# TECM: Transfer Learning-based Evidential C-Means Clustering

Lianmeng Jiao∗<EMAIL_ADDRESS>Feng Wang<EMAIL_ADDRESS>Zhun-ga Liu<EMAIL_ADDRESS>Quan Pan<EMAIL_ADDRESS>School of
Automation, Northwestern Polytechnical University, Xi’an 710072, P.R. China

###### Abstract

As a representative evidential clustering algorithm, evidential c-means (ECM)
provides a deeper insight into the data by allowing an object to belong not
only to a single class, but also to any subset of a collection of classes,
which generalizes the hard, fuzzy, possibilistic, and rough partitions.
However, compared with other partition-based algorithms, ECM must estimate
numerous additional parameters, and thus insufficient or contaminated data
will have a greater influence on its clustering performance. To solve this
problem, in this study, a transfer learning-based ECM (TECM) algorithm is
proposed by introducing the strategy of transfer learning into the process of
evidential clustering. The TECM objective function is constructed by
integrating the knowledge learned from the source domain with the data in the
target domain to cluster the target data. Subsequently, an alternate
optimization scheme is developed to solve the constraint objective function of
the TECM algorithm. The proposed TECM algorithm is applicable to cases where
the source and target domains have the same or different numbers of clusters.
A series of experiments were conducted on both synthetic and real datasets,
and the experimental results demonstrated the effectiveness of the proposed
TECM algorithm compared to ECM and other representative multitask or transfer-
clustering algorithms.

###### keywords:

Evidential clustering, transfer learning, belief function theory, credal
partition

††journal: Knowledge-Based Systems

##  1 Introduction

Clustering is a process of dividing a set of objects into groups of similar
members. Classical clustering algorithms can be roughly divided into the
following categories: partition-based methods [1, 2], hierarchy-based methods
[3, 4], density-based methods [5, 6], grid-based methods [7, 8], and graph-
based methods [9, 10]. Among these, partition-based methods are the most
popular because they are universally applicable to numerous real-world
clustering tasks. Although classical partition-based clustering algorithms
such as c-means only provide hard partitions, a large number of variants have
been proposed by incorporating fuzzy sets [11], possibility theory [12], and
rough sets [13] to characterize the uncertainty of cluster membership.
Recently, evidential clustering [14, 15, 16, 17, 18, 19, 20] has been
developed as a promising framework that uses the Dempster-Shafer belief
function theory [21, 22] as an uncertainty model. It generalizes the above-
mentioned hard, fuzzy, possibilistic, and rough partitions, and thus, has been
widely used in numerous application scenarios, including tumor segmentation
[23, 24], community detection [25, 26], and items of interest recommendation
[27, 28].

Evidential c-means (ECM) [14], as a representative evidential clustering
algorithm, can assign objects not only to a single class, but also to any
subset of a collection of classes by credal partition [29]. Therefore, ECM can
provide a deeper insight into the data for situations where the data
distribution is complex. However, this additional flexibility comes at the
expense of algorithm complexity, that is, the ECM algorithm must estimate
numerous additional parameters compared to the original c-means or fuzzy
c-means (FCM) algorithms; therefore, insufficient or contaminated data will
have a greater influence on its clustering performance. Noise is unavoidable
in practical clustering. In addition, collecting sufficient data is difficult
in certain practical applications. Such shortcomings severely restrict the
applicability of ECM. One feasible method to address this problem is transfer
learning [30, 31].

Transfer learning assumes that the data in the current scene (called the
target domain) are inadequate for the learning task, whereas other helpful
information is available from a relevant scene (called the source domain).
Currently, transfer learning is mainly integrated with the following machine
learning tasks [32]: classification [33, 34], regression [35, 36], feature
extraction [37, 38], and clustering [39, 40]. The first three tasks have been
studied extensively; however, research on integrating transfer learning with
clustering remains inadequate, although it has wide application in reality. In
the past decade, research on transfer clustering has gradually emerged.
Studies on transfer clustering can be broadly classified into the following
four categories based on the transfer strategy [41]: (1) the instance-based
method [42], where a portion of the source data is selected and transferred to
the target domain for clustering target data; (2) the feature-representation-
based method [43, 44], whose intuitive idea is to generate a superior feature
representation for the target data based on the knowledge learned from the
source data; (3) the parameter-based method [39, 40, 45, 46, 47, 48], which
assumes that two domains share similar parameters such as cluster centers; and
(4) the relational knowledge-based method [49, 50], which builds a mapping of
the relational knowledge between two domains. Among these, parameter-based
transfer clustering is the focus of the current research because it can fully
exploit the inter-domain similarity and is more suitable for integration with
popular partition-based clustering algorithms.

In this paper111A preliminary version of this paper have been presented in
BELIEF 2021 Conference [51]. This paper extends our short conference paper by
adding a comprehensive review of related work, a deep analysis of the
challenges, a more detailed description of the proposed algorithm with an
analysis of its generality, convergence, and complexity, and additional
datasets and comparison algorithms to illustrate the advantages of the
proposed algorithm., transfer learning is applied to ECM to improve the
clustering performance when the target data are insufficient or contaminated.
A parameter-based transfer strategy is adopted to develop a transfer learning-
based ECM (TECM) algorithm, that is, the parameters induced from the source
data, namely, barycenters, are appropriately used to improve the clustering
performance of the target data. The main contributions of this study are
summarized as follows.

(1) A novel TECM objective function is constructed by introducing the
barycenters learned from the source data as knowledge to guide the clustering
of the target data.

(2) An alternate optimization scheme is developed to solve the constrained
TECM objective function, which derives closed-form solutions by alternately
updating the parameters.

(3) The proposed TECM is applicable to any transfer-clustering scenario where
the source and target domains have the same or different numbers of clusters.

A series of experiments on both synthetic and real datasets are designed to
demonstrate the effectiveness of the TECM algorithm proposed in this study. In
the experiments on synthetic datasets, considering three situations (the
number of clusters in the source domain is equal to, greater than, or less
than the number of clusters in the target domain) and two application
scenarios (target data are insufficient or contaminated), six groups of
synthetic datasets are designed to illustrate the TECM clustering performance.
Furthermore, two synthetic ambiguous datasets are designed to demonstrate the
superiority of TECM in characterizing the uncertainty of cluster membership.
In the experiments on real datasets, the segmentation results of texture
images are used to illustrate the TECM clustering performance compared to ECM
and other representative multitask or transfer-clustering algorithms. The
experimental results on the synthetic and real datasets demonstrate that the
proposed TECM achieves superior clustering performance compared to the other
methods on the three popular validity indices.

The remainder of this paper is organized as follows. Section 2 comprehensively
reviews of the related work. The ECM algorithm is summarized and analyzed in
Section 3. A detailed description of the proposed TECM is presented in Section
4. In Section 5, we evaluate the proposed TECM using both synthetic and real
datasets. Section 6 concludes the paper and outlines the directions for future
research.

##  2 Related work

In the past decade, transfer-learning techniques have been integrated with
classical clustering algorithms. Based on their transfer strategy, the
existing transfer-clustering algorithms can be categorized as follows.

Instance-based method. In [42], the unlabeled heterogeneous data from
different domains are first projected into a common subspace, and then other
informative source data are selected and transferred to the target domain to
improve the clustering performance of the target data using an algorithm
called DicTrans. However, because of the differences in data distribution
between the two domains, it is difficult to determine how to select the source
data and the amount of source data to be transferred.

Feature-representation-based method. In [43], based on co-occurrence data, a
co-transfer-clustering algorithm was presented to simultaneously address
multiple individual clustering tasks. A collective nonnegative matrix
factorization algorithm is used to obtain a common subspace for all domains.
Then, data in different domains are mapped into this subspace and the data
from the different domains are clustered simultaneously using a symmetric
nonnegative matrix factorization algorithm. In [44], a transfer-clustering
algorithm called shared hidden space transfer fuzzy c-means was proposed. In
this algorithm, source data and target data are mapped into a common subspace
by attempting to identify a projected matrix with the strategy of maximum mean
discrepancy; then, all data in the common subspace are clustered
simultaneously.

Parameter-based method. In [40], a representative parameter-based transfer-
clustering algorithm called transfer fuzzy c-means (TFCM) clustering was
proposed using the transfer-learning strategy. The cluster centers obtained
from the source domain via classical FCM clustering are used to construct the
corresponding objective function to promote the clustering performance of the
target domain. In [46, 47, 48], as in [40], the cluster centers obtained from
the source domain are used as transfer knowledge to guide the clustering of
the target domain. In [45], a novel transfer-clustering algorithm called
transfer distributed EM clustering (TDEM) was proposed, where each node is
considered as both the source and target domains; these nodes can learn from
each other to complete the clustering task in distributed peer-to-peer (P2P)
networks. In [39], a transfer maximum entropy clustering algorithm was
proposed and the corresponding objective function was constructed using the
cluster centers and membership matrix obtained from the source data. However,
this algorithm is only applicable to situations where the source and target
domains have the same number of clusters.

Relational knowledge-based method. In [49], transfer spectral clustering
(TSC), a representative relational knowledge-based transfer-clustering
algorithm, was proposed. Based on the assumption that embedded feature
information can connect two clustering tasks, TSC can simultaneously improve
the clustering performance in two domains. Furthermore, co-clustering is
employed to control the knowledge transfer between the two tasks. In [50], a
novel transfer-clustering algorithm called topic constraint transfer
clustering was proposed. First, several clusters are obtained from the source
data using an efficient clustering algorithm. Subsequently, several topics are
extracted from each cluster. Then, the similarities between the target data
and extracted topics are discovered to assist in the clustering of the target
data. Finally, a semi-supervised clustering algorithm is applied to the target
data.

Apart from the above four categories of transfer-clustering algorithms, our
task is also related to multitask clustering, which addresses multiple
clustering tasks simultaneously and uses mutual cooperation to improve the
performance of all the clustering tasks. Representative multitask clustering
algorithms include learning the shared subspace for multi-task clustering
(LSSMTC) and combining k-means (CombKM) [53]. LSSMTC aims to learn a subspace
shared by all tasks, through which the knowledge of the tasks can be
transferred to each other; CombKM clusters all data from different tasks
simultaneously using c-means.

Among these transfer or multitask clustering methods, parameter-based transfer
clustering is the focus of the current research because it is more suitable
for integration with popular partition-based clustering algorithms. In
partition-based clustering algorithms, the cluster membership of each object
depends on its distance to the cluster centers; therefore, the parameter
“cluster centers” is the most reliable and intuitive transfer knowledge. In
addition, parameter-based transfer clustering typically obtains acceptable
transfer performance because it can fully utilize the inter-domain similarity.
Therefore, the parameter-based transfer strategy is adopted in this study to
develop an ECM transfer-clustering version.

##  3 ECM: Basics and challenges

Belief function theory [21, 22] is a formal framework for approximate
reasoning. In this theory, the set $\Omega$ = $\\{w_{1},...,w_{c}\\}$
containing all possible results of a discrimination problem is called the
frame of discernment. A mass function $m$ is used to express uncertain
information, which is a mapping function from $2^{\Omega}$ to [0, 1] that
satisfies the following equations:

$m(\emptyset)=0,\;\sum\limits_{A\in{2^{\Omega}}}{m(A)=1}.$ (1)

Subsets $A$ are called focal sets if $A$ $\in$ $2^{\Omega}$ satisfies
$m(A)>0$. Based on the type of information represented, the mass function $m$
is classified into the following categories:

$\bullet$ _Bayesian_ : The focal sets of this type of mass function are
singletons. In this case, the mass function becomes the classical probability
distribution function.

$\bullet$ _Certain_ : This type of mass function has only one focal set, which
can be denoted by $m_{A}$, where $A$ is its unique focal set. This mass
function certainly indicates that the real result is in set $A$.

$\bullet$ _Vacuous_ : This type of mass function has only one focal set,
$\Omega$, which can be denoted as $m_{\Omega}$. This mass function indicates
that the actual result is unknown.

In [29], the notion of a credal partition was proposed to represent the class
membership uncertainty of each object $x_{i}$ by a mass function $m_{i}$,
which can model different situations from completely unknown to total
certainty regarding the class membership of the object.

###### Example 1

Consider an example of $n=5$ objects belonging to $c=2$ classes with
evidential membership. The mass functions of these objects are listed in Table
1, which illustrates all possible situations from complete unknown to total
certainty. The case of object $1$ corresponds to the situation of total
certainty ($m_{1}$ is certain). Object $2$ corresponds to the situation of
completely unknown ($m_{2}$ is vacuous). Object $3$ corresponds to the
situation of probabilistic uncertainty ($m_{3}$ is Bayesian). Object $4$ is a
general example of evidential membership. Mass $m_{5}(\emptyset)=1$ indicates
that the class of object $5$ is not in $\Omega$.

Table 1: Illustration of credal partition $A$ | $m_{1}(A)$ | $m_{2}(A)$ | $m_{3}(A)$ | $m_{4}(A)$ |  $m_{5}(A)$
---|---|---|---|---|---
$\emptyset$ | 0 | 0 | 0 | 0.1 | 1 [t]
$w_{1}$ | 1 | 0 | 0.4 | 0.4 | 0 [t]
$w_{2}$ | 0 | 0 | 0.6 | 0.3 | 0 [t]
$\Omega$ | 0 | 1 | 0 | 0.2 | 0 [t]

The ECM algorithm [14] is a representative clustering algorithm with credal
partition, which is an extended version of FCM in the framework of belief
function theory. The mass ${m_{ij}}={m_{i}}({A_{j}})$ associated with object
$x_{i}$ with $A_{j}$ $({A_{j}}\neq\emptyset,{\kern
1.0pt}\,{A_{j}}\subseteq\Omega)$ depends on the distance
${d_{ij}}=||{x_{i}}-{\bar{v}_{j}}|{|^{2}}$ between object $x_{i}$ and
barycenter ${\overline{v}_{j}}$ of $A_{j}$, which is greater when $d_{ij}$ is
smaller. Barycenter ${\overline{v}_{j}}$ is defined as

${\overline{v}_{j}}=\frac{1}{{{c_{j}}}}\sum\limits_{k=1}^{c}{{s_{kj}}{v_{k}}},\quad{\rm{s}}.{\kern
1.0pt}\,{\rm{t}}.\quad
s_{kj}=\left\\{\begin{array}[]{l}1\quad{\rm{if}}\;{w_{k}}\in{A_{j}}\\\
0\quad\rm{else},\end{array}\right.$ (2)

where $c_{j}$ is the cardinality of the focal set $A_{j}$ and $v_{k}$ is the
center of the singleton $w_{k}$.

For $n$ objects with $p$ dimensions, the ECM objective function with respect
to the credal partition $M=\\{{m_{1}},...,{m_{n}}\\}\in{R^{n\times{2^{c}}}}$
and cluster centers $V=\\{{v_{1}},...,{v_{c}}\\}\in R^{c\times p}$ is defined
as

$\begin{split}&\mathop{\min}{J_{ECM}(M,V)}=\sum\limits_{i=1}^{n}{\sum\limits_{\\{j/{A_{j}}\subseteq\Omega,{A_{j}}\neq\emptyset\\}}{c_{j}^{\alpha}m_{ij}^{\beta}||{x_{i}}-{{\overline{v}}_{j}}|{|^{2}}}}+\sum\limits_{i=1}^{n}{{\delta^{2}}m_{i\emptyset}^{\beta}},\\\
&{\rm{s}}.{\kern
1.0pt}\,{\rm{t}}.\quad\sum\limits_{\\{j/{A_{j}}\subseteq\Omega,{A_{j}}\neq\emptyset\\}}{{m_{ij}}+{m_{i\emptyset}}=1,\quad\forall
i=1,...,n,}\end{split}$ (3)

where ${m_{i\emptyset}}$ represents ${m_{i}}(\emptyset)$ and $c_{j}$
represents the cardinality of the focal set $A_{j}$. Parameters $\alpha$ and
$\beta$ control the penalization degree of high cardinality and credibility of
the partition, respectively, and $\delta$ controls the number of outliers.

Based on the objective function given in Eq. (3), it can be observed that the
ECM must estimate a considerably greater number of parameters ($n*2^{c}$) to
obtain a credal partition compared to other classical partition-based
algorithms ($n*c$ for FCM and $n$ for c-means). Therefore, insufficient or
contaminated data could have a greater influence on ECM compared to other
partition-based algorithms. Here, we provide an example to illustrate this
issue.

###### Example 2

Two synthetic datasets, $Synt_{1}$ and $Synt_{2}$, are used to illustrate the
ECM clustering performance for application scenarios where the data are
insufficient and contaminated, respectively. Let $\mu_{i}$ and $\Sigma_{i}$
represent the mean vector and covariance matrix of the $i$th cluster in each
dataset, respectively. Dataset $Synt_{1}$ simulates a scenario where the data
are insufficient. Dataset $Synt_{1}$ is generated with $\mu_{1}$=[0,
0]$,\bm{$\Sigma_{1}}=[1,0;0,1]$, $\mu_{2}$=[0,
3]$,\bm{$\Sigma_{2}}=[1,0;0,1]$, $\mu_{3}$=[3,
0]$,\bm{$\Sigma_{3}}=[1,0;0,1]$, $\mu_{4}$=[3,
3]$,and\bm{$\Sigma_{4}}=[1,0;0,1]$, where each cluster consists of only five
objects. The data distributions used to generate the dataset $Synt_{2}$ are
the same as those for $Synt_{1}$; however, each cluster in $Synt_{2}$ consists
of 20 objects. In addition, zero-mean Gaussian noise with a standard deviation
of 0.3 is added to $Synt_{2}$ to simulate the scenario where the data are
contaminated. The clustering results for datasets $Synt_{1}$ and $Synt_{2}$
using ECM are displayed in Fig. 1 and Fig. 2, respectively. It can be observed
from the clustering results that the real cluster profiles were not recovered
precisely, and the ambiguous objects at the boundaries were not properly
recognized.

Figure 1: Clustering results for dataset $Synt_{1}$ using ECM. Figure 2:
Clustering results for dataset $Synt_{2}$ using ECM.

The above example confirms that the ECM clustering performance is not ideal
for situations where the data are insufficient or contaminated. Therefore, it
is necessary to improve the ECM clustering performance in these situations by
exploiting sufficient high-quality data from a relevant scene.

##  4 TECM

In this section, a TECM clustering algorithm is developed to improve the
clustering performance when the target data are insufficient or contaminated.
In Section 4.1, the basic principle of TECM is introduced. In Section 4.2, a
novel TECM objective function is presented. In Section 4.3, a procedure for
solving the TECM objective function is developed. In Section 4.4, we provide
the algorithm pseudocode and analyze the generality, convergence, and
complexity of the proposed algorithm.

###  4.1 Basic principle of TECM

In transfer learning, as indicated in Fig. 3, sampled data and learned
knowledge are typically the two types of information available from the source
data. The sampled data are not appropriate for the direct clustering of the
target data because of the differences in the data distribution between these
two domains. In addition, source data are at times inaccessible because of
privacy protection in certain practical scenarios. Compared with sampled data,
learned knowledge such as feature representations, parameters, and
relationships is typically considered more insightful and noise-resistant.

In this study, the TECM clustering algorithm was developed using the learned
knowledge from sufficient data of acceptable quality in a relevant scene. The
basic principle of TECM is illustrated in Fig. 4. It can be observed that ECM
is first applied to the source data to learn the knowledge, that is,
barycenters of the clusters, which concisely represent the data distribution.
These barycenters are then used to guide the clustering of the target data.
Following this principle, we study how to construct the TECM objective
function by appropriately using the knowledge learned from the source data and
how to develop an efficient optimization scheme to solve the objective
function.

Figure 3: Available information obtained from source data for clustering of
target data. Figure 4: Basic principle of TECM.

###  4.2 TECM Objective function

In this section, the TECM objective function is constructed by introducing the
learned barycenters into the ECM objective function. The TECM objective
function with regard to the credal partition $M$, association matrix $R$, and
cluster centers $V$ is defined as

$\begin{split}&\mathop{\min}{J_{TECM}(M,R,V)}=\sum\limits_{i=1}^{n}{\sum\limits_{\\{j/{A_{j}}\subseteq\Omega_{t},{A_{j}}\neq\emptyset\\}}{c_{j}^{\alpha}m_{ij}^{\beta}||{x_{i}}-{{\bar{v}}_{j}}|{|^{2}}}}\hfill+\\\
&\lambda\sum\limits_{\\{k/{A_{k}}\subseteq\Omega_{s},{A_{k}}\neq\emptyset\\}}{\sum\limits_{\\{j/{A_{j}}\subseteq\Omega_{t},{A_{j}}\neq\emptyset\\}}{c_{j}^{\alpha}r_{kj}^{\gamma}||{{\tilde{v}}_{k}}-{{\bar{v}}_{j}}|{|^{2}}}}\hfill+\sum\limits_{i=1}^{n}{{\delta^{2}}m_{i\emptyset}^{\beta}}\hfill,\\\
&s.t.{\kern
1.0pt}\;\sum\limits_{\\{j/{A_{j}}\subseteq\Omega_{t},{A_{j}}\neq\emptyset\\}}{{m_{ij}}+{m_{i\emptyset}}=1,\;\;\forall
i=1,...,n,}\\\ &\hfill{\kern
1.0pt}\;\sum\limits_{\\{j/{A_{j}}\subseteq{\Omega_{t}},{A_{j}}\neq\emptyset\\}}{{r_{kj}}=1,\quad\forall
k/}{A_{k}}\subseteq{\Omega_{s}},{A_{k}}\neq\emptyset,\\\ \end{split}$ (4)

where ${\widetilde{v}_{k}}$ is the $k$-th barycenter obtained from the source
data and element $r_{kj}$ of $R$ denotes the association between the $j$-th
barycenter learned from the target data and $k$-th barycenter learned from the
source data. The sets $\Omega_{t}$ and $\Omega_{s}$ denote the frames of the
discernment of the target and source domains, respectively. Parameter $\gamma$
controls the credibility of the association and $\lambda$ is the balance
coefficient of transfer learning. The definitions of the other parameters
$\alpha$, $\beta$, and $\delta$ are the same as those in ECM.

The characteristics of the objective function defined in Eq. (4) are as
follows.

1) The first and third terms are inherited from the ECM objective function,
which is used to generate the credal partition of the target data by
minimizing the global inner-cluster distance by considering the membership
degrees between the objects and barycenters.

2) The second term introduces knowledge learned from the source data. The
barycenters generated from the source data provide concise yet rich class
distribution information of the relevant source domain. This term requires
that the barycenters of the target data be as close as possible to the
barycenters learned from the source data as a whole by considering their
potential associations.

3) The parameter $\lambda$ is a balance coefficient used to control the
transfer degree of the learned knowledge from the source data. If the value of
$\lambda$ is zero, TECM becomes the original ECM; if the value of $\lambda$ is
relatively large, a negative transfer could occur. Therefore, an appropriate
value of $\lambda$ must be selected. The choice of $\lambda$ depends on the
specific application; the grid search strategy [54] can be used to determine
an appropriate value. A series of experiments [40] proved that as the value of
$\lambda$ increases, the clustering performance always improves first and then
worsens. Based on this, we can first set a large search range with a large
step size, and then select the part with superior clustering performance and
reduce the step size. These steps are repeated to obtain an appropriate value
of $\lambda$ with acceptable clustering performance. Parameter $\gamma$ can be
set to $2$, similar to that in [40]. The values of the other parameters
$\alpha$, $\beta$, and $\delta$ can be set to the same values as in ECM [14].

4) The objective function is defined for the general case, where the frames of
discernment $\Omega_{s}$ and $\Omega_{t}$ can be different. By minimizing the
objective function, we can determine the optimal pairwise associations of the
barycenters (represented by the association matrix $R$) between the source and
target data. This property significantly expands the scope of application of
TECM.

###  4.3 Optimization procedure

To minimize $J_{TECM}$, an alternate optimization scheme is used in this
study, that is, optimizing one variable by setting the other variables to
fixed values. From Eq. (4), it can be observed that the objective function
$J_{TECM}$ is unconstrained to $V$, yet is constrained to $M$ and $R$. To
solve the constrained object function for $M$ and $R$, $n$ Lagrange
multipliers $\lambda_{i}$ and $k$ ($k$ is the number of focal sets in
$\Omega_{s}$) Lagrange multipliers $\eta_{k}$ are introduced to formulate the
following Lagrangian objective function $J_{L}$:

$\begin{split}{J_{L}}={J_{TECM}}&+\sum\limits_{i=1}^{n}{\lambda_{i}}\left[{1-(\sum\limits_{\\{j/{A_{j}}\subseteq\Omega_{t},{A_{j\neq\emptyset}}\\}}{{m_{ij}}+{m_{i\emptyset}}})}\right]\\\
&+\sum\limits_{\\{k/{A_{k}}\subseteq\Omega_{s},{A_{k}}\neq\emptyset\\}}\
{{\eta_{k}}}(1-\sum\limits_{\\{j/{A_{j}}\subseteq\Omega_{t},{A_{j\neq\emptyset}}\\}}{{r_{kj}}}).\end{split}$
(5)

####  4.3.1 Optimization with respect to credal partition $M$

First, let $V$ and $R$ be fixed. Taking the partial derivatives of $J_{L}$ to
$m_{ij}$, $m_{i\emptyset}$, and $\lambda_{i}$, and setting them to $0$, we
obtain

$\begin{split}\frac{{\partial J_{L}}}{{\partial{m_{ij}}}}=\beta
c_{j}^{\alpha}m_{ij}^{\beta-1}d_{ij}^{2}-{\lambda_{i}}=0,\end{split}$ (6)

$\begin{split}\frac{{\partial
J_{L}}}{{\partial{m_{i\emptyset}}}}=\beta{\delta^{2}}m_{i\emptyset}^{\beta-1}-{\lambda_{i}}=0,\end{split}$
(7)

$\begin{split}\frac{{\partial
J_{L}}}{{\partial{\lambda_{i}}}}=\sum\limits_{\\{j/{A_{j}}\subseteq\Omega_{t},{A_{j}}\neq\emptyset\\}}{{m_{ij}}+{m_{i\emptyset}}-1}=0.\end{split}$
(8)

From Eq. (6), we obtain

$\begin{split}{m_{ij}}={\left({\frac{{{\lambda_{i}}}}{\beta}}\right)^{\frac{1}{{\beta-1}}}}{\left({\frac{1}{{c_{j}^{\alpha}||{x_{i}}-{{\bar{v}}_{j}}|{|^{2}}}}}\right)^{\frac{1}{{\beta-1}}}},\end{split}$
(9)

and from Eq. (7), we obtain

${m_{i\emptyset}}={\left({\frac{{{\lambda_{i}}}}{\beta}}\right)^{\frac{1}{{\beta-1}}}}{\left({\frac{1}{{{\delta^{2}}}}}\right)^{\frac{1}{{\beta-1}}}}.$
(10)

Using Eq. (8)–Eq. (10), we obtain

$\begin{split}{\left({\frac{{{\lambda_{i}}}}{\beta}}\right)^{\frac{1}{{\beta-1}}}}={\left({\sum\limits_{j}{\frac{1}{{{{(c_{j}^{\alpha}||{x_{i}}-{{\bar{v}}_{j}}|{|^{2}})}^{\frac{1}{{\beta-1}}}}}}+\frac{1}{{{\delta^{\frac{2}{{\beta-1}}}}}}}}\right)^{-1}}.\end{split}$
(11)

Substituting Eq. (11) into Eq. (9), we obtain the updated equation for $M$:

$\begin{split}&{m_{ij}}=\frac{{{{\left({c_{j}^{\alpha}{\rm{||}}{x_{i}}-{{\bar{v}}_{j}}|{|^{2}}}\right)}^{\frac{{-1}}{{\beta-1}}}}}}{{\sum\nolimits_{{A_{t}}\neq\emptyset}{{{\left({c_{t}^{\alpha}||{x_{i}}-{{\bar{v}}_{t}}|{|^{2}}}\right)}^{\frac{{-1}}{{\beta-1}}}}+{\delta^{\frac{{-2}}{{\beta-1}}}}}}},\\\
&\forall i=1,...,n,\quad\forall
j/{A_{j}}\subseteq{\Omega_{t}},{A_{j}}\neq\emptyset,\end{split}$ (12)

and

$\begin{split}{m_{i\emptyset}}=&1-\sum\limits_{j/{A_{j}}\subseteq{\Omega_{t}},{A_{j}}\neq\emptyset}{{m_{ij}}},~{}~{}~{}~{}\forall
i=1,...,n.\end{split}$ (13)

####  4.3.2 Optimization with respect to association matrix $R$

Next, we assume that $V$ and $M$ are fixed. Taking the partial derivatives of
$J_{L}$ to $r_{kj}$ and $\eta_{k}$ and setting them to $0$, we obtain

$\frac{{\partial J_{L}}}{{\partial r{}_{kj}}}=\lambda c_{j}^{\alpha}\gamma
r_{kj}^{\gamma-1}{({{\tilde{v}}_{k}}-{v_{j}})^{2}}-{\eta_{k}}=0,$ (14)

$\frac{{\partial
J_{L}}}{{\partial{\eta_{k}}}}=\sum\limits_{\\{j/{A_{j}}\subseteq\Omega_{t},{A_{j}}\neq\emptyset\\}}{{r_{kj}}-1=0.}$
(15)

From Eq. (14), we obtain

${r_{kj}}={\left({\frac{{{\eta_{k}}}}{{{\lambda_{\gamma}}}}}\right)^{\frac{1}{{\gamma-1}}}}{\left({\frac{1}{{c_{j}^{\alpha}||{{\tilde{v}}_{k}}-{{\bar{v}}_{j}}|{|^{2}}}}}\right)^{\frac{1}{{\gamma-1}}}}.$
(16)

Using Eq. (15) and Eq. (16), we obtain

${\left[{\frac{{{\eta_{k}}}}{{\lambda\gamma}}}\right]^{\frac{1}{{\gamma-1}}}}={\left({\sum\limits_{j}{{{\left[{\frac{1}{{c_{j}^{\alpha}{{({{\widetilde{v}}_{k}}-{{\bar{v}}_{j}})}^{2}}}}}\right]}^{\frac{1}{{\gamma-1}}}}}}\right)^{-1}}.$
(17)

Substituting Eq. (17) in Eq. (16), we obtain the updated equation for $R$:

$\begin{split}&{r_{kj}}=\frac{{{{(c_{j}^{\alpha}||{{\tilde{v}}_{k}}-{{\bar{v}}_{j}}|{|^{2}})}^{\frac{{-1}}{{\gamma-1}}}}}}{{\sum\nolimits_{{A_{t}}\neq\emptyset}{{{(c_{t}^{\alpha}||{{\tilde{v}}_{k}}-{{\bar{v}}_{t}}|{|^{2}})}^{\frac{{-1}}{{\gamma-1}}}}}}},\\\
&\forall k/{A_{k}}\subseteq{\Omega_{s}},{A_{k}}\neq\emptyset,\;\;\forall
j/{A_{j}}\subseteq{\Omega_{t}},{A_{j}}\neq\emptyset.\end{split}$ (18)

####  4.3.3 Optimization with respect to cluster centers $V$

Taking the partial derivative of $J_{TECM}$ to $V$ and setting it to $0$, we
obtain

$\begin{split}&\frac{{\partial{J_{TECM}}}}{{\partial{v_{l}}}}=-2\sum\limits_{i=1}^{n}\sum\limits_{{A_{t,j}}\neq\emptyset}c_{j}^{\alpha-1}m_{ij}^{\beta}{s_{lj}}({x_{i}}-{{\overline{v}}_{j}})\\\
&-2\lambda\sum\limits_{{A_{s,k}}\neq\emptyset}{\sum\limits_{{A_{t,j}}\neq\emptyset}{c_{j}^{\alpha-1}r_{kj}^{{\gamma}}{s_{lj}}({{\widetilde{v}}_{k}}-{{\overline{v}}_{j}})}}\\\
&=-2\sum\limits_{i=1}^{n}\sum\limits_{{A_{t,j}}\neq\emptyset}c_{j}^{\alpha-1}m_{ij}^{\beta}{s_{lj}}({x_{i}}-\frac{1}{{{c_{j}}}}\sum\limits_{z}{{s_{zj}}{v_{z}}})\\\
&-2\lambda\sum\limits_{{A_{s,k}}\neq\emptyset}{\sum\limits_{{A_{t,j}}\neq\emptyset}{c_{j}^{\alpha-1}r_{kj}^{{\gamma}}{s_{lj}}({{\widetilde{v}}_{k}}-\frac{1}{{{c_{j}}}}\sum\limits_{z}{{s_{zj}}{v_{z}}})}}=0.\end{split}$
(19)

This is equivalent to the following equation:

$\begin{split}&\sum\limits_{i}{{x_{i}}}\sum\limits_{{A_{t,j}}\neq\emptyset}{c_{j}^{\alpha-1}m_{ij}^{\beta}{s_{lj}}}+\lambda\sum\limits_{{A_{s,k}}\neq\emptyset}{{{\widetilde{v}}_{k}}}\sum\limits_{{A_{t,j}}\neq\emptyset}{c_{j}^{\alpha-1}r_{kj}^{{\gamma}}{s_{lj}}}\\\
&=\sum\limits_{z}{{v_{z}}}\sum\limits_{i}{\sum\limits_{{A_{t,j}}\neq\emptyset}{c_{j}^{\alpha-2}m_{ij}^{\beta}{s_{lj}}{s_{zj}}}}+\lambda\sum\limits_{z}{{v_{z}}}\sum\limits_{{A_{s,k}}\neq\emptyset}{\sum\limits_{{A_{t,j}}\neq\emptyset}{c_{j}^{\alpha-2}r_{kj}^{{\gamma}}{s_{lj}}{s_{zj}}}}.\end{split}$
(20)

Let ${B_{\rm{1}}}\in{R^{c\times p}}$, ${B_{\rm{2}}}\in{R^{c\times p}}$,
${H_{\rm{1}}}\in{R^{c\times c}}$ and ${H_{\rm{2}}}\in{R^{c\times c}}$ be
matrices defined as

$\begin{split}{B_{{1_{lq}}}}=\sum\limits_{i=1}^{n}{{x_{iq}}}\sum\limits_{{A_{t,j}}\neq\emptyset}&{c_{j}^{\alpha-1}m_{ij}^{\beta}{s_{lj}}}=\sum\limits_{i=1}^{n}{{x_{iq}}}\sum\limits_{{w_{l}}\in{A_{t,j}}}{c_{j}^{\alpha-1}m_{ij}^{\beta}},\\\
&l=1,...,c,\quad q=1,...,p,\end{split}$ (21)

$\begin{split}{B_{{2_{lq}}}}=\sum\limits_{{A_{s,k}}\neq\emptyset}{{{\widetilde{v}}_{kq}}}\sum\limits_{{A_{t,j}}\neq\emptyset}&{c_{j}^{\alpha-1}r_{kj}^{{\gamma}}{s_{lj}}}=\sum\limits_{{A_{s,k}}\neq\emptyset}{{{\widetilde{v}}_{kq}}}\sum\limits_{{w_{l}}\in{A_{t,j}}}{c_{j}^{\alpha-1}r_{kj}^{{\gamma}}},\\\
&l=1,...,c,\quad q=1,...,p,\end{split}$ (22)

$\begin{split}{H_{{1_{lz}}}}=\sum\limits_{i}\sum\limits_{{A_{t,j}}\neq\emptyset}c_{j}^{\alpha-2}&m_{ij}^{\beta}{s_{lj}}{s_{zj}}=\sum\limits_{i}{\sum\limits_{{A_{t,j}}\supseteq\\{{w_{z}},{w_{l}}\\}}{c_{j}^{\alpha-2}m_{ij}^{\beta}}},\\\
&l=1,...,c,\quad z=1,...,c,\end{split}$ (23)

$\begin{split}{H_{{2_{lz}}}}=\sum\limits_{{A_{s,k}}\neq\emptyset}\sum\limits_{{A_{t,j}}\neq\emptyset}c_{j}^{\alpha-2}&r_{kj}^{{\gamma}}{s_{lj}}{s_{zj}}=\sum\limits_{{A_{s,k}}\neq\emptyset}{\sum\limits_{{A_{t,j}}\supseteq\\{{w_{z}},{w_{l}}\\}}{c_{j}^{\alpha-2}r_{kj}^{{\gamma}}}},\\\
&l=1,...,c,\quad z=1,...,c,\end{split}$ (24)

where $c$ denotes the number of clusters in the target domain. Using the above
definitions, $V$ is obtained by solving the following linear equation:

$\begin{split}{B_{1}}+\lambda{B_{2}}=({H_{1}}+\lambda{H_{2}})V.\end{split}$
(25)

###  4.4 Summary and analysis

With the above optimization procedure, the clustering results of the objects
can be finally obtained by the evidential membership $m_{ij}$, $i=1,...,n$,
$\forall j/{A_{j}}\subseteq{\Omega_{t}},\;{A_{j}}\neq\emptyset$, using Eqs.
(12) and (13). The computed evidential membership $M$ creates a credal
partition of the objects. The credal partition provides a general clustering
framework, which can be reduced to a traditional hard partition, fuzzy
partition [55, 56], possibilistic partition [48, 57], and rough partition [58,
59]. A summary of TECM is presented in Algorithm 1.

Algorithm 1 Transfer learning-based evidential c-means (TECM)

1:objects in target domain {$x_{1}$,…, $x_{n}$}, barycenters learned from
source domain $\widetilde{v}_{k}$,
$\forall{A_{k}}\subseteq{\Omega_{s}},{A_{k}}\neq\emptyset$, number of clusters
$c$, weighting exponents $\alpha\geqslant 0$, $\beta>1$, ${\gamma}>1$,
distance to the empty set $\delta>0$, termination threshold $\varepsilon$, and
balance coefficient of transfer learning $\lambda$.

2:credal partition $M$.

3:initialize cluster centers $V(0)$.

4:$t\leftarrow 0$; $J_{TECM}(0)\leftarrow+\infty$.

5:repeat

6: $t\leftarrow t+1$;

7: compute credal partition $M(t)$ using Eqs. (12) and (13);

8: compute cluster barycenter association matrix $R(t)$ using Eq. (18);

9: compute $B_{1}$, $B_{2}$, $H_{1}$, and $H_{2}$ using Eqs. (21)-(24);

10: compute cluster centers $V(t)$ using Eq. (25);

11: compute objective function $J_{TECM}(t)$ using Eq. (4);

12:until ($|J_{TECM}(t)-J_{TECM}(t-1)|<\varepsilon$).

Generality analysis: The proposed TECM can be reduced to the original ECM when
$\lambda=0$ and to TFCM [40] when all the evidential components are
constrained to singletons. The proposed TECM provides a fine framework for
integrating the advantages of evidential clustering and transfer learning.

Convergence analysis: The convergence of TECM can be proved by Zangwill’s
convergence theorem [60] in a similar manner to [61]. It should be noted that
although only a local optimal solution can be obtained for the TECM nonconvex
optimization problem, similar to other FCM-like algorithms, this local optimal
solution is sufficiently effective in the majority of cases.

Complexity analysis: Similar to ECM, the computational complexity of TECM is
$O(2^{c}*n)$. However, in the majority of cases, the assignment of objects to
focal sets with high cardinality is less interpretable. Therefore, in real-
world scenarios, complexity can be reduced from $O(2^{c}*n)$ to $O(c^{2}*n)$
by constraining the focal sets containing at most two clusters.

###### Remark 1

A similar study was recently presented in [52], which developed an evidential
clustering algorithm based on transfer learning. Compared to this study, our
proposed TECM algorithm has several advantages. First, the knowledge learned
from the source data is represented by the cluster centers in [52], whereas in
our proposed algorithm, it is represented by the barycenters, which can
provide richer class distribution information of the source data. Furthermore,
the clustering algorithm in [52] is only applicable to situations where the
source and target domains have the same number of clusters, whereas our
proposed TECM is also applicable to situations where the number of clusters is
different, which has a wider application scope.

##  5 Experiments

In this section, the TECM proposed in this study is evaluated on both
synthetic and real datasets. First, the comparison algorithms and indices for
the performance evaluation are introduced in Section 5.1. In Section 5.2 and
Section 5.3, the TECM clustering performance is evaluated using synthetic
datasets and texture image datasets and compared with other related
algorithms. Finally, in Section 5.4, a parameter analysis of the proposed TECM
is provided.

###  5.1 Experimental setup

In addition to ECM, six other related algorithms: LSSMTC [53], CombKM [53],
TSC [49], TDEM [45], TFCM [40], and TLEC [52], are employed for comparison.
Among these, LSSMTC and CombKM are representative multitask clustering
algorithms, and TSC, TDEM, TFCM, and TLEC are representative transfer-
clustering algorithms. The main ideas of these algorithms are described as
follows.

$\bullet$ LSSMTC [53]: The intuitive idea of LSSMTC is to learn a subspace
shared by all clustering tasks such that knowledge can be transferred through
the learned subspace. The objective of LSSMTC consists of two parts. The first
part involves clustering the data of each task individually; the second part
involves learning the shared subspace and clustering the data of all the tasks
in the shared subspace simultaneously.

$\bullet$ CombKM [53]: The intuitive idea of CombKM is to perform the c-means
algorithm on a combined dataset created by merging the data of all the tasks
together.

$\bullet$ TSC [49]: TSC assumes that embedded feature information can connect
two clustering tasks, which can simultaneously improve the clustering
performance in two domains. Therefore, the data manifold information of the
clustering tasks and feature manifold information shared between the two
clustering tasks are both introduced in TSC.

$\bullet$ TDEM [45]: TDEM was proposed to improve the clustering performance
of distributed gaussian mixture model clustering and accelerate clustering
convergence, where each node is regarded as both the source domain and target
domain, which can learn from each other to complete the clustering task in
distributed P2P networks.

$\bullet$ TFCM [40]: TFCM is a representative parameter-based transfer-
clustering algorithm that uses the cluster centers obtained from the source
domain by FCM to guide clustering in the target domain.

$\bullet$ TLEC [52]: Evidential clustering based on transfer learning (TLEC)
is another transfer-clustering version of ECM that uses the cluster centers
obtained from the source domain by ECM to guide the clustering in the target
domain. However, this algorithm requires the same number of clusters in both
the source and target domains.

The parameter settings of the involved algorithms are displayed in Table 2,
which are the recommendations of the corresponding authors of each proposal
with generally acceptable performance. For the balance coefficient of transfer
learning, $\lambda$, the grid search strategy [54] was used to determine the
optimal value. All experiments were implemented using MATLAB 2016b on a
computer with an Intel i9-10900K 3.70 GHZ CPU and 64 GB RAM.

Table 2: Parameter settings for involved algorithms Algorithm | Parameter Settings
---|---
ECM [14] |  $\alpha$ = 1, $\beta$ = 2, $\delta$ = 10
|  task number $T$ = 2
LSSMTC [53] | ${l^{*}}\in\\{2,4,8,16\\}$
| ${\lambda^{*}}\in\\{0.2,0.4,0.6,0.8\\}$
CombKM [53] |  $k$ is equal to the number of clusters
TSC [49] |  $\lambda$ = 3, $step$ = 1
TDEM [45] |  $\lambda$ $\in$ {0.5:0.1:1.5}, $\eta_{1}$=$\eta_{2}$=$10^{-10}$
|  The fuzzifers $m_{1}$, $m_{2}$ = 2,
$\varepsilon{\rm{=1}}{{\rm{0}}^{{\rm{-3}}}}$
TFCM [40] | maximum iteration = 100
|  $\lambda$ $\in$ {0, 0.1, 0.5, 1,5, 10, 50, 100, 300, 500, 1000}
TLEC [52] |  $\alpha$ = 1, $\beta$ = 2, $\delta$ = 10, $\beta_{1}$ = $\beta_{2}$ = 1
|  $\alpha$ = 1, $\beta$ = 2, $\delta$ = 10, $\gamma$ = 2,
$\varepsilon{\rm{=1}}{{\rm{0}}^{{\rm{-3}}}}$
TECM | maximum iteration = 100
|  $\lambda$ $\in$ {0, 0.1, 0.5, 1,5, 10, 50, 100, 300, 500, 1000}

The clustering performance was evaluated using three popular validity indices
[62]: accuracy ($ac$), rand index ($RI$), and normalized mutual information
($NMI$). Their definitions are as follows.

(1) $ac$

$ac=\frac{{{n_{cor}}}}{n},$

where $n$ is the total number of objects and $n_{cor}$ is the number of
objects that are clustered correctly. The greater the value of $ac$, the
better the clustering performance.

(2) $RI$

$RI=\frac{{f_{00}+f_{11}}}{{n(n-1)/2}},$

where $f_{00}$ and $f_{11}$ are the number of any two objects from the same
class belonging to different clusters and the same cluster, respectively. The
greater the value of $RI$, the better the clustering performance.

(3) $NMI$

$NMI(X;Y)=2\frac{{I(X;Y)}}{{H(X)+H(Y)}},$

where $H(\cdot)$ is the information entropy and $I(X;Y)$ is the mutual
information, with $X$ and $Y$ being the real label and clustering result,
respectively. The greater the value of $NMI$, the better the clustering
performance.

###  5.2 Synthetic dataset test

The TECM clustering performance in cases of insufficient target data and
sufficient yet contaminated target data is evaluated in this section. In
addition, two ambiguous datasets are used to illustrate the advantages of TECM
for detecting ambiguous objects.

####  5.2.1 Clustering performance evaluation

In this section, the TECM clustering performance is evaluated on synthetic
datasets in cases where the target data are insufficient or sufficient yet
contaminated, although an abundance of source data with similar data
distributions are available.

$\bullet$ Insufficient target data: Four synthetic datasets, S1-1, T1-1, S1-2,
and T1-2, were generated to verify three cases: $c_{s}=c_{t}$, $c_{s}>c_{t}$,
and $c_{s}<c_{t}$, where $c_{s}$ and $c_{t}$ are the number of clusters in the
source and target domains, respectively. The parameters for generating these
four synthetic datasets are listed in Table 3, and the distributions of these
datasets are displayed in Fig. 5.

Table 3: Parameters for generating synthetic datasets S1-1, T1-1, S1-2, and T1-2 S1-1 | mean | covariance | size
---|---|---|---
Cluster 1 | [0 0 0] | [3 0 0; 0 3 0; 0 0 3] | 200
Cluster 2 | [0 0 5] | [3 0 0; 0 3 0; 0 0 3] | 200
Cluster 3 | [0 5 0] | [3 0 0; 0 3 0; 0 0 3] | 200
T1-1 | mean | covariance | size
Cluster 1 | [0 0 0] | [4 0 0; 0 4 0; 0 0 4] | 20
Cluster 2 | [0 0 5] | [4 0 0; 0 4 0; 0 0 4] | 20
Cluster 3 | [0 5 0] | [4 0 0; 0 4 0; 0 0 4] | 20
S1-2 | mean | covariance | size
Cluster 1 | [0 0 0] | [3 0 0; 0 3 0; 0 0 3] | 200
Cluster 2 | [0 0 5] | [3 0 0; 0 3 0; 0 0 3] | 200
Cluster 3 | [0 5 0] | [3 0 0; 0 3 0; 0 0 3] | 200
Cluster 4 | [5 0 0] | [3 0 0; 0 3 0; 0 0 3] | 200
T1-2 | mean | covariance | size
Cluster 1 | [0 0 0] | [4 0 0; 0 4 0; 0 0 4] | 20
Cluster 2 | [0 0 5] | [4 0 0; 0 4 0; 0 0 4] | 20
Cluster 3 | [0 5 0] | [4 0 0; 0 4 0; 0 0 4] | 20
Cluster 4 | [5 0 0] | [4 0 0; 0 4 0; 0 0 4] | 20

(a) dataset S1-1 ($c_{s}$=3)

(b) dataset T1-1 ($c_{t}$=3)

(c) dataset S1-2 ($c_{s}$=4)

(d) dataset T1-2 ($c_{t}$=4)

Figure 5: Distributions of synthetic datasets S1-1, T1-1, S1-2, and T1-2.

$\bullet$ Sufficient yet contaminated target data: Two synthetic datasets,
T1-3 and T1-4, were generated to evaluate the TECM clustering performance in
the case of sufficient yet contaminated target data. Dataset T1-3 was
generated by adding zero-mean Gaussian noise with a standard deviation of five
to S1-1. Dataset T1-4 was generated by adding zero-mean Gaussian noise with a
standard deviation of three to S1-2. The distributions of T1-3 and T1-4 are
displayed in Fig. 6. Four datasets, S1-1, S1-2, T1-3, and T1-4, were used to
verify three cases: $c_{s}=c_{t}$, $c_{s}>c_{t}$, and $c_{s}<c_{t}$.

(a) dataset T1-3 ($c_{t}$=3)

(b) dataset T1-4 ($c_{t}$=4)

Figure 6: Distributions of synthetic datasets T1-3 and T1-4.

To minimize the influence of random initialization on the clustering
performance evaluation, each algorithm was executed ten times on each target
dataset, and then the mean value of each evaluation index was recorded. The
clustering performance evaluation results for insufficient target data and
sufficient yet contaminated target data are listed in Table 4 and Table 5,
respectively, based on the means of $ac$, $RI$, and $NMI$. In this comparison,
ECM used only target data for clustering. LSSMTC and CombKM clustered target
data and source data simultaneously, and TSC, TDEM, TFCM, TLEC, and TECM used
knowledge learned from the source data to guide the clustering of the target
data. Because LSSMTC, CombKM, TSC, TDEM, and TLEC require that the source and
target domains have the same number of clusters, parts of the experimental
results are absent.

Table 4: Clustering performance of algorithms for insufficient target data

Datasets | Metrics |  Algorithms
---|---|---
ECM | LSSMTC | CombKM | TSC | TDEM | TFCM | TLEC |  TECM
T1-1 (S1-1$\Rightarrow$T1-1) | mean_ac | 0.850 | 0.670 | 0.595 | 0.795 | 0.767 | 0.867 | 0.867 |  0.867
mean_RI | 0.825 | 0.694 | 0.682 | 0.779 | 0.763 | 0.842 | 0.842 |  0.842
mean_NMI | 0.595 | 0.337 | 0.334 | 0.471 | 0.482 | 0.626 | 0.626 |  0.626
T1-1 (S1-2$\Rightarrow$T1-1) | mean_ac | 0.750 | \ | \ | \ | \ | 0.850 | \ |  0.867
mean_RI | 0.757 | \ | \ | \ | \ | 0.825 | \ |  0.842
mean_NMI | 0.504 | \ | \ | \ | \ | 0.595 | \ |  0.626
T1-2 (S1-1$\Rightarrow$T1-2) | mean_ac | 0.738 | \ | \ | \ | \ | 0.800 | \ |  0.813
mean_RI | 0.788 | \ | \ | \ | \ | 0.831 | \ |  0.839
mean_NMI | 0.452 | \ | \ | \ | \ | 0.549 | \ |  0.559

  * *

For a $\Rightarrow$ b, a and b denote the source and target datasets,
respectively.

Table 5: Clustering performance of algorithms for sufficient yet contaminated
target data

Datasets | Metrics |  Algorithms
---|---|---
ECM | LSSMTC | CombKM | TSC | TDEM | TFCM | TLEC |  TECM
T1-3 (S1-1$\Rightarrow$T1-3) | mean_ac | 0.682 | 0.533 | 0.500 | 0.603 | 0.481 | 0.702 | 0.692 |  0.697
mean_RI | 0.683 | 0.613 | 0.603 | 0.636 | 0.525 | 0.696 | 0.688 |  0.693
mean_NMI | 0.253 | 0.140 | 0.123 | 0.168 | 0.109 | 0.274 | 0.258 |  0.267
T1-3 (S1-2$\Rightarrow$T1-3) | mean_ac | 0.498 | \ | \ | \ | \ | 0.522 | \ |  0.548
mean_RI | 0.599 | \ | \ | \ | \ | 0.604 | \ |  0.607
mean_NMI | 0.097 | \ | \ | \ | \ | 0.102 | \ |  0.105
T1-4 (S1-1$\Rightarrow$T1-4) | mean_ac | 0.619 | \ | \ | \ | \ | 0.635 | \ |  0.641
mean_RI | 0.721 | \ | \ | \ | \ | 0.728 | \ |  0.732
mean_NMI | 0.243 | \ | \ | \ | \ | 0.263 | \ |  0.264

  * *

For a $\Rightarrow$ b, a and b denote the source and target datasets,
respectively.

Based on the above six groups of experiments, the following conclusions can be
drawn.

(1) For all six groups of experiments, the TECM clustering performance was
superior to that of ECM, which indicates that the proposed TECM improved the
clustering performance with the assistance of the knowledge learned from the
source domain.

(2) In the majority of cases, the TECM clustering performance was superior to
that of the other representative multitask or transfer-clustering algorithms,
which demonstrates its competitive transfer capability.

(3) Compared with TFCM, the performance improvement of TECM was not evident
based on the indices of the hard partition. Nevertheless, the advantage of
TECM over TFCM was not only reflected in its hard-partition performance. For
certain cases with complex data distributions, TECM with credal partition can
provide a deeper insight into the data compared to TFCM. This point will be
discussed further in future studies.

####  5.2.2 Comparison between TECM and TFCM

In this section, two synthetic ambiguous datasets are used to illustrate the
superiority of TECM over TFCM in detecting ambiguous objects. The parameters
for generating the synthetic datasets T2-1 and T2-2 are listed in Table 6, and
the distributions of the datasets are displayed in Fig. 7. The number of
clusters in TECM and TFCM were set to two for T2-1 and four for T2-2; TECM had
three barycenters for T2-1 ($w_{1}$, $w_{2}$ and $w_{1}\cup w_{2}$) and 15
barycenters for T2-2 ($w_{1}$, $w_{2}$, $w_{1}\cup w_{2}$,…, $w_{1}\cup
w_{2}\cup w_{3}\cup w_{4}$).

Table 6: Parameters used to generate synthetic datasets S2-1, T2-1, S2-2, and
T2-2

S2-1 | mean | covariance | size
---|---|---|---
Cluster 1 | [0 0] | [1 0; 0 1] | 100
Cluster 2 | [1 0] | [1 0; 0 1] | 100
T2-1 | mean | covariance | size
Cluster 1 | [0 0.2] | [1 0; 0 1] | 10
Cluster 2 | [1 0.2] | [1 0; 0 1] | 10
S2-2 | mean | covariance | size
Cluster 1 | [0 0] | [1 0; 0 1] | 100
Cluster 2 | [1 0] | [1 0; 0 1] | 100
Cluster 3 | [0 1] | [1 0; 0 1] | 100
Cluster 4 | [1 1] | [1 0; 0 1] | 100
T2-2 | mean | covariance | size
Cluster 1 | [0.2 0.2] | [1 0; 0 1] | 30
Cluster 2 | [1.2 0.2] | [1 0; 0 1] | 30
Cluster 3 | [0.2 1.2] | [1 0; 0 1] | 30
Cluster 4 | [1.2 1.2] | [1 0; 0 1] | 30

(a) dataset S2-1

(b) dataset T2-1

(c) dataset S2-2

(d) dataset T2-2

Figure 7: Distributions of synthetic datasets S2-1, T2-1, S2-2, and T2-2.

Fig. 8 displays the clustering results of target data T2-1 and T2-2 obtained
by transferring the corresponding source data S2-1 and S2-2, respectively. It
can be observed that TFCM cannot cluster ambiguous objects well, whereas the
proposed TECM can discover those objects located at the boundaries of
different clusters and assign them to ambiguous sets of clusters. These
examples confirm that the TECM proposed in this study is more effective in
detecting ambiguous objects, and thus provides a deeper insight into the
distribution of the data. In addition, Table 7 indicates that TECM’s
clustering performance is superior or comparable to that of TFCM in terms of
$ac$, $RI$, and $NMI$.

(a) TFCM for dataset T2-1

(b) TECM for dataset T2-1

(c) TFCM for dataset T2-2

(d) TECM for dataset T2-2

Figure 8: Clustering results of T2-1 and T2-2 using TFCM and TECM. Table 7:
Comparison of clustering performance of TECM and TFCM on T2-1 and T2-2.

Algorithms |  T2-1 (S2-1$\Rightarrow$T2-1) |  T2-2 (S2-2$\Rightarrow$T2-2)
---|---|---
| mean_ac | mean_RI | mean_NMI | mean_ac | mean_RI |  mean_NMI
TFCM | 0.850 | 0.732 | 0.399 | 0.633 | 0.735 |  0.371
TECM | 0.850 | 0.732 | 0.399 | 0.700 | 0.766 |  0.428

  * *

For a $\Rightarrow$ b, a and b denote the source and target datasets,
respectively.

###  5.3 Texture image segmentation

In this section, five basic textures D6, D8, D11, D46, and D96 in the Brodatz
texture image dataset [63] are used to create the texture images employed in
this experiment; these texture images are then resized to 90 by 90 pixels. In
addition, Gaussian noise is added to the target images. Fig. 9 (a) displays
the source image and Figs. 9 (b)–9 (g) display the three types of target
images in our experiments. Specifically, Figs. 9 (b) and 9 (c) have the same
texture number as that in Fig. 9 (a) but have an inconsistent data
distribution. Figs. 9 (d) and 9 (e) have smaller texture numbers than those in
Fig. 9 (a)). Figs. 9 (f) and 9 (g) have larger texture numbers than those in
Fig. 9 (a)). In addition, the target images had different noise levels. The
number of textures and noise levels in each image are given in brackets. The
texture features of the images were obtained using the Gabor filter proposed
in [64]. The ideal segmentation results for the target images are displayed in
Fig. 10. In each subgraph, small squares of the same color represent the same
texture.

(a) S3-1 (K=4, $\sigma$=0)

(b) T3-1 (K=4, $\sigma$=0.05)

(c) T3-2 (K=4, $\sigma$=0.1)

(d) T3-3 (K=3, $\sigma$=0.05)

(e) T3-4 (K=3, $\sigma$=0.05)

(f) T3-5 (K=5, $\sigma$=0.05)

(g) T3-6 (K=5, $\sigma$=0.05)

Figure 9: Texture images used in the experiment.

(a) T3-1

(b) T3-2

(c) T3-3

(d) T3-4

(e) T3-5

(f) T3-6

Figure 10: Ideal segmentation of target texture images.

Fig. 11–Fig. 16 display the segmentation results for each target image using
the different algorithms. In addition, Table 8 provides the quantitative
scores of $ac$, $RI$, and $NMI$ obtained using the different algorithms. For
the same reasons as those indicated in Section 5.2, the results of LSSMTC,
CombKM, TSC, TEDM, and TLEC for certain datasets are absent. Based on the
above results, the following conclusions can be drawn.

1) Owing to the existence of noise, ECM could not achieve acceptable texture
segmentation performance on noisy target images. Conversely, benefiting from
the reference information across domains, TECM could obtain a relatively
superior performance compared to ECM. This demonstrates the effectiveness of
transfer clustering.

2) TECM and TFCM could obtain relatively superior performance compared to the
other multitask and transfer-clustering algorithms. Owing to the existence of
noise and differences in data distribution between the two domains, raw source
data were unsuitable for direct use in the clustering of the target data.
Therefore, LSSMTC and CombKM could easily encounter negative effects
unexpectedly between tasks, rather than simultaneously achieving performance
improvements.

3) TECM and TFCM were similar in terms of the clustering indices of the hard
partition. However, it can be observed from the experimental results that in
the cases where TFCM was superior, the clustering performance gap between TFCM
and TECM was small. However, in the cases where TECM was superior, the
clustering performance gap between TECM and TFCM was large, which indicates
that the proposed TECM was more powerful than TFCM overall. In addition, TECM
had an advantage over TFCM in that it provided a deeper insight into the data
when the data distribution was complex, as demonstrated in Section 5.2.2.

4) Regardless of whether the number of clusters in the source domain was equal
to, greater than, or less than the number of clusters in the target domain,
the TECM clustering performance always improved in comparison with ECM.
Therefore, if the structure of the source data is partially similar to that of
the target data, the learned knowledge from the source data will be beneficial
for clustering the target data.

(a) ECM

(b) LSSMTC

(c) CombKM

(d) TSC

(e) TDEM

(f) TFCM

(g) TLEC

(h) TECM

Figure 11: Segmentation results of T3-1 obtained by eight clustering
algorithms.

(a) ECM

(b) LSSMTC

(c) CombKM

(d) TSC

(e) TDEM

(f) TFCM

(g) TLEC

(h) TECM

Figure 12: Segmentation results of T3-2 obtained by eight clustering
algorithms.

(a) ECM

(b) TFCM

(c) TECM

Figure 13: Segmentation results of T3-3 obtained by three clustering
algorithms.

(a) ECM

(b) TFCM

(c) TECM

Figure 14: Segmentation results of T3-4 obtained by three clustering
algorithms.

(a) ECM

(b) TFCM

(c) TECM

Figure 15: Segmentation results of T3-5 obtained by three clustering
algorithms.

(a) ECM

(b) TFCM

(c) TECM

Figure 16: Segmentation results of T3-6 obtained by three clustering
algorithms. Table 8: Clustering performance comparisons of six algorithms on
T3-1 to T3-6

Datasets | Metrics |  Algorithms
---|---|---
ECM | LSSMTC | CombKM | TSC | TDEM | TFCM | TLEC |  TECM
T3-1 (S3-1$\Rightarrow$T3-1) | mean_ac | 0.705 | 0.664 | 0.646 | 0.690 | 0.805 | 0.793 | 0.729 |  0.836
mean_RI | 0.815 | 0.751 | 0.749 | 0.721 | 0.822 | 0.818 | 0.764 |  0.865
mean_NMI | 0.536 | 0.472 | 0.443 | 0.380 | 0.586 | 0.579 | 0.480 |  0.629
T3-2 (S3-1$\Rightarrow$T3-2) | mean_ac | 0.722 | 0.604 | 0.561 | 0.677 | 0.718 | 0.735 | 0.685 |  0.742
mean_RI | 0.786 | 0.726 | 0.715 | 0.723 | 0.770 | 0.794 | 0.733 |  0.828
mean_NMI | 0.463 | 0.401 | 0.368 | 0.406 | 0.417 | 0.480 | 0.424 |  0.503
T3-3 (S3-1$\Rightarrow$T3-3) | mean_ac | 0.774 | \ | \ | \ | \ | 0.918 | \ |  0.916
mean_RI | 0.785 | \ | \ | \ | \ | 0.895 | \ |  0.901
mean_NMI | 0.550 | \ | \ | \ | \ | 0.734 | \ |  0.700
T3-4 (S3-1$\Rightarrow$T3-4) | mean_ac | 0.814 | \ | \ | \ | \ | 0.814 | \ |  0.840
mean_RI | 0.806 | \ | \ | \ | \ | 0.806 | \ |  0.830
mean_NMI | 0.566 | \ | \ | \ | \ | 0.566 | \ |  0.564
T3-5 (S3-1$\Rightarrow$T3-5) | mean_ac | 0.800 | \ | \ | \ | \ | 0.879 | \ |  0.877
mean_RI | 0.875 | \ | \ | \ | \ | 0.916 | \ |  0.916
mean_NMI | 0.671 | \ | \ | \ | \ | 0.755 | \ |  0.738
T3-6 (S3-1$\Rightarrow$T3-6) | mean_ac | 0.874 | \ | \ | \ | \ | 0.902 | \ |  0.890
mean_RI | 0.910 | \ | \ | \ | \ | 0.925 | \ |  0.920
mean_NMI | 0.707 | \ | \ | \ | \ | 0.751 | \ |  0.737

  * *

For a $\Rightarrow$ b, a and b denote the source and target datasets,
respectively.

###  5.4 Parameter analysis

The balance coefficient of transfer learning $\lambda$ is an important
parameter in the TECM algorithm, which significantly influences the algorithm
performance. In this section, the influence of $\lambda$ on the TECM
clustering performance is discussed. Here, the synthesis datasets described in
Table 3 were used to study the variation in the algorithm performance with
$\lambda$. The performance of the algorithm was evaluated using $ac$.

Fig. 17 indicates that the TECM clustering performance first improves and then
decreases with an increase in $\lambda$. When the value of $\lambda$ is
relatively small, the clustering performance is not significantly improved
compared to ECM because the transferred knowledge has a small role. When the
value of $\lambda$ is relatively large, the clustering performance is even
worse than that of ECM because the clustering is completely dependent on the
knowledge learned from the source data, which results in a negative transfer.
In practice, the grid search strategy can be used to determine an effective
value of $\lambda$ by optimizing an internal clustering index.

Figure 17: TECM clustering performance with different values of $\lambda$.

##  6 Conclusions

In this study, a TECM clustering algorithm was proposed to improve the
clustering performance of the target data with the help of barycenters learned
from the source data for situations where target data are insufficient or
contaminated. The proposed algorithm is applicable to situations where the
source and target domains have the same or different numbers of clusters. In
synthetic and real dataset experiments, the proposed TECM achieved
approximately 5$\%$ and 7$\%$ performance improvement over ECM for
insufficient and contaminated target data on average, respectively. In
addition, compared with other related multitask or transfer-clustering
algorithms, the proposed TECM performed the best overall.

In the future, this research topic can be further explored by integrating TECM
with deep learning such that the clustering algorithm can be better applied to
raw data such as images and texts.

## Acknowledgments

This work was funded by the National Natural Science Foundation of China
(Grant Nos. 62171386, 61801386, and 61790552), Key Research and Development
Program in Shaanxi Province of China (Grant No. 2022GY-081), and China
Postdoctoral Science Foundation (Grant Nos. 2019M653743 and 2020T130538).

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