above optimization problem separable, an auxiliary variable
$\mathcal{Y}$ is introduced as follows:

$\displaystyle\min_{\\{\mathbf{E}_{v},\mathbf{Z}_{v}\\}_{v=1}^{V}}\quad\sum_{v=1}^{V}\lambda\|\mathbf{E}_{v}\|_{2,1}$
(11)
$\displaystyle\quad\text{s.~{}t.}\quad\mathbf{X}_{v}=\mathbf{X}_{v}\mathbf{Z}_{v}+\mathbf{E}_{v},v=1,\cdots,V,$
$\displaystyle\quad\qquad\hat{\mathcal{Z}}=\mathcal{Y},\mathcal{Y}=f(\mathbf{B},\mathcal{W},\mathcal{U}).$

The Lagrangian function can be formulated as follows:

$\displaystyle\operatorname{L}\left(\\{\mathbf{Z}_{v},\mathbf{E}_{v},\Lambda_{v}\\}_{v=1}^{V},\mathcal{Y},\Gamma\right)$
(12)
$\displaystyle=\sum_{v=1}^{V}(\lambda\|\mathbf{E}_{v}\|_{2,1}+\langle\Lambda_{v},\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}_{v}-\mathbf{E}_{v}\rangle$
$\displaystyle+\frac{\mu_{2}}{2}\|\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}_{v}-\mathbf{E}_{v}\|_{\operatorname{F}}^{2})+\frac{\mu_{1}}{2}\|\hat{\mathcal{Z}}-\mathcal{Y}\|_{\operatorname{F}}^{2}+\langle\Gamma,\hat{\mathcal{Z}}-\mathcal{Y}\rangle,$

under the constraint $\mathcal{Y}=f(\mathbf{B},\mathcal{W},\mathcal{U})$,
where $\\{\Lambda_{v}\\}_{v=1}^{V}$ and $\Gamma$ are Lagrangian multipliers
and $\mu_{1},\mu_{2}$ are penalty factors. According to the ADMM framework,
problem (12) can be divided into several subproblems where each variable can
be alternately updated with other variables fixed.
Update $\\{\mathbf{Z}_{v}\\}_{v=1}^{V}$: With other variables fixed, the
solution of $\mathbf{Z}_{v}$ can be obtained by setting the derivative of the
objective function in (12) with respect to $\mathbf{Z}_{v}$ to zero, as
follows:

$\displaystyle-\mathbf{X}_{v}^{\operatorname{T}}\Lambda_{v}-\mu_{2}\mathbf{X}_{v}^{\operatorname{T}}(\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}_{v}-\mathbf{E}_{v})$
(13)
$\displaystyle+\mu_{1}(\mathfrak{R}_{v}^{-1}(\hat{\mathcal{Z}}-\mathcal{Y})+\mathfrak{R}_{v}^{-1}(\Gamma))=\mathbf{0},$

where $\mathfrak{R}_{v}^{-1}$ is the inverse operator along the $v$-th view,
i.e., $\mathfrak{R}_{v}^{-1}(\hat{\mathcal{Z}})=\mathbf{Z}_{v}$. The solution
of $\mathbf{Z}_{v}$ can be rewritten as:

$\displaystyle\mathbf{Z}_{v}=$
$\displaystyle(\mu_{1}\mathbf{I}+\mu_{2}\mathbf{X}_{v}^{\operatorname{T}}\mathbf{X}_{v})^{-1}(\mathbf{X}_{v}^{\operatorname{T}}\Lambda_{v}+\mu_{2}\mathbf{X}_{v}^{\operatorname{T}}\mathbf{X}_{v}-$
(14)
$\displaystyle\mu_{2}\mathbf{X}_{v}^{\operatorname{T}}\mathbf{E}_{v}+\mathfrak{R}_{v}^{-1}(\mu_{1}\mathcal{Y}-\Gamma)).$

Update $\\{\mathbf{E}_{v}\\}_{v=1}^{V}$: The solution of $\mathbf{E}_{v}$ can
be obtained by optimizing

$\frac{\lambda}{\mu_{2}}\|\mathbf{E}_{v}\|_{2,1}+\frac{1}{2}\|\mathbf{E}_{v}-(\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}_{v}+(1/\mu_{2})\Lambda_{v})\|_{\operatorname{F}}^{2}.$
(15)

Letting
$\mathbf{D}=\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}_{v}+(1/\mu_{2})\Lambda_{v}$,
and according to [34], we have

$\mathbf{E}_{v}(:,n)=\left\\{\begin{array}[]{l}\frac{\left\|\mathbf{D}(:,n)\right\|_{2}-\frac{\lambda}{\mu_{2}}}{\left\|\mathbf{D}(:,n)\right\|_{2}}\mathbf{D}(:,n),\quad\left\|\mathbf{D}(:,n)\right\|_{2}>\frac{\lambda}{\mu_{2}}\\\
\mathbf{0},~{}~{}~{}~{}~{}~{}~{}~{}\qquad\qquad\qquad\text{
otherwise,}\end{array}\right.$ (16)

where $n=1\cdots,N$, $N$ is the number of samples.
Update $\mathcal{Y}$: With respect to $\mathcal{Y}$, the problem (12) can be
transferred into the following formulation:

$\displaystyle\min_{\mathcal{Y}}\quad\frac{\mu_{1}}{2}\left\|\mathcal{Y}-\left(\hat{\mathcal{Z}}+\frac{1}{\mu_{1}}\Gamma\right)\right\|_{\operatorname{F}}^{2}$
(17) $\displaystyle~{}\text{s. t.
}\quad\mathcal{Y}=f(\mathbf{B},\mathcal{W},\mathcal{U}).$

It can be solved by Algorithm 1, where the input self-representation tensor is
$\hat{\mathcal{Z}}+\frac{1}{\mu_{1}}\Gamma$.
Update Lagrangian multipliers:

$\Gamma=\Gamma+\mu_{1}(\hat{\mathcal{Z}}-\mathcal{Y}).$ (18)
$\Lambda_{v}=\Lambda_{v}+\mu_{2}\left(\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}_{v}-\mathbf{E}_{v}\right),v=1,\cdots,V.$
(19)

After finishing updating the variables, the affinity matrix of multi-view data
can be obtained by
$\mathbf{S}=\frac{1}{V}\sum_{v=1}^{V}\left|\mathbf{Z}_{v}\right|+\left|\mathbf{Z}_{v}^{\operatorname{T}}\right|$,
which will be used in spectral clustering algorithm[39] for the final
clustering result. The optimization processing of MERA-MSC is summarized in
algorithm 2. When
$\max_{v,v=1,\cdots,V}\left\|\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}^{t}_{v}-\mathbf{E}^{t}_{v}\right\|_{\infty}\leq\varepsilon$
and
$\max_{v,v=1,\cdots,V}\left\|\mathbf{Z}^{t}_{v}-\mathbf{Y}^{t}_{v}\right\|_{\infty}\leq\varepsilon$,
$\varepsilon=10^{-6}$, our algorithm stops.

The computational complexity of the MERA-MSC algorithm for one iteration is
summarized in Table 2. Overall, the main computational complexity is
$\operatorname{O}(TN^{3}(V+1))$ where $T$ is the number of iterations in
Algorithm 2.

Algorithm 2 MERA-MSC

1: Input: Multi-view data $\\{\mathbf{X}_{v}\\}_{v=1}^{V},\lambda$

2: Initialize:
$\mathcal{Y}=\Gamma=0;\mathbf{Z}_{v}=\mathbf{E}_{v}=\Lambda_{v}=\mathbf{0},v=1,\cdots,V;\mu_{1}=10^{-4};\mu_{2}=5^{-4};\varepsilon=10^{-6};\eta=2;\mu_{1}^{\max}=\mu_{2}^{\max}=10^{10};t=1,T=50$

3: while $t\leq T$ do

4: for $v=1:V$ do

5: Update $\mathbf{Z}_{v}^{t}$ via equation (14)

6: Update $\mathbf{E}_{v}^{t}$ via equation (16)

7: Update $\Lambda_{v}^{t}$ via equation (19)

8: end for

9: Update $\mathcal{Y}^{t}$ via equation (17)

10: Update $\Gamma^{t}$ by equation (18)

11: $\mu_{1}=\min\left(\eta\mu_{1},\mu_{1}^{\max}\right)$,
$\mu_{2}=\min\left(\eta\mu_{2},\mu_{2}^{\max}\right)$;

12: Check convergence conditions

13: end while

14: Calculate affinity matrix:
$\mathbf{S}=\frac{1}{V}\sum_{v=1}^{V}\left|\mathbf{Z}_{v}\right|+\left|\mathbf{Z}_{v}^{\operatorname{T}}\right|$

15: Apply spectral clustering algorithm using $\mathbf{S}$

16: Output: Clustering result

Table 1: The computational complexity for one iteration, where $D$, $N$, and $V$ are the dimension of features, the number of samples, and views, respectively. Variables | $\mathbf{Z}_{v}$ | $\mathbf{E}_{v}$ | $\mathcal{Y}$
---|---|---|---
Complexity | $\operatorname{O}(N^{3})$ | $\operatorname{O}(DN^{2})$ | $\operatorname{O}(N^{3}V)$
Main Complexity | $\operatorname{O}(N^{3}(1+V))$ |

##  4 EXPERIMENTS

###  4.1 Experimental Settings

####  4.1.1 Multi-view Datasets Description

Five well-known multi-view datasets are chosen to evaluate the effectiveness
of MERA-MSC: Yale111http://vision.ucsd.edu/content/yale-face-database contains
165 samples of 15 clusters, with 11 samples per subject. We choose
3304-dimension(D) LBP, 6750-D Gabor, and 4096-D intensity as 3 views.
MSRC-v5[44] consists of 210 image samples collected from 7 clusters with 5
views, including 254-D CENT, 24-D CMT, 512-D GIST, 576-D HOG, 256-D LBP.
Extended YaleB222http://vision.ucsd.edu/leekc/ExtYaleDatabase/ExtYaleB.html
collects 2432 face image samples from 38 individuals, where each one has 64
images. Following [15], we choose the first 10 classes with 3 views, including
1024-D intensity, 1239-D LBP, and 256-D Gabor. Since data points in one
subspace are very close to other subspaces, clustering on this dataset can be
a challenge [8]. Notting-Hill [55] contains 4660 face samples with 5 classes.
We choose intensity 2000-D, 3304-D LBP, and 6750-D Gabor as 3 views. Due to
the large number of samples, we will downsample it as 2330 samples with 5
classes. BDGP [4] contains 2500 samples from 5 classes with 4 views, including
1000-D lateral, 500-D dorsal, 250-D ventral, and 79-D texture.

The statistical information of the above datasets is summarized in Table 2.
Note that each self-representation tensor of size $N\times N\times V$ is
rearranged into a 5th-order tensor for low-rank MERA approximation. The last
column of Table 2 displays the sizes of the 5th-order self-representation
tensors corresponding to different datasets. More information about this
rearrangement can be found in Subsection 3.1.

Table 2:  Statistical information of different multi-view datasets.

Datasets | Samples | Views | Clusters |  5-D ($I_{1},\cdots,I_{5}$)
---|---|---|---|---
Yale | 165 | 3 | 15 | (11,15,15,11,3)
MSRC-v5 | 210 | 5 | 7 | (15,14,15,14,5)
Extended YaleB | 640 | 3 | 10 | (32,20,20,32,3)
Notting-Hill | 2330 | 3 | 5 | (233,10,233,10,3)
BDGP | 2000 | 4 | 5 | (50,50,50,50,4)

####  4.1.2 Compared Clustering Algorithms

In order to compare clustering performance, we select nine state-of-the-art
methods including one single-view method: LRRbest [22], seven self-
representation based multi-view methods: multi-view low-rank representation
(MLRR) [2021, TCYB] [5], latent multi-view subspace clustering(L-MSC) [2017,
CVPR] [54], exclusivity-consistency regularized multi-view subspace clustering
(ECMSC) [2017, CVPR] [43], low-rank tensor constrained multi-view subspace
clustering (LTMSC) [2015, ICCV] [53], t-SVD based multi-view subspace
clustering (t-SVD-MSC) [2018, IJCV] [49], weighted tensor Schatten $p$-norm
based multi-view subspace clustering (WTSNM) [2021, TCYB] [47], multi-view
subspace clustering tailored tensor low-rank representation (MVSC-TLRR) [2021,
TCSVT] [15], and one updated tensor based multi-view clustering method:
tensorized bipartite graph learning for multi-view clustering (TBGL-MVC)
[2022, TPAMI] [45]. All tests are accomplished on a desktop computer with 2.4
GHz Quad-Core Intel Core i5 Processor and 16 GB 2133 MHz LPDDR3 Memory.

####  4.1.3 Evaluation Metrics

Six standard evaluation metrics are considered to evaluate the performance in
our experiments, including F-score, Precision, Recall, normalized mutual
information (NMI), adjusted rand index (ARI), and accuracy (ACC). The larger
values of these metrics indicate better clustering performance. The detailed
information can refer to [30].

####  4.1.4 Parameter settings

Our MERA-MSC has two free parameters, one for MERA decomposition, namely MERA
rank $R$, and the other is the balancing parameter $\lambda$. We fix one and
tune the other via brute force search. For example, we first fix
$\lambda=0.01$ and tune parameters $R$ on the Yale dataset, where $R$ varies
from {4, 8, 12, 16, 20, 24, 28}. The clustering results are reported in Fig.
6(6(a)), where the ACC almost approaches 1 and performs stably when $R$ ranges
in $[4,8]$. In addition, we fix $R=6$ to choose the parameter $\lambda$ from
{0, 0.0001, 0.01, 0.1, 0.5, 1}. From Fig. 6(6(b)), we can observe the best
clustering performance in terms of six metrics is achieved in a wide range of
$\lambda$, i.e., $\lambda\in[0.001,0.1]$.

(a) $\lambda=0.001$

(b) $R=6$

Figure 6: The change of clustering performance as parameters $\lambda$ and $R$
vary. (a) $R$ changed with $\lambda=0.001$; (b) $\lambda$ changed with $R=6$.

###  4.2 Clustering Performance Analysis

Table 3 shows the clustering performance of different methods on five multi-
view datasets in terms of F-score, Precision, Recall, AR, NMI, and ACC, where
the best and the second best results are highlighted in bold and underlined,
respectively. For MSRC datasets, several multi-view methods, including WTSNM,
MVSC-TLRR, TBGL-MVC, and MERA-MSC, all reached the best performance with all
metrics achieved 1. Meanwhile, our MERA-MSC also gives the best performance on
Yale datasets and outperforms all compared baselines, especially on the
extended YaleB dataset. Specifically, our MERA-MSC provides 2.7% Precision
improvement on the Yale dataset; 6.9% NMI improvement on the extended YaleB
dataset, 6.6% AR improvement on the Notting Hill dataset, and 0.4% F-score on
the BDGP dataset.

Moreover, tensor-based MSC clustering methods, including t-SVD-MSC, WTSNM,
MVSC-TLRR and MERA-MSC, generally perform better than matrix-based ones. It
verifies the capability of low-rank tensor decomposition to explore high-order
correlations from different views. However, considering the extended YaleB
dataset, which is hard to cluster due to the effects of high illumination
variation; only MVSC-TLRR and the proposed MERA-MSC perform well. Compared to
other tensor-based MSC methods, these two methods investigate both inter- and
intra-view information throughout the self-representation tensor learning
process. The introduction of intra-view information exploration can further
enhance the clustering performance. In addition, compared with MVSC-TLRR, our
method performs better. This is mainly because low-rank MERA decomposition can
adaptively learn both inter-/intra-view from the self-representation tensor
and provides a powerful and flexible representation. But for MVSC-TLRR, the
contributions from inter-view and intra-view are required to be balanced by
weight selection and thus reduce its flexibility.

Table 3: Clustering results, e.g., mean value (Standard Deviation) on Yale,
MSRC-v5, Extended YaleB, Notting Hill, and BDGP datasets.

Datasets | Methods | F-score | Precision | Recall | NMI | ARI | ACC
---|---|---|---|---|---|---|---
MSRC-v5 ($R$=2,$\lambda$=0.001) | LRR-best | 0.5469(0.0025) | 0.5469(0.0027) | 0.5469(0.0027) | 0.5658(0.0049) | 0.4734(0.0029) | 0.6862(0.0015)
MLRR | 0.6674(0.0035) | 0.6604(0.0031) | 0.6745(0.0031) | 0.6828(0.0028) | 0.6131(0.0040) | 0.7800(0.0038)
L-MSC | 0.6351(0.0074) | 0.6197(0.0087) | 0.6512(0.0087) | 0.6597(0.0086) | 0.5745(0.0088) | 0.7714(0.0045)
ECMSC | 0.6924(0.0000) | 0.6804(0.0000) | 0.7048(0.0000) | 0.7185(0.0000) | 0.6418(0.0000) | 0.7857(0.0000)
LTMSC | 0.7148(0.0035) | 0.7023(0.0036) | 0.7276(0.0036) | 0.7444(0.0037) | 0.6679(0.0041) | 0.8338(0.0027)
tSVD-MSC | 0.9806(0.0000) | 0.9803(0.0000) | 0.9810(0.0000) | 0.9785(0.0000) | 0.9775(0.0000) | 0.9905(0.0000)
WTSNM | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000)
MVSC-TLRR | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000)
TBGL-MVC | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000)
MERA-MSC | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000)
Yale ($R$=6,$\lambda$=0.001) | LRR-best | 0.5483(0.0212) | 0.5483(0.0202) | 0.5483(0.0202) | 0.7111(0.0146) | 0.5180(0.0226) | 0.7061(0.0146)
MLRR | 0.4884(0.0154) | 0.4725(0.0145) | 0.5055(0.0145) | 0.6569(0.0112) | 0.4540(0.0163) | 0.6327(0.0192)
L-MSC | 0.5144(0.0096) | 0.4689(0.0129) | 0.5699(0.0129) | 0.7131(0.0058) | 0.4796(0.0107) | 0.6709(0.0076)
ECMSC | 0.4843(0.0000) | 0.4428(0.0000) | 0.5345(0.0000) | 0.7029(0.0000) | 0.4475(0.0000) | 0.6545(0.0000)
LTMSC | 0.6005(0.0114) | 0.5785(0.0122) | 0.6242(0.0122) | 0.7500(0.0080) | 0.5735(0.0122) | 0.7297(0.0042)
tSVD-MSC | 0.9155(0.0657) | 0.9037(0.0753) | 0.9280(0.0753) | 0.9536(0.0364) | 0.9099(0.0701) | 0.9394(0.0525)
WTSNM | 0.9725(0.0313) | 0.9628(0.0504) | 0.9833(0.0504) | 0.9875(0.0104) | 0.9706(0.0335) | 0.9800(0.0294)
MVSC-TLRR | 0.9746(0.0000) | 0.9734(0.0000) | 0.9758(0.0000) | 0.9848(0.0000) | 0.9729(0.0000) | 0.9879(0.0000)
TBGL-MVC | 0.6475(0.0000) | 0.5626(0.0000) | 0.7624(0.0000) | 0.8531(0.0000) | 0.6208(0.0000) | 0.8606(0.0000)
MERA-MSC | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000) | 1.0000(0.0000)
Extended YaleB ($R$=10,$\lambda$=1) | LRR-best | 0.4750(0.0008) | 0.4750(0.0007) | 0.4750(0.0007) | 0.6014(0.0013) | 0.4163(0.0009) | 0.5827(0.0005)
MLRR | 0.1789(0.0021) | 0.1756(0.0023) | 0.1822(0.0023) | 0.1889(0.0035) | 0.0872(0.0025) | 0.2561(0.0058)
L-MSC | 0.2791(0.0023) | 0.2291(0.0028) | 0.3571(0.0028) | 0.4453(0.0035) | 0.1807(0.0030) | 0.4644(0.0010)
ECMSC | 0.3707(0.0000) | 0.2841(0.0000) | 0.5332(0.0000) | 0.6282(0.0000) | 0.2777(0.0000) | 0.6312(0.0000)
LTMSC | 0.4154(0.0049) | 0.4063(0.0055) | 0.4250(0.0055) | 0.5424(0.0040) | 0.3499(0.0056) | 0.5109(0.0051)
tSVD-MSC | 0.4584(0.0050) | 0.4231(0.0053) | 0.5001(0.0053) | 0.6113(0.0040) | 0.3936(0.0057) | 0.6005(0.0028)
WTSNM | 0.4426(0.0043) | 0.4070(0.0047) | 0.4851(0.0047) | 0.5964(0.0033) | 0.3757(0.0050) | 0.5875(0.0028)
MVSC-TLRR | 0.9139(0.0013) | 0.9125(0.0013) | 0.9153(0.0013) | 0.9123(0.0009) | 0.9045(0.0014) | 0.9558(0.0008)
TBGL-MVC | 0.2394(0.0000) | 0.1560(0.0000) | 0.5140(0.0000) | 0.4984(0.0000) | 0.1038(0.0000) | 0.4641(0.0000)
MERA-MSC | 0.9535(0.0000) | 0.9497(0.0000) | 0.9574(0.0000) | 0.9653(0.0000) | 0.9484(0.0000) | 0.9766(0.0000)
Notting Hill ($R$=15,$\lambda$=0.0001) | LRR-best | 0.8047(0.0000) | 0.8047(0.0000) | 0.8047(0.0000) | 0.7456(0.0000) | 0.7520(0.0000) | 0.8811(0.0000)
MLRR | 0.8394(0.0008) | 0.8301(0.0006) | 0.8490(0.0006) | 0.8006(0.0008) | 0.7936(0.0010) | 0.8446(0.0007)
L-MSC | 0.7967(0.0000) | 0.8033(0.0000) | 0.7903(0.0000) | 0.8250(0.0000) | 0.7402(0.0000) | 0.7777(0.0000)
ECMSC | 0.6684(0.0000) | 0.5455(0.0000) | 0.8627(0.0000) | 0.6919(0.0000) | 0.5465(0.0000) | 0.7300(0.0000)
LTMSC | 0.8320(0.0003) | 0.8386(0.0004) | 0.8255(0.0004) | 0.7949(0.0004) | 0.7853(0.0004) | 0.9036(0.0002)
tSVD-MSC | 0.8901(0.0000) | 0.9096(0.0000) | 0.8715(0.0000) | 0.8743(0.0000) | 0.8601(0.0000) | 0.9421(0.0000)
WTSNM | 0.9101(0.0000) | 0.9280(0.0000) | 0.8929(0.0000) | 0.8939(0.0000) | 0.8854(0.0000) | 0.9532(0.0000)
MVSC-TLRR | 0.8766(0.0000) | 0.8929(0.0000) | 0.8609(0.0000) | 0.8555(0.0000) | 0.8427(0.0000) | 0.9343(0.0000)
TBGL-MVC | 0.7586(0.0000) | 0.7436(0.0000) | 0.7742(0.0000) | 0.8123(0.0000) | 0.6890(0.0000) | 0.8476(0.0000)
MERA-MSC | 0.9619(0.0000) | 0.9552(0.0000) | 0.9687(0.0000) | 0.9450(0.0000) | 0.9511(0.0000) | 0.9700(0.0000)
BDGP ($R$=10,$\lambda$=0.0002) | LRR-best | 0.7448(0.0000) | 0.7448(0.0000) | 0.7448(0.0000) | 0.8020(0.0000) | 0.6711(0.0000) | 0.7004(0.0000)
MLRR | 0.6838(0.0000) | 0.6113(0.0000) | 0.7758(0.0000) | 0.7006(0.0000) | 0.5928(0.0000) | 0.7064(0.0000)
L-MSC | 0.7354(0.0000) | 0.6673(0.0000) | 0.8191(0.0000) | 0.7762(0.0000) | 0.6601(0.0000) | 0.6912(0.0000)
ECMSC | 0.7562(0.0000) | 0.717(0.0000) | 0.800(0.0000) | 0.761(0.0000) | 0.691(0.0000) | 0.789(0.0000)
LTMSC | 0.4325(0.0000) | 0.4281(0.0000) | 0.4369(0.0000) | 0.3889(0.0000) | 0.2891(0.0000) | 0.5272(0.0000)
tSVD-MSC | 0.9889(0.0000) | 0.9888(0.0000) | 0.9890(0.0000) | 0.9814(0.0000) | 0.9861(0.0000) | 0.9944(0.0000)
WTSNM | 0.9901(0.0000) | 0.9901(0.0000) | 0.9911(0.0000) | 0.9840(0.0000) | 0.9881(0.0000) | 0.9951(0.0000)
MVSC-TLRR | 0.9842(0.0000) | 0.9841(0.0000) | 0.9842(0.0000) | 0.9719(0.0000) | 0.9802(0.0000) | 0.9920(0.0000)
TBGL-MVC | 0.3303(0.0000) | 0.2000(0.0000) | 0.9487(0.0000) | 0.0410(0.0000) | 0.0007(0.0000) | 0.2236(0.0000)
MERA-MSC | 0.9913(0.000) | 0.9912(0.0000) | 0.9913(0.0000) | 0.9860(0.0000) | 0.9891(0.0000) | 0.9956(0.0000)

Figure 7:  Visual illustration of latent representation v.s. learning
iteration on Yale data, where each feature is visualized by t-SNE [36], and
the value indicates the clustering accuracy (ACC) by K-means. View 1 is
intensity features; View 2 is LBP features; Global is the fusion features from
three different views. The four columns from left to right represent the
clustering performance of the 0-th, 1-st, 10-th, and 50-th iterations,
respectively.

###  4.3 Discussions

####  4.3.1 Model Analysis

Fig. 7 shows how the clustering performance of the proposed MERA-MSC varies
during the learning iteration on the Yale dataset. From the first column of
Fig. 7, we can observe the clustering performance on all views is superior to
that on a single view. Interestingly, when the low-rank MERA approximation is
first used to explore inter/intra-view correlations, the clustering accuracy
(ACC) increases by around 11%, as illustrated in the second column of Fig. 7.
Iteratively, the correlation within views and across views mutually benefits
from the low-rank MERA approximation. This allows us to learn a better latent
structure from multi-view data, ultimately achieving a clustering accuracy of
100%. Moreover, the performance improvement indicates that the utilization of
information from both within and across views plays a significant role in
enhancing clustering accuracy. Additionally, the low-rank MERA approximation
effectively explores the inter-view and intra-view information in the self-
representation tensor simultaneously.

####  4.3.2  Convergence Analysis

Fig. 8 shows the convergence curves of MERA-MSC on different datasets, where
the X-axis means the number of iterations and the Y-axis is the values of
reconstruction error (RE) and match error (ME), which are defined by RE =
$\max_{v,v=1,\cdots,V}\|\mathbf{X}_{v}-\mathbf{X}_{v}\mathbf{Z}_{v}-\mathbf{E}_{v}\|_{\infty}$,
and ME = $\max_{v,v=1,\cdots,V}\|\mathbf{Z}_{v}-\mathbf{Y}_{v}\|_{\infty}$,
respectively. It can be observed the RE of MERA-MSC drops very fast at the
first 10 iterations and eventually approaches 0 on all datasets. In
particular, the larger size of the datasets, the faster the RE and ME change.

(a) RE v.s. iteration

(b) ME v.s. iteration

Figure 8: The convergence performance of MERA-MSC. Table 4: Clustering results
on CCV, Caltech-all and ALOI datasets.

Methods | F-score | Precision | Recall | NMI | ARI | ACC | CPU Time (s)
---|---|---|---|---|---|---|---
ALOI ($R$=4, $\lambda$=0.02, $M$=10)
SMSC | 0.7617 | 0.7129 | 0.8185 | 0.8990 | 0.7591 | 0.8273 | 365.7435
LMVSC | 0.3874 | 0.3498 | 0.4341 | 0.6964 | 0.3806 | 0.4897 | 93.0113
SMVSC | 0.2089 | 0.1366 | 0.4437 | 0.6033 | 0.1967 | 0.3251 | 423.8729
FPMVS-CAG | 0.1795 | 0.1053 | 0.6062 | 0.6334 | 0.1654 | 0.3121 | 180.5025
FastMICE | 0.6713 | 0.6223 | 0.7289 | 0.8455 | 0.6678 | 0.7543 | 20.3759
sMERA-MVC | 0.7130 | 0.6107 | 0.8598 | 0.9114 | 0.7096 | 0.7424 | 19.5916
Caltech-all ($R$=8, $\lambda$=0.1, $M$=9)
SMSC | 0.1817 | 0.2935 | 0.1316 | 0.4679 | 0.1671 | 0.2448 | 269.1425
LMVSC | 0.1685 | 0.2810 | 0.1203 | 0.4374 | 0.1541 | 0.2075 | 92.7087
SMVSC | 0.1895 | 0.1200 | 0.4496 | 0.3485 | 0.1516 | 0.2906 | 356.5180
FPMVS-CAG | 0.2040 | 0.1425 | 0.3589 | 0.3379 | 0.1704 | 0.2654 | 669.0376
FastMICE | 0.1775 | 0.3209 | 0.1227 | 0.4370 | 0.1644 | 0.2090 | 34.8471
sMERA-MVC | 0.3807 | 0.5807 | 0.2834 | 0.7533 | 0.3690 | 0.4517 | 56.7210
CCV ($R$=2, $\lambda$=0.5, $M$=6)
SMSC | 0.1615 | 0.1599 | 0.1633 | 0.2184 | 0.1096 | 0.2714 | 128.4686
LMVSC | 0.1202 | 0.1069 | 0.1374 | 0.1694 | 0.0592 | 0.2073 | 59.2433
SMVSC | 0.1294 | 0.1235 | 0.1359 | 0.1540 | 0.0734 | 0.2110 | 31.4677
FPMVS-CAG | 0.1316 | 0.1174 | 0.1496 | 0.1597 | 0.0716 | 0.2282 | 28.7811
FastMICE | 0.1320 | 0.1387 | 0.1260 | 0.1795 | 0.0816 | 0.2306 | 10.3551
sMERA-MVC | 0.3603 | 0.3686 | 0.3526 | 0.5087 | 0.3221 | 0.4632 | 6.0318

####  4.3.3 Scalability analysis

Most tensor-based MSC methods involve constructing an $N\times N$ relational
graph with a computational complexity of $O(VN^{3})$, where $N$ and $V$
represent the number of instances and views, respectively. This process is
time-consuming, particularly for large-scale multi-view data, as the
computational time increases with the number of samples.

To extend the MERA-MSC method, we have developed a scalable MERA based multi-
view clustering (sMERA-MVC) algorithm. This approach replaces the self-
representation learning step, denoted as
$\mathbf{X}_{v}=\mathbf{X}_{v}\mathbf{Z}_{v}+\mathbf{E}_{v}$, in the MSC
framework (equation (10)), with anchor learning
$\mathbf{X}_{v}=\mathbf{A}_{v}\mathbf{C}_{v}+\mathbf{E}_{v}$ [33, 41]. The
revised framework is illustrated below:

$\displaystyle\min_{\\{\mathbf{E}_{v},\mathbf{C}_{v}\\}_{v=1}^{V}}\quad\sum_{v=1}^{V}\lambda\|\mathbf{E}_{v}\|_{2,1}$
(20)
$\displaystyle\text{s.~{}t.~{}}\mathbf{X}_{v}=\mathbf{A}_{v}\mathbf{C}_{v}+\mathbf{E}_{v},\mathbf{A}_{v}^{\operatorname{T}}\mathbf{A}_{v}=\mathbf{I}_{M},v=1,\cdots,V,$
$\displaystyle\quad\hat{\mathcal{C}}=f(\mathbf{B},\mathcal{W},\mathcal{U}),\hat{\mathcal{C}}=\mathfrak{R}(\mathbf{C}_{1},\mathbf{C}_{2},\cdots,\mathbf{C}_{V}).$

Here, $\mathbf{A}_{v}\in\mathbb{R}^{D_{v}\times M}$ represents the anchor
matrix, $M$ represents the number of anchors, and
$\mathbf{C}_{v}\in\mathbb{R}^{M\times N}$ represents the anchor graph. The
anchor graph captures the relationships between $N$ data points and $M$
anchors (where $M\ll N$) to represent the relationships among all data points,
effectively reducing the computational complexity to $O(VNM^{2})$.
$\hat{\mathcal{C}}=f(\mathbf{B},\mathcal{W},\mathcal{U})$ means that low-rank
MERA approximation is considered on the anchor graph tensor to well explore
the inter/intra-view correlations within multi-view data.

To validate the effectiveness and efficiency of sMERA-MVC, we conducted a
comparative analysis with five fast MVC methods on three large-scale datasets.
Below, we introduce the datasets used and outline the methods used for
comparison.
Datasets discription: Amsterdam Library of Object Image
(ALOI)333http://elki.dbs.ifi.lmu.de/wiki/DataSets/MultiView [12] has a
collection of 110250 images of 1000 small objects. Following the approach in
[21], we use the first 100 classes as our testing dataset and obtained 10800
samples with four views. Columbia Consumer Video
(CCV)444https://www.ee.columbia.edu/ln/dvmm/CCV/ [16] is a rich database of
YouTube videos containing 20 semantic categories. We remove the last 73
samples in our experiment, resulting in 6700 samples. Caltech-all
555http://www.vision.caltech.edu/Image Datasets/Caltech101/ [9] has 9144
samples and five views. In our experiment, we remove the last 24 samples to
obtain 9120 samples.
Methods discription: Sparse multi-view spectral clustering (SMSC) [2020,
Neurocomputing] [13], large-scale multi-view subspace clustering (LMVSC)
[2020, AAAI] [17], scalable multi-view subspace clustering with unified
anchors (SMVSC) [2021, ACM MM] [33], fast Parameter-free multiview subspace
clustering with consensus anchor guidance (FPMVS-CAG) [2021, TIP] [41], fast
multi-view clustering via ensembles (FastMICE) [2023, TKDE] [14].
Performance and Analysis on Large-Scale Multi-view Datasets: Table 4 shows the
clustering performance of different methods on three large-scale multi-view
datasets in terms of F-score, Precision, Recall, AR, NMI, ACC, and CPU time.
We can observe that sMERA-MVC consistently outperforms the other five methods
regarding clustering performance. In particular, for ALOI, SMSC performs well
in most clustering accuracy metrics, with the sMERA-MVC method following
closely behind. Notably, sMERA-MVC achieves a significant advantage in terms
of CPU time, running approximately 30 times faster than the SMSC. For Caltech-
all, sMERA-MVC demonstrates superior performance across all evaluation
metrics, except for recall and computation time, surpassing all comparison
baselines. Notably, sMERA-MVC excels in all aspects of the CCV dataset, while
the performance of the other five methods is comparatively weaker. This
further implies that the low-rank MERA approximation exhibits excellent
performance in exploring the shared latent structure across views and within
individual views in multi-view data.

##  5 CONCLUSION

In this paper, we propose a low-rank MERA based MSC algorithm, where the self-
representation tensor can be adaptively learned from the multi-view data and
low-rank MERA approximation in each iteration. Benefiting from the interaction
among orthogonal/semi-orthogonal factors of low-rank MERA approximation, the
correlations present across and within views can be effectively explored.
Numerical experiments on five well-known datasets show that MERA-MSC
outperforms all state-of-the-art methods in clustering performance evaluated
on six different metrics. Besides, we have developed an accelerated and
scalable MERA based multi-view clustering algorithm for large-scale multi-view
data and verified its effectiveness and efficiency on three large-scale
datasets.

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# Novel imprint of a vector doublet

Phung Van Dong  Corresponding author.
<EMAIL_ADDRESS>Phenikaa Institute for Advanced Study and
Faculty of Basic Science, Phenikaa University, Yen Nghia, Ha Dong, Hanoi
100000, Vietnam  Duong Van Loi  Phenikaa Institute for Advanced Study and
Faculty of Basic Science, Phenikaa University, Yen Nghia, Ha Dong, Hanoi
100000, Vietnam  Le Duc Thien  Department of Natural Science, Hoang Van Thu
Specialized High School, Thinh Lang, Hoa Binh 350000, Vietnam  Pham Ngoc Thu
Faculty of Mathematics-Physics-Informatics, Tay Bac University, Quyet Tam, Son
La 360000, Vietnam

###### Abstract

It is shown that the presence of a vector doublet is suitable to address
neutrino mass, dark matter, and the recent muon anomalous magnetic moment.

##  I Motivation

Neutrino mass and dark matter are the two leading questions in science, which
require the new physics beyond the conventional standard model Zyla:2020zbs .
Additionally, the muon anomalous magnetic moment recently measured also
reveals a significant deviation at 4.2$\sigma$ from the standard model
prediction Abi:2021gix .

Within the attempts to solve simultaneously the first two issues, the
scotogenic setup Ma:2006km  is the most compelling. Indeed, it introduces an
inert scalar doublet and three sterile neutrinos, which couple to the usual
lepton doublets, producing the relevant neutrino masses through the one-loop
contribution of these new fields, while the lightest of which, either a
neutral inert scalar or a sterile neutrino, is viably to be a dark matter
candidate. However, an exact $Z_{2}$ parity, that necessarily makes all the
new fields odd, plays the crucial role for the model properly working as well
as stabilizing the dark matter. The origin of the $Z_{2}$ was left as an open
question. Further, it is found that the minimal scotogenic contributes
insignificantly (and negative) to the muon $g-2$ Chen:2019nud .

We suggest that if a new vector doublet is proposed instead of the inert
scalar doublet, it cannot develop a vacuum expectation value due to the
Lorentz invariance, in opposition to the inert scalar case that potentially
breaks the $Z_{2}$, inducing an unwanted tree-level neutrino mass.
Additionally, the standard model symmetry demands that the vector doublet
couples to normal matter only through new fermions; for such reason, the
sterile neutrinos are presented.111Indeed, the alternative choice includes new
fermion triplets or (irrelevant) doublets, but not interested, since they are
not universally supported by the gauge completion models. Furthermore, the
Lagrangian of the vector doublet realizes a matter parity, instead of the
$Z_{2}$, which commonly arises as a residual gauge symmetry in the theories
that let the vector doublet be a part of the extended gauge field.

With the effective theory at hand, we give a novel recognition of a scotogenic
mechanism with the vector doublet, leading to the neutrino mass, dark matter,
and the muon $g-2$ manifestly. The new physics predicted is appropriate to the
collider bounds. A brief discussion on the UV-completion theory is delivered
at the end.

##  II Proposal

We introduce a vector doublet to the standard model,

$V_{\mu}=\left(\begin{array}[]{c}V^{0}_{\mu}\\\
V^{-}_{\mu}\end{array}\right)\sim(1,2,-1/2),$ (1)

which has the $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ quantum numbers
like the ordinary lepton doublets, $\psi_{aL}=(\nu_{aL}\
e_{aL})^{T}\sim(1,2,-1/2)$, where $a=1,2,3$ is a family index. Notice that
$\tilde{V}_{\mu}\equiv i\sigma_{2}V^{*}_{\mu}\sim(1,2,1/2)$ transforms as the
usual Higgs doublet, $\phi=(\phi^{+}\ \phi^{0})^{T}\sim(1,2,1/2)$, and the
introduction of either $V$ or $\tilde{V}$ from outset is physically
equivalent. Such a vector doublet is commonly hinted from numerous electroweak
gauge extensions, such as the 3-3-1 Singer:1980sw ; Montero:1992jk ;
Foot:1994ym , trinification Babu:1985gi , $SU(6)$ and $E_{6}$, as well as
gauge-Higgs unification Fairlie:1979at ; Hosotani:1983xw , that all contain
the higher weak isospin, $SU(3)_{L}$, as a sub- or residual group
Chizhov:2009fc .

The Lagrangian of the vector doublet according to the standard model symmetry
takes the form,

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\frac{1}{2}F^{\dagger}_{\mu\nu}F^{\mu\nu}+m^{2}_{V}V^{\dagger}_{\mu}V^{\mu}-\frac{1}{\xi}(D_{\mu}V^{\mu})^{\dagger}(D_{\nu}V^{\nu})$
(2)
$\displaystyle+\alpha_{1}(V^{\dagger}_{\mu}V^{\mu})(V^{\dagger}_{\nu}V^{\nu})+\alpha_{2}(V^{\dagger}_{\mu}V^{\nu})(V^{\dagger}_{\nu}V^{\mu})$
$\displaystyle+\alpha_{3}(V^{\dagger}_{\mu}V^{\nu})(V^{\dagger\mu}V_{\nu})+i\beta_{1}V_{\mu}^{\dagger}A^{\mu\nu}V_{\nu}+i\beta_{2}V_{\mu}^{\dagger}B^{\mu\nu}V_{\nu}$
$\displaystyle+\lambda_{1}(\phi^{\dagger}\phi)(V^{\dagger}_{\mu}V^{\mu})+\lambda_{2}(\phi^{\dagger}V_{\mu})(V^{\dagger\mu}\phi)$
$\displaystyle+\frac{1}{2}[\lambda_{3}(\phi V_{\mu})(\phi V^{\mu})+H.c.],$

where $F_{\mu\nu}=D_{\mu}V_{\nu}-D_{\nu}V_{\mu}$,
$D_{\mu}=\partial_{\mu}+igA_{\mu}+ig^{\prime}B_{\mu}$, and $A_{\mu}$
($A_{\mu\nu}$) and $B_{\mu}$ ($B_{\mu\nu})$ are the gauge field (field
strength) of $SU(2)_{L}$ and $U(1)_{Y}$, respectively.222Here $\phi
V=\phi^{+}V^{-}-\phi^{0}V^{0}\sim 1$ under $SU(2)_{L}$. The couplings of other
forms, e.g. $(\phi V)(\phi V)\sim 3\times 3$ and $(\phi V)^{\dagger}(\phi
V)\sim 1\times 1$ or $3^{*}\times 3$, are reducible to the given interactions,
thus suppressed. Notice that the nontrivial $1/\xi$ term is simply an allowed
gauge-invariant coupling, not a gauge fixing, since $V_{\mu}$ is not a gauge
field. Surely from (2), the vector fields obey the Klein-Gordon equation as
well as the Lorentz condition manifestly. One would replace $A\rightarrow
T_{j}A_{j}$ and $B\rightarrow YB$ for practical computation, in which $T_{j}$
($j=1,2,3$) and $Y$ are the weak isospin and hypercharge, respectively.

This theory preserves an accidental $Z_{2}$ symmetry, which transforms
$V_{\mu}\rightarrow-V_{\mu}$, while leaving the standard model fields
unchanged, in agreement with Saez:2018off . The $Z_{2}$ symmetry is identical
to the matter parity

$P=(-1)^{3(B-L)+2s}$ (3)

to be a residual gauge symmetry of the 3-3-1-1 gauge group Dong:2013wca  or
flipped trinification Dong:2017zxo  breaking, there the vector doublet has
$B-L=-1$. Unlike all other couplings, the $\lambda_{3}$ interaction violates
$B-L$, but conserves $P$, to be induced by breaking $B-L=2$. That said, it
should be radically smaller than the $B-L$ conserving couplings, such as

$\lambda_{3}\ll\lambda_{1,2},\alpha_{1,2,3},\beta_{1,2},1/\xi.$ (4)

On the other hand, the scalar-vector mixing terms of type, $i\kappa\phi
D_{\mu}V^{\mu}+H.c.$, disappear due to $Z_{2}$ conservation, or they are
automatically eliminated if $V_{\mu}$ originates as a part of gauge field of a
higher gauge symmetry.

The vector doublet does not couple to the standard model fermions due to
$\bar{f}_{L}\gamma^{\mu}f_{R}X_{\mu}=0$ for $X=V$ or $\tilde{V}$ as suitable.
However, we introduce three fermion singlets,

$N_{aL}\sim(1,1,0),$ (5)

to be the variants of the standard model neutrinos. The fermion singlets
couple to the standard model lepton doublets through the vector doublet,

$\mathcal{L}\supset h_{ab}\bar{\psi}_{aL}\gamma^{\mu}N_{bL}V_{\mu}+H.c.$ (6)

The new fermions transform under $Z_{2}$ as $N_{aL}\rightarrow-N_{aL}$, or
they have $B-L=0$ under the matter parity, and they possess mass terms,

$\mathcal{L}\supset-\frac{1}{2}M_{ab}N_{aL}N_{bL}+H.c.$ (7)

Here we assume $M=\mathrm{diag}(M_{1},M_{2},M_{3})$ to be flavor diagonal,
without loss of generality. We label the physical states and masses, say
$N_{k},M_{k}$, in the mass basis.

After electroweak symmetry breaking, the vector doublet components
$V^{0}_{\mu}\equiv(V_{1\mu}+iV_{2\mu})/\sqrt{2}$ and $V^{-}_{\mu}$ are
separated in mass, such as

$\displaystyle m^{2}_{V^{-}}$ $\displaystyle=$ $\displaystyle
m^{2}_{V}+(\lambda_{1}+\lambda_{2})\frac{v^{2}}{2},$ (8) $\displaystyle
m^{2}_{V_{1}}$ $\displaystyle=$ $\displaystyle
m^{2}_{V}+(\lambda_{1}+\lambda_{3})\frac{v^{2}}{2},$ (9) $\displaystyle
m^{2}_{V_{2}}$ $\displaystyle=$ $\displaystyle
m^{2}_{V}+(\lambda_{1}-\lambda_{3})\frac{v^{2}}{2}.$ (10)

Here $\lambda_{3}$ has been assumed to be real; otherwise, its phase can be
removed by redefining appropriate $\phi,V$ fields. Moreover, the Feynman
propagators of the vector fields are similar to massive gauge fields in a
$R_{\xi}$ gauge due to the contribution of the $1/\xi$ coupling. On the other
hand, the addition of the vector fields violates the unitarity condition of
the $S$-matrix and the theory only works well below some cut-off scale,
$\Lambda\sim m_{V}$, making constraint on the new physics couplings, naively

$\frac{g^{2}_{\mathrm{NP}}}{4\pi}\lesssim\frac{\Lambda^{2}}{s}\sim 1.$ (11)

We take $\Lambda\sim\sqrt{s}\sim 1$ TeV to be the current energy of colliders,
where the standard model is still good.

The addition of $V_{\mu}$ and $N_{aL}$ presents three important results,
arranged in order.

##  III Scotogenic scheme

The scotogenic model Ma:2006km  is an idea of radiative-seesaw neutrino-mass
generation, through one-loop diagram mediated by an inert Higgs doublet and
three sterile neutrinos. It is recognized in this work in which the doublet
vector plays the role of the inert Higgs doublet instead, with the similar
interactions such as $\lambda_{3}$ in (2) and $h_{ab}$ in (6), including the
Majorana nature of $N_{aL}$ in (7).333Notice that Ref.
CarcamoHernandez:2018vdj  introduced, by contrast, highly nonrenormalizable
(dimension-12) operators connecting $V_{\mu}$ to leptons in a theory under
discrete symmetries, without possessing the $1/\xi$ coupling or a gauge
completion; as a result, the neutrino mass diagram was divergent, defined by a
cutoff.

Figure 1:  One-loop contribution to neutrino mass.

The neutrino mass generation diagram is depicted in Fig. 1. Hence, the
neutrino mass matrix, that is induced in the form of
$\mathcal{L}\supset-\frac{1}{2}\bar{\nu}_{aL}(M_{\nu})_{ab}\nu^{c}_{bL}$,
reads

$\displaystyle-i(M_{\nu})_{ab}P_{R}$ $\displaystyle=$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}\left(i\frac{h_{ak}}{\sqrt{2}}\gamma^{\mu}P_{L}\right)\frac{i}{/\\!\\!\\!p-M_{k}}\left(-i\frac{h_{bk}}{\sqrt{2}}\gamma^{\nu}P_{R}\right)\frac{-i}{p^{2}-m^{2}_{V_{1}}}\left(g_{\mu\nu}-\frac{(1-\xi)p_{\mu}p_{\nu}}{p^{2}-\xi
m^{2}_{V_{1}}}\right)$ (12)
$\displaystyle+\int\frac{d^{4}p}{(2\pi)^{4}}\left(-\frac{h_{ak}}{\sqrt{2}}\gamma^{\mu}P_{L}\right)\frac{i}{/\\!\\!\\!p-M_{k}}\left(\frac{h_{bk}}{\sqrt{2}}\gamma^{\nu}P_{R}\right)\frac{-i}{p^{2}-m^{2}_{V_{2}}}\left(g_{\mu\nu}-\frac{(1-\xi)p_{\mu}p_{\nu}}{p^{2}-\xi
m^{2}_{V_{2}}}\right).$

Notice that the real ($V_{1}$) and imaginary ($V_{2}$) parts of $V^{0}$ give
opposite contributions, since the product of the two neutrino vertices to each
of $V_{1,2}$ changes sign, and the difference between the contributions is
only due to the $V_{1,2}$ mass splitting. It is straightforward to obtain,

$\displaystyle(M_{\nu})_{ab}$ $\displaystyle=$
$\displaystyle\frac{h_{ak}h_{bk}M_{k}}{32\pi^{2}}\left[f(M^{2}_{k},m^{2}_{V_{1}})-f(M^{2}_{k},m^{2}_{V_{2}})\right],$
(13)

where $f(x,y)$ equals to

$\displaystyle\frac{y}{x-y}\left[\frac{(1+\xi)(3-\xi)x-\xi(3+\xi)y}{x-\xi
y}\ln\frac{y}{x}-\frac{x-y}{x-\xi y}\xi^{2}\ln\xi\right].$

Assume

$m^{2}_{V_{1}}-m^{2}_{V_{2}}=\lambda_{3}v^{2}\ll m^{2}_{0}\equiv
m^{2}_{V}+\lambda_{1}v^{2}/2,$ (14)

where $m^{2}_{0}=(m^{2}_{V_{1}}+m^{2}_{V_{2}})/2$. Additionally, assume

$\xi\gg\mathrm{Max}(M_{1,2,3}/m_{0})^{2}.$ (15)

We find

$(M_{\nu})_{ab}\simeq-\frac{\lambda_{3}\xi}{32\pi^{2}}\frac{h_{ak}h_{bk}M_{k}v^{2}}{m^{2}_{0}}.$
(16)

The scale $m^{2}_{0}/M_{k}$ as in inverse seesaw is significantly reduced by
the loop factor ($1/16\pi^{2}$) and the small coupling $(\lambda_{3}\xi)$.
Indeed, suppose Max$(M_{1,2,3})\sim 10$ GeV and $\lambda_{3}\xi\sim 10^{-7}$,
slightly smaller than the electron Yukawa coupling.444$\lambda_{3}$ is
associated with the $B-L$ breaking scale that determines the matter parity
Dong:2013wca ; Dong:2017zxo , so it is proportional to $\lambda_{3}\sim
m_{0}/\Lambda_{B-L}\lesssim 10^{-8}$, given that $\Lambda_{B-L}\gtrsim
10^{11}$ GeV. The neutrino masses are recovered, i.e. $M_{\nu}\sim 0.1$ eV,
requiring only $h_{ak}h_{bk}\sim 1$ and $v/m_{0}\sim 1/3$, appropriate to the
following dark matter density and muon $g-2$. Further, the charged leptons
considered are flavor diagonal, while $\nu_{a}$ are related to the mass
eigenstates $\nu_{i}$ by the PMNS matrix, $\nu_{a}=U_{ai}\nu_{i}$.

##  IV Dark matter

The dark matter candidate is the lightest field of either the fermion singlets
$N_{1,2,3}$ or the neutral vectors $V_{1},V_{2}$, stabilized by the accidental
symmetry $Z_{2}$. The dark matter phenomenology of the vector candidate was
discussed in Saez:2018off  in a model without $N_{1,2,3}$. It is noted that
the fermion singlets $N_{1,2,3}$ may govern the vector dark matter observables
through the interaction in (6). This work does not consider such case, instead
we interpret the lightest fermion singlet (assumed $N_{1}$, thus
$M_{1}<M_{2,3},m_{V_{1,2}},m_{V^{-}}$) as dark matter. Its annihilation is
given in the diagram in Fig. 2.

Figure 2:  Dark matter annihilation to normal matter.

The total annihilation cross-section, $N_{1}N^{c}_{1}\rightarrow
l^{\prime}l^{c}(\nu_{l^{\prime}}\nu^{c}_{l})$, is evaluated as

$\langle\sigma
v\rangle\simeq\frac{\sum_{l,l^{\prime}}|h^{*}_{l1}h_{l^{\prime}1}|^{2}}{8\pi
M^{-2}_{1}}\left(1+\frac{8}{x_{F}}\right)\left(\frac{1}{m^{4}_{V^{-}}}+\frac{1}{m^{4}_{0}}\right),$
(17)

where $M_{1}$ is the $N_{1}$ mass, $h_{l1}$ and $h_{l^{\prime}1}$ for
$l,l^{\prime}=e,\mu,\tau$ are the $h$ couplings in (6) that equally couple
$N_{1}$ to a pair of charged leptons $(ll^{\prime})$ or neutrino flavors
$(\nu_{l}\nu_{l^{\prime}})$, and $x_{F}=M_{1}/T_{F}\sim 20$–25 at the freeze-
out temperature.

Assuming $m_{V^{-}}\simeq m_{0}$, it leads to

$\displaystyle\langle\sigma v\rangle$ $\displaystyle\simeq$ $\displaystyle 1\
\mathrm{pb}\left(\frac{\sum_{l}|h_{l1}|^{2}}{4\pi}\right)\left(\frac{\sum_{l^{\prime}}|h_{l^{\prime}1}|^{2}}{4\pi}\right)$
(18)
$\displaystyle\times\left(\frac{88M_{1}}{m_{0}}\right)^{2}\left(\frac{800\
\mathrm{GeV}}{m_{0}}\right)^{2},$

where the benchmark value of $m_{0}$ is given in the following section. The
dark matter gets the correct density if the couplings
$\sum_{l}|h_{l1}|^{2}/4\pi$ are proportional to unity and the dark matter mass
$M_{1}\sim m_{0}/88\sim 9$ GeV.

##  V Muon $g-2$

In the standard model, the anomalous magnetic moment of muon,
$a_{\mu}=(g-2)_{\mu}/2$, receives the radiative contributions from the
electromagnetic, hadronic, and electroweak parts, now established
Aoyama:2020ynm ,

$a_{\mu}(\mathrm{SM})=116591810(43)\times 10^{-11}.$ (19)

The recent measurement of $a_{\mu}$ presents an exciting hint for the new
physics beyond the standard model Abi:2021gix . The new result combined with
the previous E821 measurement Bennett:2006fi  yields a deviation from the
standard model prediction at 4.2$\sigma$:

$a_{\mu}(\mathrm{Exp})-a_{\mu}(\mathrm{SM})=(251\pm 59)\times 10^{-11}.$ (20)

If this deviation is further confirmed, it might rule out many new physics
models, due to the fact that the deviation is bigger than the electroweak
contributions of the standard model coming from $W,Z,H$ mediators, i.e.
$a_{\mu}(\mathrm{EW})=153.6(1.0)\times 10^{-11}$. Indeed, the new physics
constrained is potentially in tension with the electroweak precision test and
the other collider bounds.

Figure 3:  Doublet vector contribution to the muon $g-2$.

We suggest to solve this question by a diagram exchanged by the doublet vector
depicted in Fig. 3. Assuming $m_{\mu}\ll m_{V^{-}},M_{k}$, one obtains

$\displaystyle\Delta a_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi^{2}}\frac{m^{2}_{\mu}}{m^{2}_{V^{-}}}\sum_{k}|h_{\mu
k}|^{2}$ (21) $\displaystyle\times$
$\displaystyle\int^{1}_{0}dxx\frac{x(1+x)m^{2}_{V^{-}}+(1-x)(1-\frac{x}{2})M^{2}_{k}}{xm^{2}_{V^{-}}+(1-x)M^{2}_{k}}.$

The integral is proportional to 1 for $m_{V^{-}}\gg M_{k}$. Hence,

$\Delta a_{\mu}\simeq 2.5\times 10^{-9}\left(\frac{800\
\mathrm{GeV}}{m_{V^{-}}}\right)^{2}\left(\frac{\sum_{k}|h_{\mu
k}|^{2}}{4\pi}\right),$ (22)

comparable to the muon $g-2$ in (20), given that $m_{V^{-}}\simeq 800$ GeV and
$\sum_{k}|h_{\mu k}|^{2}/4\pi\simeq 1$ as the unitarity bound.

##  VI Summary and remark

Unlike the photon and other gauge fields, massive vector fields are always
quantized properly with three physical degrees of freedom due to automatic
Lorentz condition and a propagator analogous to that in a gauge theory with
unitary gauge. The standard model can contain such a vector doublet with a
coupling to the electroweak gauge bosons, $\frac{1}{\xi}|D^{\mu}V_{\mu}|^{2}$,
the Higgs field, $\lambda_{3}(\phi V_{\mu})^{2}$, and the three singlet
variants of light neutrinos, $h\bar{\psi}_{L}\gamma^{\mu}N_{L}V_{\mu}$, but
not to ordinary fermions. Here $V_{\mu}$ and $N_{L}$ are dark, ensured by the
$Z_{2}$ or $P$ matter parity.

The scotogenic producing small neutrino mass is recognized with the exchange
of the dark neutral vector $V^{0}$ and the dark fermion $N$. Whereas, a
similar scotogenic producing the observed muon $g-2$ is realized with the
mediator of the dark charged vector $V^{\pm}$ and the dark fermion $N$, given
that $m_{V}\sim 800$ GeV. The lightest fermion singlet $N_{1}$ properly
contributes to the dark matter abundance if its mass is $M_{1}\sim 9$ GeV.

The fermion dark matter does not interact with nucleons in the direct
detection experiments, while it can scatter with electrons in the XENON1T
experiment Aprile:2020tmw , but gives a tiny recoil energy $2m_{e}v^{2}\sim 1$
eV as well as small signal strength as suppressed by $m^{2}_{e}/m^{4}_{0}$.
The predicted vector doublet mass $m_{V}\sim 800$ GeV would satisfy all the
high energy collider bounds, given that the $B-L$ conserving couplings in (2)
are perturbative, as shown in Saez:2018off . So we do not refer to such
constraints here.

On the other hand, a mono-photon event may be created at the LEPII recoiled
against the missing energy carried by a pair of dark matter $N_{1}N_{1}$,
governed by the effective interaction,

$\frac{|h_{e1}|^{2}}{m^{2}_{V}}(\bar{e}_{L}\gamma^{\mu}N_{1L})(\bar{N}_{1L}\gamma_{\mu}e_{L}).$
(23)

Using the Fierz identities, we can translate this interaction to the vector
and axial vector operators studied in Ref. Fox:2011fx , implying a bound on
$m_{V}>(450$–$500)\sqrt{\pi}=797$–886 GeV, for $|h_{e1}|^{2}/4\pi\sim 1$,
which is in good agreement with the relic density and muon $g-2$ bound in this
model.

A crucial difference between our approach and potential $U(1)$ gauge
extensions is that the vector doublet by its non-Hermitian nature behaves as a
charged current linking distinct flavors, even in a gauge completion, in
similarity to the $W$ boson. Whereas, the Abelian gauge fields often conserve
flavors, hence there are a few parameters in such theories, such as the gauge
boson mass, kinetic mixing, and coupling constant. As a result, a portion of
them may encounter tensions in predicting all the mentioned experiments,
although they are very predictive Bauer:2018onh .

The UV-completion is related to the search for a gauge symmetry that contains
the doublet $V_{\mu}\sim 2$ in its adjoint gauge representation. The smallest
of which is the 3-3-1-1 gauge symmetry Dong:2013wca , because the $SU(3)_{L}$
adjoint is decomposed as $8=3\oplus 2\oplus 2^{*}\oplus 1$ under $SU(2)_{L}$,
which includes $V_{\mu}$ and $V^{*}_{\mu}$, as mentioned. In this case,
$SU(3)_{L}$ is broken by a new Higgs boson unified with a Goldstone boson
doublet of $V_{\mu}$ in a multiplet. Additionally, the fermion singlets are
unified with the usual lepton doublets in $SU(3)_{L}$ triplets, such as
$(\nu_{aL}\ e_{aL}\ N_{aL})^{T}$. $N_{aL}$ may have respective right-handed
partners, $N_{aR}$, and all the fields, $N$’s and $V$’s, are odd under the
matter parity as a residual 3-3-1-1 symmetry Dong:2015yra . All the couplings
in Eq. (2) are restricted by the gauge symmetry and the model is now
predictive. Particularly, the $\lambda_{3}$ coupling is induced by the gauge
interaction of a scalar multiplet, e.g. an octet, that contains a Higgs
doublet. Although $h_{ab}$ is restricted to the gauge coupling, the flavor
change happens due to a mixing of $N_{aL}$, which shifts
$h_{ak}\rightarrow(g/\sqrt{2})V_{ak}$ relating $N_{aL}$ to mass eigenstates
$N_{k}$. Here note that the $N_{aL}$ mass can be induced by a seesaw, with the
Dirac mass connecting $N_{L}$ to $N_{R}$ given by a $SU(3)_{L}$ breaking,
while $N_{R}$ has a large Majorana mass by itself. The $1/\xi$ coupling
becomes a gauge fixing canceled out by the contribution of the Goldstone boson
modes of $V_{\mu}$ in the $R_{\xi}$ gauge for relevant physical processes. It
is noteworthy that in this gauge completion, the inert scalar doublet in the
usual scotogenic sense becomes the Goldstone modes of $V_{\mu}$, hence the
vector doublet governs the neutrino mass and other observables instead. All
this is out of the scope of the work, to be published elsewhere.

## Acknowledgments

This research is funded by Vietnam National Foundation for Science and
Technology Development (NAFOSTED) under grant number 103.01-2019.353.

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