# TCGF: A unified tensorized consensus graph framework for multi-view
representation learning

Xiangzhu Meng  Wei Wei∗ Qiang Liu  IEEE  Member  Shu Wu  IEEE  Senior Member
Liang Wang  IEEE  Fellow Xiangzhu Meng and Wei Wei are corresponding authors
(denoted by $\ast$)X. Meng is with the Center for Research on Intelligent
Perception and Computing, Institute of Automation, Chinese Academy of
Sciences, Beijing, China (xiangzhu.meng@cripac.ia.ac.cn)W. Wei is with Center
for Energy, Environment & Economy Research, School of Management, Zhengzhou
University (weiwei123@zzu.edu.cn)Q. Liu, W. Shu and L. Wang are with Center
for Research on Intelligent Perception and Computing Institute of Automation,
Chinese Academy of Sciences, and School of Artificial Intelligence, University
of Chinese Academy of Sciences (qiang.liu, shu.wu<EMAIL_ADDRESS>
###### Abstract

Multi-view learning techniques have recently gained significant attention in
the machine learning domain for their ability to leverage consistency and
complementary information across multiple views. However, there remains a lack
of sufficient research on generalized multi-view frameworks that unify
existing works into a scalable and robust learning framework, as most current
works focus on specific styles of multi-view models. Additionally, most multi-
view learning works rely heavily on specific-scale scenarios and fail to
effectively comprehend multiple scales holistically. These limitations hinder
the effective fusion of essential information from multiple views, resulting
in poor generalization. To address these limitations, this paper proposes a
universal multi-view representation learning framework named Tensorized
Consensus Graph Framework (TCGF). Specifically, it first provides a unified
framework for existing multi-view works to exploit the representations for
individual view, which aims to be suitable for arbitrary assumptions and
different-scales datasets. Then, stacks them into a tensor under alignment
basics as a high-order representation, allowing for the smooth propagation of
consistency and complementary information across all views. Moreover, TCGF
proposes learning a consensus embedding shared by adaptively collaborating all
views to uncover the essential structure of the multi-view data, which
utilizes view-consensus grouping effect to regularize the view-consensus
representation. To further facilitate related research, we provide a specific
implementation of TCGF for large-scale datasets, which can be efficiently
solved by applying the alternating optimization strategy. Experimental results
conducted on seven different-scales datasets indicate the superiority of the
proposed TCGF against existing state-of-the-art multi-view learning methods.

###### Index Terms:

Multi-view learning, Unified framework, Consensus graph, Low-rank tensor
representation, Large-scale datasets, Iterative alternating strategy

##  I Introduction

With the rapid development of the big data era, more and more data can be
obtained from different domains or described from various perspectives, which
have gained extensive attention from researchers in recent years. For
examples, the document could be translated as different versions via various
languages [1]; an image could be represented by different visual descriptors
[2, 3] to reveal its color, texture, and shape information. Differing from
single view, multi-view data are complementary to each other, which motivates
the development of multi-view learning [4]. Considering the diversity of
multiple views, it is essential for multi-view learning to exploit how to
properly uncover rich information across views for improving practical
performance.

Multi-view learning methods have been extensively investigated in various
applications, including classification [5, 6], clustering [7, 8], and
reidentification [9, 10]. This paper primarily focuses on the background of
unsupervised multi-view learning without label information, such as clustering
tasks. Graph-based multi-view methods [11, 12, 13, 8, 14, 15, 16, 17] are
prevalent since graphs can effectively represent data structures. The most
representative group of graph-based multi-view methods aim to fuse multiple
graphs into a common latent space shared by all views. For example, Multiple
Kernel Learning (MKL) [11, 14] proposes a natural way to integrate different
views by directly combining different views for learning a common
representation. Unlike MKL, parameter-free multi-view learning methods [12]
provide a self-weighting strategy to fuse multiple graph information without
additional parameters. Furthermore, learning a shared graph among all views is
an efficient way to integrate the diverse information within multi-view data,
e.g., Graph-based Multi-view Clustering (GMC) [8] and Multiview Latent
Proximity Learning (MLPL) [16]. To handle large-scale multi-view datasets,
bipartite graph-based fast methods [18, 19] have been proposed to obtain the
consensus bipartite graph by linearly combining the bipartite graphs, which
are adaptively learned from the corresponding views. Another group of
representative multi-view learning methods is self-representation based multi-
view learning, which learns a self-representation matrix to act as the
affinity, such as Low-Rank Representation (LRR) [20, 21], Sparse Subspace
Clustering (SSC) [22, 23], etc. Motivated by LRR and SSC, multi-view works
[24, 25, 26, 27] are further developed to learn affinity matrices based on the
self-representation property. For example, MDcR [24] utilizes the
complementarity information of multiple views based on the Hilbert Schmidt
Independence Criterion (HSIC) as a regularization term to enhance the
correlations across views and explores the correlations within each view
jointly. The work [25] learns a shared affinity representation for multi-view
subspace clustering by simultaneously considering the diversity regularization
and a rank constraint.

Apart from the works on multi-view learning mentioned earlier, recent tensor-
based multi-view learning methods [26, 28, 29, 30, 31] have also played an
important role in exploring diversity and complementary information across
views. These methods attempt to capture high-order correlations among multiple
views in tensor space by applying tensor nuclear norms [32, 33, 34] and
tensor-Singular Value Decomposition (t-SVD) [35]. For example, T-SVD-MSC [26]
models self-representation matrices of different views as tensors, which
explores the consistency information of multiview data in tensor space. The
work [28] aggregates multiple subspace representations into a third-order
tensor, imposing a low-rank constraint by combining the nuclear norms of all
matrices unfolded along each view. Furthermore, LTBPL [30] utilized weighted
tensor nuclear norm to recover the comprehensiveness in the low-rank
constrained tensor stacked by multiple low-rank probability affinity matrices,
and links consensus indicator graphs with view-specific representations
carrying different adaptive confidences. The work [29] incorporates feature
space based missing-view inferring with low-rank tensor constraint to recover
the missing views and explore the full information of the recovered views and
available views. IMVTSC-MVI [31] utilizes tensorial modeling to capture both
the pairwise correlations between samples and the higher-order correlations
between features, resulting in a more accurate and robust multi-view
clustering method.

###  I-A Motivations

Despite the significant progress made by current multi-view learning methods
in practical applications, there are still several limitations that require
further attention. One such limitation is the heavy dependence of some methods
on pre-defined adjacency matrices, which hinders their flexibility and limits
their applicability to various multi-view scenarios. For example, many graph-
based methods rely heavily on predefined similarity matrices of different
views. However, due to the complexity and unknown prior information about the
geometric structure of views, it remains an open problem to manually construct
a suitable similarity matrix, often resulting in suboptimal performance.
Meanwhile, we also observe that most multi-view learning works rely heavily on
specific-scale scenarios, which prevents them from comprehensively
understanding different-scales multi-view scenarios. For instance, bipartite
graph-based multi-view methods are often used to efficiently process large-
scale multi-view datasets, but their performance is not as promising when
applied to normal-scale scenarios. Similarly, graph-based or tensor-based
multi-view learning methods also encounter difficulties in dealing with large-
scale datasets, owing to their computational complexity of squared or cubic
with the data size.

Given the success of multi-view learning methods in achieving impressive
performance, it is evident that views fusion and tensor construction play
important and essential roles in promoting correlation and consistency among
multiple views. Therefore, how to propose a more flexible and robust multi-
view learning method based on these mentioned earlier works is essential yet
full of challenges problem. To tackle these challenges, this paper attempts to
propose a unified multi-view learning framework that can transform a wide
range of existing multi-view learning approaches into a unified formulation,
thereby enhancing its flexibility and suitability for various multi-view
scenarios.

###  I-B Contributions

To simultaneously address the above limitations, this paper proposes a novel
multi-view representation learning framework named Tensorized Consensus Graph
Framework (TCGF). It first provides a unified framework for existing multi-
view works to learn the view-specific basics and representations, and then
stacks representations of all views into a tensor as a high-order
representation. Then, TCGF utilizes weighted tensor singular value
decomposition (t-SVD) based tensor rank minimization to exploit the
complementary information among different views. Different existing tensor-
based muli-view learning works, it’s scalable to construct the weighted t-SVD
operator based on the axis for basics or instances, enhancing its potential
ability to uncover complementary information. Moreover, the view-consensus
grouping effect is formulated between the view-specific representations and
consensus representation to capture the consensus information across different
views, which enforces the multi-view smooth regularization on shared space.
Consequently, TCGF not only leverages most existing embedding works into a
unified formulation but simultaneously considers the diversity, complementary
and consensus information across multiple views. Notably, we additionally
observe that it’s feasible to select the appropriate embedding manner as well
as its basics to handle different-scale multi-view datasets. To
comprehensively validate the effectiveness of TCGF, we conduct massive
experiments on seven multi-view datasets. Experimental results demonstrate
that the proposed TCGF can outperform the current state-of-the-art multi-view
learning methods in most situations. The major contributions in this paper can
be summarized as follows:

  * •

We propose a novel multi-view representation learning framework named TCGF to
learn the shared embedding, which serves as a unified framework for existing
multi-view works.

  * •

The axis-free weighted t-SVD operator is designed to exploit the complementary
information among different views, which further extends the applications of
tensor rank minimization.

  * •

We propose the view-consensus grouping effect to regularize the view-consensus
representation, which enables the discovery of essential structure information
within the multi-view data.

  * •

The experimental results conducted on seven different-scale datasets
demonstrate that TCGF is not only capable of maintaining or surpassing the
performance of other state-of-the-art multi-view methods, but also adaptable
to datasets with different scales.

##  II Related work

Existing multi-view methods can be divided into two categories according to
the means of calculating the affinity representation, i.e., graph-based and
self-representation-based multi-view models. Besides, tensor-based multi-view
methods also play an important role in exploring complemenary information
across views.

###  II-A Graph-based Multi-view Learning

The graph-based learning framework involves learning an affinity matrix
$\bm{S}^{v}$ that encodes the similarity between different samples in the
$v$th view. This affinity matrix $\bm{S}^{v}$ is learned by minimizing the
distance between samples in the latent space, which can be formulated as

$\mathop{\min}\limits_{{\bm{S}^{v}}\in\mathbf{\bm{C}}}\sum\limits_{i=1}^{N}{\sum\limits_{j=1}^{N}}{d(\bm{x}^{v}_{i},\bm{x}^{v}_{j})\bm{S}_{i,j}^{v}}+\lambda\bm{\Omega}(\bm{S}^{v}),$
(1)

where $d(\bm{x}^{v}_{i},\bm{x}^{v}_{j})$ denotes the distance between two
samples, which can be calculated by $L_{1}$-norm, Euclidean distance,
Mahalanobis distance, etc. $\mathbf{\bm{C}}$ and $\bm{\Omega}(\bm{S}^{v})$
stands by the constraint and normalization terms on the affinity matrix
$\bm{S}^{v}$, respectively. Based on the view-specific affinity matrix
$\bm{S}^{v}$, graph-based multi-view methods aim to learn an intrinsic
representation, capturing both consistent and complementary information among
multiple views. The most representative group of multi-view methods [36, 11,
12, 13, 8, 37, 14, 15, 16] aim to fuse multiple features or graphs into one
common latent space shared by all views. Multiple Kernel Learning (MKL) [11,
14] is also a natural way to integrate different views based on the direct
combination of different views and learns a common low-dimensional
representation. Different from MKL, parameter-free multi-view learning methods
[12] provide a self-weighting strategy to fuse multiple graph information
without additional parameters. Besides, learning a shared graph among all
views is also an efficient manner to integrate the diversity information
within multi-view data, e.g. Graph-based Multi-view Clustering (GMC) [8] and
Multiview Latent Proximity Learning (MLPL) [16]. Due to the squared
computational complexity of graph-based works, these methods might be
inefficient in dealing with large-scale multi-view datasets. For this reason,
bipartite graph-based multi-view methods [18, 19, 38] have aroused widespread
research interest to reduce both the computational complexity and storage
complexity, where the bipartite graph can well present the relationship
between $N$ samples and $K$ ($K\ll N$) anchors. It is worth noting that the
performance of the aforementioned graph-based methods heavily depends on the
quality of the predefined view-specific affinity matrix $\bm{S}^{v}$. However,
it is still an open problem to manually construct a suitable similarity matrix
for each view due to the complex and unknown geometric structure of multi-view
data, which limits their applicability.

###  II-B Self-representation based Multi-view Learning

Another graph-based ones are using the so-called self-expressiveness property,
which are developed to learn affinity matrices based on the self-
representation property. Specifically, the self-representation method is an
important subspace learning technology based on generating a large number of
representative samples, which can be formulated as

$\mathop{\min}\limits_{{\bm{S}^{v}}\in\mathbf{\bm{C}}}{\left\|\bm{X}^{v}-\bm{X}^{v}\bm{S}^{v}\right\|}_{2}+\lambda\bm{\Omega}(\bm{S}^{v}),$
(2)

where the normalization term $\bm{\Omega}(\bm{S}^{v})$ on the affinity matrix
$\bm{S}^{v}$ is the core component. For example, the work [20] adopts the low-
rank constraint as normalization term $\bm{\Omega}(\bm{S}^{v})$, which aims to
approximately recover the row space with theoretical guarantees to remove
arbitrary sparse errors. Motivated by LRR [20, 21] and SSC [22, 23], multi-
view works [24, 25, 26] are further developed to learn affinity matrices based
on the self-representation property. For example, MDcR [24] utilizes the
complementarity information of multiple views based on the Hilbert Schmidt
Independence Criterion (HSIC) as a regularization term to enhance the
correlations across views and explores the correlations within each view
jointly. The work [25] learns a shared affinity representation for multi-view
subspace clustering by simultaneously considering the diversity regularization
and a rank constraint. Even though the above self-representation multi-view
works obtain impressive performance and efficiency, there still exist the
following limitation in most works. Most of them aim to study a common
representation or the pairwise correlations between views, leading to the loss
of comprehensiveness and deeper higher-order correlations among multi-view
data, and hence miss important underlying semantic information.

Figure 1: Flowchart of the proposed TCGF. Given a collection of samples with
$M$ views, e.g., $\\{\bm{X}^{(1)},\bm{X}^{(2)},\ldots,\bm{X}^{(M)}\\}$. TCGF
first explores the view-specific graph $\bm{G}^{v}$ by removing the
disturbance of the noise error $\bm{E}^{(v)}$ in the initialized graph
$\bm{S}^{(v)}$. After stacking view-specific graphs
$\\{\bm{G}^{(1)},\bm{G}^{(2)},\ldots,\bm{G}^{(M)}\\}$ into one tensor
$\bm{\mathcal{G}}$, the tensor $\bm{\mathcal{G}}$ can be updated by utilizing
the t-SVD based weighting tensor multi-rank minimization. Based on the graph
agreement term, the consensus graph $\bm{G}_{F}$ can be obtained by fusing the
view-specific graphs $\\{\bm{G}^{(1)},\bm{G}^{(2)},\ldots,\bm{G}^{(M)}\\}$
with adaptively allocated weights $\bm{\alpha}$. Taking the view-specific
graph learning, tensorized graph learning and consensus graph learning into
one whole framework, we can obtain the final latent embedding $\bm{F}$ shared
by all views.

###  II-C Tensor-based Multi-view learning

To capture the high-order correlations among multiple views, tensor-based
multi-view learning methods [26, 28, 29, 30, 31] have been also developed in
recent years, which play an important role in effectively exploring the
comprehensive information among multiple views. The core idea of tensor-based
multi-view learning methods is to model high-order correlations across views
in tensor space by the low-rank constraint based on different tensor nuclear
norms [32, 33, 34]. For example, T-SVD-MSC [26] provides a tensor low-rank
constraint on the stacked subspace representation matrices to capture the
high-order complementary information among multiple views via introducing the
tensor singular value decomposition. The work [28] aggregates multiple
subspace representations into a third-order tensor, imposing a low-rank
constraint by combining the nuclear norms of all matrices unfolded along each
view. The work [29] studied the t-SVD based weighted tensor nuclear norm
minimization to shrink different matrix singular values with the corresponding
confidence. Moreover, TMvC[31] proposes the high-order graph to uncover the
essential information stored in multiple views, which adopts the low-rank
constraint of horizontal and vertical directions to better uncover the inter-
view and inter-class correlations between multi-view data. The work [29]
incorporates feature space-based missing-view inferring with low-rank tensor
constraint to recover the missing views and explore the full information of
the recovered views and available views. IMVTSC-MVI [31] utilizes tensorial
modeling to capture both the pairwise correlations between samples and the
higher-order correlations between features, resulting in a more accurate and
robust multi-view clustering method. Although these tensor-based approaches
have achieved promising effects, most of these methods mainly focus on graph-
based settings, resulting in cubic computational complexity. Meanwhile, they
usually fail to simultaneously consider the inter-view and intra-view
relationships among samples.

##  III The Proposed Method

In this section, we first introduce the main notations and definitions used in
this paper. Then, we provide the construction process of the proposed TCGF in
detail. Correspondingly, we provide a typical implement for our proposed
framework for large-scale datasets. For clarity, the flow chart of TCGF is
shown in Fig. 1.

###  III-A Notations and Problem Definition

We use bold calligraphy letters for tensors, e.g.,
$\bm{\bm{\mathcal{A}}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$, bold
upper case letters for matrices, e.g., $\bm{A}\in\mathbb{R}^{n_{1}\times
n_{2}}$. $\bm{A}(i,j)$ denotes the $(i,j)$th entry of the matrix $\bm{A}$.
$\bm{A}(i,:)$ denotes the $i$th row of the matrix $\bm{A}$.
$\bm{\mathcal{A}}^{T}\in\mathbb{R}^{n_{2}\times n_{1}\times n_{3}}$ denotes
the transpose tensor of tensor $\bm{\mathcal{A}}$. The fast Fourier
transformation (FFT) along the third axis of a tensor $\bm{\mathcal{A}}$ and
its inverse operation are
$\bm{\mathcal{A}}_{f}=fft(\bm{\mathcal{A}},\left[\right],3)$ and
$\bm{\mathcal{A}}=ifft(\bm{\mathcal{A}}_{f},\left[\right],3)$, respectively.
The block vectorizing and its inverse operation of $\bm{\mathcal{A}}$ are
$bvec(\bm{\mathcal{A}})=[\bm{\mathcal{A}}^{(1)},\bm{\mathcal{A}}^{(2)},\cdots,\bm{\mathcal{A}}^{(n_{3})}]\in\mathbb{R}^{n_{1}n_{2}\times
n_{3}}$ and $fold(bvec(\bm{\mathcal{A}}))=\bm{\mathcal{A}}$.
$bcirc(\bm{\mathcal{A}})\in\mathbb{R}^{n_{1}n_{2}\times n_{1}n_{3}}$ denotes
block circulant matrix of tensor $\bm{\mathcal{A}}$. The $L_{1}$ norm of
matrix $\bm{A}$ is denoted as $\left\|\bm{A}\right\|_{1}$. $tr(\bm{A})$
denotes the trace of matrix $\bm{A}$. $\bm{\bm{\mathcal{I}}}$ denotes the
$n_{1}\times n_{2}\times n_{3}$ identify tensor. $\bm{I}$ denotes the identity
matrix. $\bm{1}$ denotes a vector whose elements are equal to 1.

The tensor singular value decomposition operation (t-SVD) and tensor nuclear
norm are defined as follows.

Definition 1 (t-product): For two tensors
$\bm{\mathcal{A}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ and
$\bm{\mathcal{A}}\in\mathbb{R}^{n_{2}\times n_{4}\times n_{3}}$, the t-product
$\bm{\mathcal{A}}\ast\bm{\mathcal{B}}=fold(bcirc(\bm{\mathcal{A}})bvec(\bm{\mathcal{B}}))$
is the $n_{1}\times n_{4}\times n_{3}$ tensor.

Definition 2 (t-SVD): The t-SVD of tensor
$\bm{\mathcal{A}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ is defined as
$\bm{\mathcal{A}}=\bm{\mathcal{U}}\ast\bm{\mathcal{S}}\ast\bm{\mathcal{V}}^{T}$,
where $\bm{\mathcal{U}}\in\mathbb{R}^{n_{1}\times n_{1}\times n_{3}}$ and
$\bm{\mathcal{V}}\in\mathbb{R}^{n_{2}\times n_{2}\times n_{3}}$ are two
orthogonal tensors, i.e.,
$\bm{\mathcal{U}}\ast\bm{\mathcal{U}}^{T}=\bm{\mathcal{U}}^{T}\ast\bm{\mathcal{U}}=\bm{\mathcal{I}}$
and
$\bm{\mathcal{V}}\ast\bm{\mathcal{V}}^{T}=\bm{\mathcal{V}}^{T}\ast\bm{\mathcal{V}}=\bm{\mathcal{I}}$.
$\bm{\mathcal{S}}\in\mathbb{R}^{n_{1}\times n_{2}\times n_{3}}$ is a
f-diagonal tensor, whose entire frontal slices are diagonal matrices.

Definition 3 (Weighted t-SVD based tensor nuclear norm): For a tensor
$\bm{\mathcal{A}}$, the weighted t-SVD based tensor nuclear norm
$\bm{\mathcal{A}}_{\bm{\omega},\ast}$ is defined as by the weighted sum of
singular values of all the frontal slices of $\bm{\mathcal{A}}_{f}$, i.e.,
$\bm{\mathcal{A}}_{\bm{\omega},\ast}=\sum_{i=1}^{min\\{n_{1},n_{2}\\}}\sum_{j=1}^{n_{3}}\bm{\omega}_{i}|\bm{\mathcal{S}}_{f}^{(j)}(i,i)|$.
$\bm{\omega}$ is the weighted coefficients.

###  III-B Model Formulation

####  III-B1 View-specific Graph Learning

Given a multi-view dataset consisting of $M$ views, the data in the $v$th view
($1\leq v\leq M$) can be denoted as
$\bm{X}^{(v)}=\\{\bm{x}_{1}^{(v)},\bm{x}_{2}^{(v)},\ldots,\bm{x}_{N}^{(v)}\\}$,
in which $N$ is the number of samples. $\bm{S}^{(v)}\in\mathbb{R}^{N\times N}$
denotes the initialized graph in the $v$th view, which reflect the
relationships among samples. For the construction of specific-view graph
$\bm{S}^{(v)}$, there are different manners to be flexibility adopted, such as
similarity and self-representation graphs. We use $\bm{\Omega}(\bm{S}^{(v)})$
to generally represent the objective function that constructs the original
graph $\bm{S}^{(v)}$. Notably, there are usually noise and redundant
information in the original multi-view data, which might result in the errors
$\bm{E}^{(v)}$ for view-specific graph $\bm{S}^{(v)}$. To eliminate the
efforts of error, the basic model of graph learning can be formulated as
follows:

$\begin{split}&\mathop{\min}\limits_{\bm{G}^{(v)},\bm{E}^{(v)}}\sum_{v=1}^{M}{\bm{\Omega}(\bm{S}^{(v)})}+\lambda_{E}\sum_{v=1}^{M}{\left\|\bm{E}^{(v)}\right\|_{1}},\\\
&s.t.\bm{S}^{(v)}=\bm{G}^{(v)}+\bm{E}^{(v)},\bm{G}^{(v)}\geq
0,\bm{G}^{{(v)}^{T}}\bm{1}=\bm{1},\\\ \end{split}$ (3)

where the constraint $\bm{G}^{{(v)}^{T}}\bm{1}=\bm{1}$ guarantees that the sum
of graph weights between each sample and other all samples 1. $\lambda_{E}$ is
the trade-off parameter.

####  III-B2 Tensorized Graph Learning

Inspired by tensor nuclear norm, which well exploits complementary information
and spatial structure embedded in tensor, we utilize the weighted t-SVD based
tensor nuclear norm to investigate the high-order correlations among multiple
views. Thus, we stack the affinity matrices of all views into a tensor
$\bm{\mathcal{G}}\in\mathbb{R}^{N\times N\times M}$. To better investigate the
correlations and largely reduce the computational complexity, we further
rotate $\bm{\mathcal{G}}$ into a $N\times M\times N$ tensor, where
$\bm{\mathcal{G}}(:,v,:)=\bm{G}^{(v)}$. After rotation, the dimension of the
weighted coefficient $\bm{\omega}$ decreases from $N$ to $M$ ($M\ll N$), in
which finetuned parameters $\bm{\omega}$ correlates with the view number.
Considering the influence of noise on $\bm{\mathcal{G}}$, we learn a low-rank
tensor $\bm{\mathcal{Z}}$ to approximate $\bm{\mathcal{G}}$ as follows:

$\begin{split}&\mathop{\min}\limits_{\bm{\mathcal{Z}}}\quad\bm{\mathcal{Z}}_{\bm{\omega},\ast},\\\
&s.t.\quad\bm{\mathcal{Z}}=\bm{\mathcal{G}},\
\bm{\mathcal{G}}=\Phi\left(\bm{G}^{(1)},\cdots,\bm{G}^{(M)}\right),\\\
\end{split}$ (4)

where $\Phi(\cdot)$ merges graphs of multiple views into a tensor and then
rotates along the third axis.

####  III-B3 Consensus Embedding Learning

As multi-view features are extracted from the same objects, different graphs
$\\{\bm{G}^{(v)}\\}$ should contain some similar information. Thus, it’s
essential to capture both consistent and complementary information among
multiple views. For this reason, we propose the shared embedding
$\bm{F}\in\mathbb{R}^{N\times d}$ to uncover the rich information in each
view. To preserve the locality in the learned graph $\bm{G}^{(v)}$ in each
view, we exploit the subspace-wise grouping effect [39] in the learned graph
$\bm{G}^{(v)}$ by means of a unified framework.

Definition 4 (Subspace-wise Grouping Effect): Given a set of d-dimensional
data points
$\bm{X}=[\bm{x}_{1},\bm{x}_{2},\cdots,\bm{x}_{n}]\in\mathcal{R}^{d\times n}$,
a self-representation matrix
$\bm{Z}=[\bm{z}_{1},\bm{z}_{2},\cdots,\bm{z}_{n}]\in\mathcal{R}^{n\times n}$
has the grouping effect if $\left|\bm{x}_{i}-\bm{x}_{j}\right|^{2}\rightarrow
0$ then $\left|\bm{z}_{i}-\bm{z}_{j}\right|^{2}\rightarrow 0$.

According to the above definition, we can formulate the view-consensus term as
$tr\left(\bm{F}^{T}\bm{G}^{(v)}\bm{F}\right)$. To further improve its
flexibility, we propose the consensus graph based on shared embedding
$\bm{F}$. Meanwhile, considering different views have different contributions
in learning $\bm{G}_{\bm{F}}$, we adaptively assign weight
$\bm{\alpha}_{v}\leq 0$ for the $w$th view. The above considerations can be
formed as follows:

$\begin{split}&\mathop{\max}\limits_{\bm{F}}\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}tr\left(\bm{F}^{T}\bm{F}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1}\right)},\\\
&s.t.\quad\sum_{v=1}^{M}\bm{\alpha}^{(v)}=1,\bm{\alpha}^{(v)}\geq
0,\bm{F}^{T}\bm{F}=\bm{I},\\\ \end{split}$ (5)

where $\gamma$ is a hyper-parameter. $\bm{D}^{(v)}$ is a diagonal matrix whose
diagonal elements are $\bm{D}^{(v)}(i,i)=\sum_{i=j}^{N}\bm{D}^{(v)}(j,i)$.
$tr\left(\bm{G}_{\bm{F}}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1}\right)$ can be seen as
the graph agreement term to measure the consistence between the consensus
graph $\bm{G}_{\bm{F}}$ and the normalized graph
$\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1}$. Notably, the construction manner for graph
$\bm{G}_{\bm{F}}$ is also to be flexibly chosen, such as linear kernel
function $\bm{G}_{\bm{F}}=\bm{F}\bm{F}^{T}$.

####  III-B4 Overall Framework of TCGF

In the above subsections, we discuss how to learn the intrinsic, consistent
and complement information in the multi-view data. To this end, we seamlessly
couple the above learning processes, and the overall objective function is
formulated as follows:

$\begin{split}&\mathop{\min}_{\bm{F},\bm{\mathcal{Z}},\bm{S}^{(v)},\bm{G}^{(v)},\bm{E}^{(v)},\bm{\alpha}^{(v)}}\underbrace{\sum_{v=1}^{M}{\bm{\Omega}(\bm{S}^{(v)})}+\lambda_{E}\sum_{v=1}^{M}{\left\|\bm{E}^{(v)}\right\|_{1}}}_{View-
specific\ Term}\\\
&-\underbrace{\lambda_{C}\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}tr(\bm{G}_{\bm{F}}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1})}}_{Consensus\
Graph\
Learning}+\underbrace{\lambda_{R}\bm{\mathcal{Z}}_{\bm{\omega},\ast}}_{Tensorized\
Term},\\\ &s.t.\quad\bm{S}^{(v)}=\bm{G}^{(v)}+\bm{E}^{(v)},\bm{G}^{(v)}\geq
0,\bm{G}^{T}\bm{1}=\bm{1},\\\
&\qquad\sum_{v=1}^{M}\bm{\alpha}^{(v)}=1,\bm{\alpha}^{(v)}\geq
0,\bm{F}^{T}\bm{F}=\bm{I},\\\
&\qquad\bm{\mathcal{Z}}=\bm{\mathcal{G}},\bm{\mathcal{G}}=\Phi(\bm{G}^{(1)},\cdots,\bm{G}^{(M)}),\\\
\end{split}$ (6)

where $\lambda_{E}$ and $\lambda_{T}$ are trade-off parameters. Observed from
the model in Eq. (6), the shared embedding $\bm{F}$ and view-specific graph
$\bm{G}^{(v)}$ that is constrained by the low-rank tensor $\bm{\mathcal{G}}$
can be simultaneously learned in a unified framework. The first aspect
maintains consensus information among different views to obtain the shared
embedding, via fusing multiple graph agreement terms with different adaptive
weights. The second aspect is to eliminate the efforts of error, which can
learn more robust graph $\bm{G}^{(v)}$. The final aspect depicts the low-rank
property and higher-order correlations of affinity tensor $\bm{\mathcal{G}}$
to exploit complementary information among views.

###  III-C Specific Implement for Large-scale Datasets

Due to the computational complexity of the aforementioned methods being
squared or cubic with the data size, thus they are inefficient in handling
large-scale datasets. To solve this issue, we attempt to provide a specific
implement to extend the proposed TCGF into the scenario of large-scale
datasets.

Inspired by bipartite graph, we attempt to reduce the scale of initialized
graph $\bm{S}^{(v)}$ by randomly selecting the subset of samples as anchors.
Then, we construct the bipartite graph $\bm{B}^{(v)}$ between samples and
anchors to substitute the whole graph $\bm{S}^{(v)}$. However, directly using
those anchors is difficult to cover the entire data point clouds of data and
characterize the intrinsic structure of data. To solve this issue, we propose
a novel anchors selection scheme based on the information volume, which is
simple and efficient. We combine all views into one view, and then execute the
SVD decomposition to obtain the singular values for all samples. In this way,
we select the $K$ most representative samples according to their singular
values. Besides, we also find that K-means is also utilized to find those
samples with high information volume in some situations. After that, we can
construct the bipartite graph $\bm{B}^{(v)}\in\mathcal{R}^{N\times K}$ to re-
initialize the view-specific graph. Moreover, to further control the
computational cost, we generate similarity-induced graph to construct the
fixed bipartite graph $\bm{B}^{(v)}$. According to the above considerations,
we can readily extend the proposed TCGF into the scenario of large-scale
datasets, which can be formulated as follows:

$\begin{split}&\mathop{\min}_{\bm{\Theta}}-\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}tr(\bm{G}_{\bm{F}}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1})}+\lambda_{E}\sum_{v=1}^{M}{\left\|\bm{E}^{(v)}\right\|_{1}}\\\
&+\lambda_{R}\bm{\mathcal{Z}}_{\bm{\omega},\ast},\\\
&s.t.\quad\bm{B}^{(v)}=\bm{G}^{(v)}+\bm{E}^{(v)},\bm{G}^{(v)}\geq
0,\bm{G}^{{(v)}^{T}}\bm{1}=\bm{1},\\\
&\qquad\sum_{v=1}^{M}\bm{\alpha}^{(v)}=1,\bm{\alpha}^{(v)}\geq
0,[\bm{F}_{S};\bm{F}_{A}]^{T}[\bm{F}_{S};\bm{F}_{A}]=\bm{I},\\\
&\qquad\bm{\mathcal{Z}}=\bm{\mathcal{G}},\bm{\mathcal{G}}=\Phi(\bm{G}^{(1)},\cdots,\bm{G}^{(M)}),\\\
\end{split}$ (7)

where
$\bm{\Theta}=\\{\bm{F}_{S},\bm{F}_{A},\bm{\mathcal{Z}},\bm{G}^{(v)},\bm{E}^{(v)},\bm{\alpha}^{(v)}\\}$
denotes the set of solved variables. $\bm{F}_{S}\in\mathcal{R}^{N\times d}$
and $\bm{F}_{A}\in\mathcal{R}^{K\times d}$ represent the shared embedding of
samples and anchors, respectively.

####  III-C1 Optimization

Inspired by the augmented Lagrange multiplier method, the corresponding
augmented Lagrangian function of the Eq. (6) can be formulated as follows:

$\scriptsize\begin{split}&\bm{\mathcal{L}}(\bm{F}_{S},\bm{F}_{A},\bm{\mathcal{Z}},\bm{G}^{(v)},\bm{E}^{(v)},\bm{\alpha}^{(v)})=\\\
&-\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}tr(\bm{F}^{T}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1}\bm{F})}+\lambda_{E}\sum_{v=1}^{M}{\left\|\bm{E}^{(v)}\right\|_{1}}+\lambda_{R}\bm{\mathcal{Z}}_{\bm{\omega},\ast}\\\
&+\sum_{v=1}^{M}\left(\left\langle\bm{Y}^{(v)},\bm{S}^{(v)}-\bm{G}^{(v)}-\bm{E}^{(v)}\right\rangle+\frac{\mu}{2}\left\|\bm{S}^{(v)}-\bm{G}^{(v)}-\bm{E}^{(v)}\right\|_{F}^{2}\right)\\\
&+\left\langle\bm{\mathcal{Y}},\bm{\mathcal{G}}-\bm{\mathcal{Z}}\right\rangle+\frac{\rho}{2}\left\|\bm{\mathcal{G}}-\bm{\mathcal{Z}}\right\|_{F}^{2},\\\
\end{split}$ (8)

where $\bm{Y}^{(v)}$ and $\bm{\mathcal{Y}}$ represent Lagrange multipliers.
$\mu$ and $\rho$ are the penalty parameters. We adopt the Augmented Lagrangian
Multiplier (ALM) with the Alternative Direction Minimizing (ADM) optimization
algorithm for solving the above optimization problem, and the updating rules
of varying variables are as follows.

$\bullet$ Updating $\bm{F}_{S}$ and $\bm{F}_{A}$. For the convenient of
optimization, we employ linear kernel $\bm{F}_{S}\bm{F}_{A}^{T}$ to construct
the consensus graph $\bm{G}_{\bm{F}}$. By fixing the other variables,
$\bm{F}_{S}$ and $\bm{F}_{A}$ can be updated by solving the following problem:

$\begin{split}&\mathop{\max}\limits_{\bm{F}_{S},\bm{F}_{A}}\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}tr\left(\bm{G}_{\bm{F}}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1}\right)},\\\
&s.t.\quad\bm{F}_{S}^{T}\bm{F}_{S}+\bm{F}_{A}^{T}\bm{F}_{A}=\bm{I}.\\\
\end{split}$ (9)

The Eq. (9) has the closed-form solutions
$\bm{F}_{S}=\frac{\sqrt{2}}{2}\bm{U}$ and
$\bm{F}_{A}=\frac{\sqrt{2}}{2}\bm{V}$, in which $\bm{U}$ and $\bm{V}$ are the
leading $d$ left and right singular vectors of the matrix
$\sum_{v=1}^{M}{(\bm{\alpha}^{(v)})}^{\gamma}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1}$.

$\bullet$ Updating $\bm{\mathcal{Z}}$. In this case, $\bm{\mathcal{Z}}$ can
updated by solving the following problem:

$\begin{split}&\mathop{\min}_{\bm{\mathcal{Z}}}\frac{1}{2}\left\|\bm{\mathcal{G}}+\frac{1}{\rho}\bm{\mathcal{Y}}-\bm{\mathcal{Z}}\right\|_{F}^{2}+\frac{\lambda_{R}}{\rho}\bm{\mathcal{Z}}_{\bm{\omega},\ast}.\\\
\end{split}$ (10)

The optimal solution of the Eq. (10) is
$\bm{\Gamma}_{\frac{\lambda_{R}}{\rho}}[\bm{\mathcal{G}}+\frac{1}{\rho}\bm{\mathcal{Y}}]$.
More details are placed in the Appendix A.

$\bullet$ Updating $\bm{G}^{(v)}$. In this case, $\bm{G}^{(v)}$ can updated by
solving the following problem:

$\begin{split}&\mathop{\min}_{\bm{G}^{(v)}}\left\langle\bm{Y}^{(v)},\bm{S}^{(v)}-\bm{G}^{(v)}-\bm{E}^{(v)}\right\rangle+\frac{\mu}{2}\left\|\bm{S}^{(v)}-\bm{G}^{(v)}-\bm{E}^{(v)}\right\|_{F}^{2}\\\
&\qquad+\left\langle\bm{\mathcal{Y}}^{(v)},\bm{{G}}^{(v)}-\bm{\mathcal{Z}}^{(v)}\right\rangle+\frac{\rho}{2}\left\|\bm{{G}}^{(v)}-\bm{\mathcal{Z}}^{(v)}\right\|_{F}^{2}\\\
&\qquad{-{(\bm{\alpha}^{(v)})}^{\gamma}tr(\bm{G}_{\bm{F}}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1})},\\\
&s.t.\quad\bm{G}^{(v)}\geq 0,\quad\bm{G}^{{(v)}^{T}}\bm{1}=\bm{1},\\\
\end{split}$ (11)

where $\bm{\mathcal{Z}}^{(v)}=\bm{\mathcal{Z}}(:,v,:)$ and
$\bm{\mathcal{Y}}^{(v)}=\bm{\mathcal{Y}}(:,v,:)$. It can be shown that the
above formula has a closed connection, and the corresponding proof is placed
in the Appendix B.

$\bullet$ Updating $\bm{E}^{(v)}$. In this case, $\bm{E}^{(v)}$ can updated by
solving the following problem:

$\begin{split}&\mathop{\min}\limits_{\bm{E}^{(v)}}\frac{1}{2}\left\|\bm{E}^{(v)}-\bm{\Gamma}^{(v)}\right\|_{F}^{2}+\frac{\lambda_{E}}{\mu}{\left\|\bm{E}^{(v)}\right\|_{1}},\\\
\end{split}$ (12)

where $\bm{\Gamma^{(v)}}=\bm{S}^{(v)}-\bm{G}^{(v)}-\frac{1}{\mu}\bm{Y}^{(v)}$.
Based on the proximal gradient-decent method, the optimal solution
$\bm{E}^{(v)}$ of Eq. (12) is
$max(|\bm{\Gamma^{(v)}}|-\frac{\lambda_{E}}{\mu},0)$.

$\bullet$ Updating $\bm{\alpha}^{(v)}$. In this case, $\bm{\alpha}^{(v)}$ can
be updated by solving the following problem:

$\begin{split}&\mathop{\max}\limits_{\bm{\alpha}}\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}tr\left(\bm{G}_{\bm{F}}\bm{G}^{(v)}{\bm{D}^{(v)}}^{-1}\right)},\\\
&s.t.\quad\sum_{v=1}^{M}\bm{\alpha}^{(v)}=1,\bm{\alpha}^{(v)}\geq 0.\\\
\end{split}$ (13)

Using the Lagrange multiplier method, we can obtain the closed-form solutions
of the Eq. (13). More details are placed in the Appendix C.

$\bullet$ Updating Lagrange multipliers and penalty parameters. Lagrange
multipliers and penalty parameters can be updated as follows:

$\begin{split}&\bm{Y}^{(v)}:=\bm{Y}^{(v)}+\mu(\bm{S}^{(v)}-\bm{G}^{(v)}-\bm{E}^{(v)}),\\\
&\bm{\mathcal{Y}}:=\bm{\mathcal{Y}}+\rho(\bm{\mathcal{G}}-\bm{\mathcal{Z}}),\\\
&\mu:=\eta\mu,\\\ &\rho:=\eta\rho,\\\ \end{split}$ (14)

where $\eta>1$ is used to boost the convergence speed [40].

####  III-C2 Time Complexity Analysis

In this part, the computational complexity analysis of solving the problem in
Eq. (8) is provided. The main computation consists of five parts, which are
corresponding to the updating process in the optimization section. The time
complexities in iteratively updating these variables are $\bm{O}(MNK+K^{2}N)$,
$\bm{O}(MNKlog(MN)+M^{2}NK)$, $\bm{O}(MNKd+MNKlog(K))$, $\bm{O}(MNK)$, and
$\bm{O}(MNK)$, respectively. Therefore, the main time complexity in each
iteration is $\bm{O}(N(MK+K^{2}+MKlog(MN)+M^{2}K+MKd+MKlog(K)+2MK))$. While
$K\ll N$, the main complexity of TCGF is linear to $N$.

####  III-C3 Convergence analysis

Since the model in Eq. (8) is not a joint convex problem of all variables, it
still remains an open problem to require a globally optimal solution.
Fortunately, by means of the alternating optimization algorithm, the proposed
model can be solved. Due to the convex property and optimal solution of each
sub-problem, the optimization can be shown to converge. In what follows, we
would introduce the convergence of each sub-problem.

For updating $\bm{F}_{S}$ and $\bm{F}_{A}$, sub-problem in the Eq. (9) is
equal to the following equation:

$\begin{split}&\mathop{\max}\limits_{\bm{F}\bm{F}^{T}=\bm{I}}\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}tr\left(\bm{F}\bm{L}^{(v)}\bm{F}^{T}\right)},\\\
\end{split}$ (15)

where $\bm{F}=[\bm{F}_{S};\bm{F}_{A}]\in\mathcal{R}^{(N+K)\times d}$ and
$\bm{L}^{(v)}=[\bm{0}\
(\bm{G}^{(v)}\bm{D}^{{(v)}^{-1}});{(\bm{G}^{(v)}\bm{D}^{{(v)}^{-1}})}^{T}\
\bm{0}]\in\mathcal{R}^{(N+K)\times(N+K)}$. Obviously, the Hessian matrix
$\sum_{v=1}^{M}{{(\bm{\alpha}^{(v)})}^{\gamma}\bm{L}^{(v)}}$ of the above
equation is positive semi-definite. Thus, sub-problem in the Eq. (9) is
strictly convex.

For updating $\bm{\mathcal{Z}}$,
$\bm{\mathcal{Z}}=\bm{\Gamma}_{\frac{\lambda_{R}}{\rho}}[\bm{\mathcal{G}}+\frac{1}{\rho}\bm{\mathcal{Y}}]$
is a closed-form solution, thus the sub-problem in the Eq. (10) is a convex
function.

For updating $\bm{G}^{(v)}$, the second order derivative of this function in
Eq. (11) with respect to $\bm{G}^{(v)}(i,:)$ is equal to 1, thus it’s easy to
check that the objective function of sub-problem in the eq. (11) is also a
convex function.

For updating $\bm{E}^{(v)}$, it’s readily showed that the objective value in
the Eq. (12) is monotonically decreased due to the convergence property of the
proximal gradient-decent method [41].

For updating $\bm{\alpha}^{(v)}$, the sub-problem in the Eq. (13) is a linear
convex function, and a closed-form solution can be assigned to
$\bm{\alpha}^{(v)}$.

##  IV Experiments and Analysis

In this section, we report the experimental results that have been conducted
to evaluate the performance of the proposed TCGF model using seven real-world
datasets. Additionally, we provide detailed analysis to illustrate the
effectiveness and robustness of the proposed TCGF.

(a) Caltech101 dataset.

(b) ALOI dataset.

(c) MSRC dataset.

(d) YoutubeFace dataset.

Figure 2: Some examples images in datasets.

###  IV-A Experiment Settings

####  IV-A1 Datasets

We evaluate our proposed framework on six benchmark datasets
MSRC111http://archive.ics.uci.edu/ml, NGs222http://lig-
membres.imag.fr/grimal/data.html.,
Hdigit333https://cs.nyu.edu/roweis/data.html,
Caltech101444https://data.caltech.edu/records/mzrjq-6wc02/files/caltech-101.zip,
ALOI_100555https://elki-project.github.io/datasets/multi_view,
ALOI_1K666https://elki-project.github.io/datasets/multi_view and
YoutubeFace777https://www.cs.tau.ac.il/wolf/ytfaces/. Ngs is a document
dataset, and the resting datasets are image datasets, in which some samples in
image datasets are shown in Fig. 2. The detailed information of these datasets
is summarized as follows:

  * 1.

MSRC contains 240 samples from 8 classes, where each category includes 30
images. Following the work [42], 7 classes containing Tree, building,
airplane, cow, face, car and bicycle, are selected. Five image descriptors
[43] are utilized to extract multi-view features, which are colour moment with
dimension 24, local binary pattern with dimension 256, HOG with dimension 576,
GIST with dimension 512 and CENTRIST with dimension 254.

  * 2.

NGs is the subset of the 20 Newsgroups data set that is a collection of
approximately 20,000 newsgroup documents, partitioned (nearly) evenly across
20 different newsgroups. It consists of 500 newsgroup documents. Each raw
document was pre-processed with three different methods (giving three views),
and was annotated with one of five topical labels.

  * 3.

Hdigit contains 10000 handwritten digit images from two sources, including
MNIST and USPS Handwritten Digits. These digit images belong to 10 subjects
(0-9). In our experiments, MNIST and USPS Handwritten Digits can be used as
two views, whose dimensions are 784 and 256, respectively.

  * 4.

Caltech101 contains 9144 images of 101 categories. Following related multi-
view clustering works, we select 20 categories with 2,386 images as dataset.
Six image descriptors can be employed to extract multi-view features for each
image, including GABOR with dimension 48, WM with dimension 48, CENT with
dimension 254, HOG with dimension 984, GIST with dimension 512 and local
binary pattern with dimension 256.

  * 5.

ALOI_1K contains 110250 images of 1000 small objects. For each image, we use
three image descriptors to extract its features as 3 views, which are RGB
color histograms with dimension 77, HSV/HSB color histograms with dimension
13, color affinity with dimension 64, Haralick feature with dimension 125.

  * 6.

ALOI_100 is a subset of ALOI_1K, in which We select 100 classes with three
views. For each category, 1080 images are selected, thus it contains 10800
images in our experiments.

  * 7.

YoutubeFace contains complicated backgrounds, dynamic postures, and various
obstacles, which is produced from YouTube with 101499 face images of 31
groups. Five image descriptors are utilized to extract their features as
multi-view features.

For the convenient of readers, the statistical characteristics of six used
multi-view datasets can be summarized in Table I.

TABLE I: The statistic information of eight real-world datasets. Dataset | Samples | Classes | Views | Dimensions
---|---|---|---|---
MSRC | 210 | 7 | 5 | 24$\backslash$256$\backslash$576$\backslash$512$\backslash$254
NGs | 500 | 20 | 3 | 2000$\backslash$2000$\backslash$2000
Caltech101 | 2386 | 6 | 20 | 48$\backslash$40$\backslash$254$\backslash$1984$\backslash$512$\backslash$928
Hdigit | 10000 | 10 | 2 | 784$\backslash$256
ALOI_100 | 10800 | 100 | 4 | 77$\backslash$13$\backslash$64$\backslash$125
ALOI_1K | 110250 | 1000 | 4 | 77$\backslash$13$\backslash$64$\backslash$125
YoutubeFace | 101499 | 31 | 5 | 64$\backslash$512$\backslash$64$\backslash$647$\backslash$838
TABLE II: Clustering performance comparisons in terms of ACC on seven real-world datasets. Datasets | SC | GNMF | Co-reg | MDcR | AMGL | GMC | GFSC | LMVSC | FPMVS | VCGA | LTBPL | TCGF
---|---|---|---|---|---|---|---|---|---|---|---|---
Small-scale multi-view datasets
MSRC | 0.776 | 0.148 | 0.814 | 0.468 | 0.738 | 0.748 | 0.742 | 0.343 | 0.786 | 0.834 | 0.705 | 1.000
NGs | 0.631 | 0.296 | 0.368 | 0.590 | 0.308 | 0.982 | 0.511 | 0.298 | 0.738 | 0.980 | 0.994 | 1.000
General-scale multi-view datasets
Hdigit | 0.526 | 0.935 | 0.816 | 0.866 | 0.901 | 0.998 | 0.895 | 0.854 | 0.910 | 0.918 | 0.999 | 1.000
Caltech101 | 0.442 | 0.439 | 0.412 | 0.465 | 0.557 | 0.456 | 0.702 | 0.384 | 0.655 | - | 0.694 | 0.789
ALOI_100 | 0.492 | 0.261 | 0.521 | 0.279 | 0.600 | 0.571 | 0.281 | 0.566 | 0.342 | - | 0.626 | 0.673
Large-scale multi-view datasets
ALOI_1K | 0.209 | 0.152 | OOM | OOM | OOM | OOM | OOM | 0.366 | 0.218 | OOM | OOM | 0.411
YoutubeFace | 0.139 | 0.092 | OOM | OOM | OOM | OOM | OOM | 0.148 | 0.241 | OOM | OOM | 0.267

####  IV-A2 Compared Methods

In order to show the excellent performance of the proposed TCGF, this paper
introduces nine state-of-the-art multi-view algorithms as comparing methods,
including Co-reg [44], MDcR [24], AMGL [12], GMC [8], GFSC [15], LMVSC [18],
FPMVS [19], VCGA [45] and LTBPL [30]. Moreover, two typical single-view
methods SC [46] and GNMF [47] are adopted to show the advantages of multi-view
clustering algorithms, which utilize the most informative view. The details of
these compared methods can be summarized as follows:

  * 1.

SC is a standard spectral clustering method for clustering each single view,
which is used for recognizing the different confidences of different views.

  * 2.

GNMF is a method that attempts to explore a matrix factorization to respect
the graph structure, and then consider the geometric structure in the data.

  * 3.

Co-reg is a multiview spectral clustering method proposed in work, which
regularizes different views to be close to each other.

  * 4.

MDcR is a multi-view dimensionality reduction method, which explores the
correlations of different views based on HSIC term.

  * 5.

AMGL is an auto-weighted multiple graph learning method, which could allocate
ideal weight for each view automatically.

  * 6.

GMC is a multi-view graph-based method to learn the common graph shared by all
views, and directly gives the final clusters.

  * 7.

GFSC is a multi-view spectral embedding based on multi-graph fusion to
approximate the original graph of individual view.

  * 8.] LMVSC first learn a smaller graph for each view, and then integrates those graphs to transform original multi-view problems into single-view scenario.

  * 9.

FPMVS jointly learns anchor selection and subspace graph construction into a
unified optimization formulation to promote clustering quality, which can
automatically learn an optimal anchor subspace graph without any extra hyper-
parameters.

  * 10.

VCGA first constructs the view-specific graphs and the shared graph from
original multi-view data and hidden latent representation, and then the view-
specific graphs of different views and the consensus graph are aligned into an
informative target graph.

  * 11.

LTBPL stacks multiple low-rank probability affinity matrices in a low-rank
constrained tensor to recover their comprehensiveness and higher-order
correlations, and links consensus indicator graph with view-specific
representation carrying different adaptive confidences.

For a fair comparison, we download the released code of comparison algorithms
from their original websites. Since all methods need to utilize k-means or
connected component algorithms to get the final clustering results, which can
be disturbed by the initialized states. Thus, we repeatedly run 10 times
clustering experiments to eliminate the randomness in initialization for all
compared methods and report the average performance. For the selection for
hyper-parameters, the details od the proposed TCGF are placed in the section
IV-D, and we tune these hyper-parameters following by corresponding papers for
compared methods.

####  IV-A3 Evaluation Metrics

Various metrics have been proposed from diverse perspectives to evaluate the
quality of the obtained data clusters. In general, larger metrics indicate
better clustering performances. To facilitate a comprehensive comparison,
three metrics [48] \- accuracy (ACC), normalized mutual information (NMI), and
purity (PUR) - that are commonly used in the clustering field.

###  IV-B Experimental Results and Analysis

Performance comparison of different methods. The highest performance is
highlighted in boldface. The best two scores are highlighted in bold. ’OOM’
means out of memory. From experimental results in Tables II-IV, we have the
following observations.

  * •

The proposed TCGF achieves the best performance in terms of three metrics
against other counterparts in most circumstances. Taking the results on MSRC,
NGs, and Hdigit datasets for instances, TCGF has been considered as the
strongest multi-view learning algorithm, which obtained 100% in terms of ACC,
NMI and PUR. This indicates that TCGF can be more effective and suitable for
multiview features, which can well uncover the intrinsic rich information in
multi-view features.

  * •

For the ALOI_1K and YoutubeFace datasets that contains over 100,000 samples,
most of multi-view learning methods suffer from out-of-memory errors, such as
GMC and LTBPL. The main reason is that the computational complexity of these
works is squared or cubic with the data size. However, the proposed TCGF is
capable of handling such large-scale datasets, which can obtain comparable and
better performance against LMVSC and FPMVS. Meanwhile, the proposed TCGF
performs these two methods in other datasets. It implies the effectiveness and
efficiency of TCGF, which is applicable on both normal- and large-scale multi-
view datasets.

TABLE III: Clustering performance comparisons in terms of NMI on seven real-world datasets. Datasets | SC | GNMF | Co-reg | MDcR | AMGL | GMC | GFSC | LMVSC | FPMVS | VCGA | LTBPL | TCGF
---|---|---|---|---|---|---|---|---|---|---|---|---
Small-scale multi-view datasets
MSRC | 0.625 | 0.023 | 0.719 | 0.402 | 0.714 | 0.771 | 0.668 | 0.246 | 0.686 | 0.751 | 0.628 | 1.000
NGs | 0.562 | 0.105 | 0.119 | 0.501 | 0.191 | 0.939 | 0.451 | 0.101 | 0.592 | 0.935 | 0.979 | 1.000
General-scale multi-view datasets
Hdigit | 0.468 | 0.862 | 0.689 | 0.777 | 0.883 | 0.993 | 0.854 | 0.808 | 0.816 | 0.849 | 0.999 | 1.000
Caltech101 | 0.269 | 0.378 | 0.587 | 0.564 | 0.385 | 0.552 | 0.815 | 0.283 | 0.632 | - | 0.577 | 0.806
ALOI_100 | 0.687 | 0.466 | 0.699 | 0.536 | 0.681 | 0.735 | 0.542 | 74.95 | 0.648 | - | 0.655 | 0.762
Large-scale multi-view datasets
ALOI_1K | 0.544 | 0.530 | OOM | OOM | OOM | OOM | OOM | 0.704 | 0.555 | OOM | OOM | 0.679
YoutubeFace | 0.121 | 0.071 | OOM | OOM | OOM | OOM | OOM | 0.133 | 0.243 | OOM | OOM | 0.102
TABLE IV: Clustering performance comparisons in terms of PUR on seven real-world datasets. Datasets | SC | GNMF | Co-reg | MDcR | AMGL | GMC | GFSC | LMVSC | FPMVS | VCGA | LTBPL | TCGF
---|---|---|---|---|---|---|---|---|---|---|---|---
Small-scale multi-view datasets
MSRC | 0.627 | 0.171 | 0.713 | 0.492 | 0.795 | 0.790 | 0.761 | 0.381 | 0.786 | 0.835 | 0.714 | 1.000
NGs | 0.689 | 0.328 | 0.368 | 0.693 | 0.330 | 0.982 | 0.504 | 0.818 | 0.738 | 0.980 | 0.994 | 1.000
General-scale multi-view datasets
Hdigit | 0.345 | 0.935 | 0.660 | 0.866 | 0.912 | 0.996 | 0.914 | 0.875 | 0.910 | 0.918 | 0.999 | 1.000
Caltech101 | 0.492 | 0.607 | 0.754 | 0.515 | 0.677 | 0.555 | 0.631 | 0.435 | 0.736 | - | 0.725 | 0.877
ALOI_100 | 0.493 | 0.288 | 0.519 | 0.512 | 0.634 | 0.507 | 0.753 | 0.598 | 0.348 | - | 0.640 | 0.687
Large-scale multi-view datasets
ALOI_1K | 0.238 | 0.193 | OOM | OOM | OOM | OOM | OOM | 0.426 | 0.242 | OOM | OOM | 0.435
YoutubeFace | 0.280 | 0.267 | OOM | OOM | OOM | OOM | OOM | 0.282 | 0.328 | OOM | OOM | 0.302

###  IV-C Ablation Study

In this section, the ablation study is conducted to evaluate the effects of
consensus graph learning, view-specific graph learning, and tersorized graph
learning. Specifically, for the each test, the corresponding term is removed
while retaining the other terms. For notation simplicity, we denote these
three tests as TCGF-v1, TCGF-v2, and TCGF-v3, respectively. These tests are
performed on MSRC and NGs datasets, and the results of clustering preformance
comparison in terms of ACC, NMI and PUR are reported in Table V. According to
the table, we can observe that TCGF achieves superior clustering performance
compared with its variants in all testing cases. To this end, the ablation
study demonstrates the necessity of the proposed model, which simultaneously
takes view-specific graph learning, tersorized graph learning, and consensus
graph learning into consideration.

TABLE V: Comparison results of TCGF and its variants. Datasets | Variants | ACC | NMI | RUR
---|---|---|---|---
MSCRv1 | TCGF-v1 | 0.910 | 0.821 | 0.910
TCGF-v2 | 0.971 | 0.942 | 0.971
TCGF-v3 | 0.819 | 0.728 | 0.819
TCGF | 1.000 | 1.000 | 1.000
NGs | TCGF-v1 | 0.878 | 0.791 | 0.878
TCGF-v2 | 0.952 | 0.866 | 0.952
TCGF-v3 | 0.664 | 0.592 | 0.696
TCGF | 1.000 | 1.000 | 1.000

###  IV-D Hyper-parameter Analysis

In this subsection, hyper-parameter analysis is conducted to investigate the
effects of two parameters $\lambda_{E}$ and $\lambda_{R}$ on MSRC and NGs
datasets with different settings, in which the experimental results in terms
of ACC and NMI are reported in Figs. 3-4. Through these experimental results
in Figs. 3-4, we can observe that for different datasets, the selections of
$\lambda_{E}$ and $\lambda_{R}$ are different, and the best values of these
two parameters vary from one dataset to another. However, there exists a wide
range for each hyper-parameter in which relatively stable and good results can
be readily obtained. Meanwhile, We set $\lambda_{E}$ and $\lambda_{R}$ to
those values that make the proposed TCGF has the best results according to the
experiments.

Figure 3: Hyper-parameter analysis on MSRC dataset.

Figure 4: Hyper-parameter analysis on NGs dataset.

###  IV-E Visualization

Additional visualization results conducted on MSRC, NGs and Hdigits datasets
are shown in Fig.5, which adopt t-SNE [49] to project features into the
2-dimensional subspace. Obviously, the distributions of original data are
disordered. After TCGF is conducted, samples can be readily separated into
several clusters, which further validates the effectiveness of TCGF.

(a) Original Features in MSCRv1 (b) Original Features in NGs (c) Original
Features in Hdigits (d) Learnt Embedding by TCGF in MSCRv1

(e) Learnt Embedding by TCGF in NGs (f) Learnt Embedding by TCGF in Hdigits

Figure 5: Visualization of Original Features and Learnt Embedding on MSCRv1,
NGs, and Hdigits datasets.

To show the superiority of the learned consensus graph, we visualized the
consensus graphs of TCEF on MSRC, NGs and Hdigits datasets, as shown in Fig.
6. As shown in Fig. 6, we can find clear and complete block diagonal
structures. Thus, TCEF can adaptively promote the learning of consensus graphs
towards better attributes. Although there are some ineluctable noisy values in
the consensus graphs, TCEF can achieve good results on multi-view clustering
due to the superior attributes of the learned consensus graph. In this way, we
can verify that how to construct a consensus graph is of vital importance for
multi-view clustering.

(a) MSCRv1

(b) NGs

(c) Hdigits

Figure 6: The Learned Consensus Graphs of TCGF on MSCRv1, NGs, and Hdigits
Datasets.

##  V Conclusion

In this paper, we propose a novel unified consensus embedding framework for
multi-view representation learning, termed Tensorized Consensus Graph
Framework (TCGF). TCGF aims to be served as a universal learning framework for
existing multi-view works under arbitrary assumptions and is applicable for
multi-view applications with varying scales. TCGF firstly proprovides a
unified framework to exploit the representations for individual view, enabling
it to be suitable for different-scales datasets. Then, it stacks these
representations into a tensor under alignment basics as a high-order
representation, allowing for the smooth propagation of consistency and
complementary information across all views. Additionally, TCGF proposes
learning a consensus embedding shared by adaptively collaborating all views to
uncover the essential structure of the multi-view data, which utilizes view-
consensus grouping effect to regularize the view-consensus representation. The
proposed TCGF is evaluated through extensive experiments on seven different-
scales datasets, demonstrating its effectiveness and superiority to maintain
or outperform other sota multi-view methods.

## Acknowledgements

The authors would like to thank the anonymous reviewers for their insightful
comments and suggestions to significantly improve the quality of this paper.

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|  Xiangzhu Meng received his B.S. degree from Anhui University, in 2015, and
the Ph.D. degree in Computer Science and Technology from Dalian University of
Technology, in 2021. Now, he is a postdoctoral researcher with the Center for
Research on Intelligent Perception and Computing, Institute of Automation,
Chinese Academy of Sciences, China. He regularly publishes papers in
prestigious journals, including TNNLS, KBS, EAAI, INSC, APIN, etc. In
addition, he serves as a reviewer for the conferences of ACM MM 2022 and
MICAAI 2023. His research interests include multi-modal learning, and data
mining.
---|---
|  Wei Wei received the B.S. degree from the School of Mathematics and
Information Science, Henan University, in 2012, and the Ph.D. degree in
Institute of Systems Engineering from the Dalian University of Technology, in
2018. He is an associate professor of the Center for Energy, Environment &
Economy Research, College of Tourism Management, Zhengzhou University. His
research interests include energy policy analysis, text mining, machine
learning, and artificial intelligence.
---|---
|  Qiang Liu received his B.S. degree from is an Associate Professor in Center
for Research on Intelligent Perception and Computing (CRIPAC), Institute of
Automation, Chinese Academy of Sciences (CASIA). He received his PhD degree in
pattern recognition from CASIA. Currently, his research interests include data
mining, recommender systems, text mining, knowledge graph, and causal
inference. He has published more than 30 papers in top-tier journals and
conferences, such as ITKDE, AAAI, IJCAI, NeurIPS, WWW, SIGIR, CIKM and ICDM.
---|---
|  Shu Wu received his B.S. degree from Hunan University, China, in 2004, M.S.
degree from Xiamen University, China, in 2007, and Ph.D. degree from
Department of Computer Science, University of Sherbrooke, Quebec, Canada, all
in computer science. He is an Associate Professor with the Center for Research
on Intelligent Perception and Computing (CRIPAC) at National Laboratory of
Pattern Recognition (NLPR), Institute of Automation, Chinese Academy of
Sciences (CASIA). He has published more than 70 papers in the areas of data
mining and information retrieval in international journals and conferences,
such as IEEE TKDE, IEEE THMS, AAAI, ICDM, SIGIR, and CIKM. His research
interests include data mining and information retrieval.
---|---
|  Liang Wang received both the B.E. and M.S. degrees from Anhui University in
1997 and 2000, respectively, and the Ph.D. degree from the Institute of
Automation, Chinese Academy of Sciences (CASIA) in 2004. Currently, he is a
full professor of the Hundred Talents Program at the National Lab of Pattern
Recognition, CASIA. His major research interests include machine learning,
pattern recognition, and computer vision. He has widely published in highly
ranked international journals such as IEEE TPAMI, IEEE TKDE and IEEE TIP, and
leading international conferences such as CVPR, ICCV and ICDM. He is an IEEE
and IAPR Fellow.
---|---