# Partially latent factors based multi-view subspace learning

Run-kun Lu<EMAIL_ADDRESS>Jian-wei Liu<EMAIL_ADDRESS>Ze-yu Liu
<EMAIL_ADDRESS>Jin-zhong Chen<EMAIL_ADDRESS>China
Special Equipment Inspection and Research Institute, Beijing 100029, China
Department of Automation, College of Information Science and Engineering,
China University of Petroleum, Beijing, 260 Mailbox, Changping District,
Beijing 102249, China

###### Abstract

Multi-view subspace clustering always performs well in high-dimensional data
analysis, but is sensitive to the quality of data representation. To this end,
a two stage fusion strategy is proposed to embed representation learning into
the process of multi-view subspace clustering. This paper first propose a
novel matrix factorization method that can separate the coupling consistent
and complementary information from observations of multiple views. Based on
the obtained latent representations, we further propose two subspace
clustering strategies: feature-level fusion and subspace-level hierarchical
strategy. Feature-level method concatenates all kinds of latent
representations from multiple views, and the original problem therefore
degenerates to a single-view subspace clustering process. Subspace-level
hierarchical method performs different self-expressive reconstruction
processes on the corresponding complementary and consistent latent
representations coming from each view, i.e. the prior constraints imposed on
different types of subspace representations are related to the appropriate
input factors. Finally, extensive experimental results on real-world datasets
demonstrate the superiority of our proposed methods by comparing against some
state-of-the-art subspace clustering algorithms.

###### keywords:

multi-view subspace clustering , matrix factorization , representation
learning

††journal: journalt1t1footnotetext: This work was supported by the Science
Foundation of China University of Petroleum, Beijing (No.
2462020YXZZ023).t2t2footnotetext: We note that a shorter conference version of
this paper appeared in International Joint Conference on Neural Networks
(2021). Our initial conference paper only propose the subspace-level
hierarchical method MVSC-CSLFS. This manuscript propose another feature-level
fusion algorithm MVSC-CSLF. Besides, the proposed algorithms are analyzed and
described in more details, and the performances are verified on more datasets.

##  1 Introduction

In real-world applications, an object can be observed from multiple
perspectives, which are denoted as views or modalities. In particular, readers
acquire information from the title, text, and picture while reading the
newspaper; or multi-feature learning always generates multiple kinds of
features using different approaches, e.g., extracting diverse image features
using LBP [1], HOG [2], and SIFT [3] operators. In contrast to typical machine
learning approaches, multi-view learning is expert in modeling the
relationships between views or modalities [4]. A fundamental assumption behind
such a paradigm is that different views share some semantics called consistent
or consensus information, and each view also has unique semantic called
complementary or view-specific information [5]. Obviously, the above
assumptions conform to real-world physical law, e.g., each camera in a multi-
camera system has a shared field of view and its own unique viewpoint.
Therefore, multi-view learning could leverage both consistency and
complementarity to explore more compact and complete information from multiple
observations.

Subspace clustering is one of the most attractive research topics in the field
of multi-view learning, especially for algorithms based on self-expressiveness
[6, 7, 8, 9, 10, 11, 12, 13, 14], which state each data point in a union of
subspaces can be represented by a linear combination of all other data points
[15]. Existing multi-view subspace clustering usually focus on two
perspectives: the prior constraint on subspace representation and the
construction method of latent representation. To the former, different effects
can be obtained by introducing various prior constrains on subspace
representation, such as [9] imposes sparse and low rank constraints on
corresponding subspace representations simultaneously, then fuses all obtained
coefficient matrices with an average operation to construct the final
adjacency matrix; [13] integrates multiple coefficient matrices into a common
subspace representation through learning with Laplacian regularization. In
addition, for the construction method of latent representation, researchers
often attempt to mine the underlying complementary and consistent information
between views to obtain more complete and non-redundant latent
representations, and thus improving performance of the learning task. For
instance, [16] discovers a latent embedding representation involved
complementary information from multiple views; and [14] first concatenates all
views together and handles the view-specific corruptions with $L_{2,1}$ norm
so that the model can leverage the consistent semantics to improve the
clustering performance. Compared to learning compact and comprehensive latent
representation from multiple views, most algorithms are more concerned with
the prior norms and subspace representation learning process. However, self-
expressive subspace clustering requires the data should be well distributed
inside the subspace [15], and it is difficult to satisfy the condition with
real-world datasets. Therefore, it makes sense to introduce representation
learning in the process of multi-view subspace clustering, which not only
provides a good representation for subspace clustering, but also can utilize
the specific task to guide the algorithm to explore the underlying semantics
from multiple observations.

Motivated by this, we propose a two-stage strategy that can link
representation learning with specific task (multi-view clustering). Firstly, a
novel matrix factorization algorithm is introduced to separate the coupling
consistent and complementary information from multiple observations. Among
them, the consistent one involves the common information shared by all views,
and complementary one is the unique view-specific information corresponding to
each view. And based on this, we further propose two self-expressive learning
methods: Multi-View Subspace Clustering with Consistent and view-Specific
Latent Factors (MVSC-CSLF), and Multi-View Subspace Clustering with Consistent
and view-Specific Latent Factors and Subspaces (MVSC-CSLFS). The first method
MVSC-CSLF concatenates the learned multiple representations and applies it to
least squares regression subspace clustering algorithm [17, 18], which
indicates that competitive clustering performance can be obtained even using a
simple clustering strategy with a good representation. And the second
algorithm MVSC-CSLFS imposes various prior constraints on different kinds of
subspace representations, i.e., the self-expressive reconstruction task guides
the matrix factorization process to separate the coupled consistent and
complementary information, and the appropriate prior terms imposed on them
derive better clustering performance. Finally, adaptive weights corresponding
to multiple views are introduced to the learning process, which improves the
model’s ability to accommodate noisy or unreliable views. Extensive
experiments on real-world datasets demonstrate that even the simple method
MVSC-CSLF outperforms than many baseline algorithms, and with more suitable
subspace clustering strategy, MVSC-CSLFS has achieved superior performance on
most datasets.

Our contributions in this study are summarized as follows:

(1) Our proposed matrix factorization approach helps MVSC-CSLF and MVSC-CSLFS
to obtain more compact and comprehensive latent representations by considering
consistent and complementary information of multiple views, simultaneously. In
particular, the method could separate the coupled consistent and complementary
information to make latent representations have more explicit semantics.

(2) MVSC-CSLFS regularizes subspace representations with different norms
according to the types of learned semantics, which makes the appropriate prior
constraint applied on suitable subspace representation.

(3) Our proposed methods both introduce dynamic weights learning process for
each view, which eliminates the influence of unreliable views.

The remainder of this paper is organized as follows: section 2 introduces the
related work and notation definitions of this study; section 3 first
elaborates our proposed MVSC-CSLF and MVSC-CSLFS, then introduces their
optimization method in details; section 4 gives detailed introduction and
discussion of experimental results; section 5 concludes this study and
discusses some future works.

##  2 Related Work and Notations

###  2.1 Related work

Self-expressive subspace clustering has several advantages over traditional
subspace clustering methods:

(1) the subspace dimensionality does not need to be set in advance;

(2) the global geometric information of the data can be exploited;

(3) the computational overhead is relatively small;

(4) the clustering accuracy can be theoretically guaranteed under certain
conditions.
And the method has been widely used in real-world applications, such as the
Internet-of-Things (IoT) [19], human motion segmentation [20, 21], and face
recognition [22, 23, 24]. In particular, Sparse Subspace Clustering (SSC) [25]
and Low-Rank Representation (LRR) are foundational studies, which provide a
standard learning paradigm:

$\begin{array}[]{l}\mathop{\min}\limits_{\mathbf{C}}R\left(\mathbf{C}\right)\\\
s.t.\;\mathbf{X}=\mathbf{XC},\;\mathbf{C}\in\Omega,\\\ \end{array}$

where $\mathbf{C}$ is the subspace representation, and $R\left(\cdot\right)$
is the prior constraint on it. Then the adjacency matrix can be obtained using
the following formula:

$\frac{\left|\mathbf{C}\right|+\left|\mathbf{C}^{\text{T}}\right|}{2},$

and one can substitutes it into standard spectral clustering to obtain
clustering result. Based on the above process, plenty of improved algorithms
for self-expressive subspace clustering have been proposed recent years [26,
27, 28, 29, 30, 31].

Besides, multi-view subspace clustering has gradually become an important
method because it can model the relationships between views, and explore the
underlying common and view-specif information of subspace representations
corresponding to different views. Specifically, [9, 32] both impose low-rank
and sparse constraints on the subspace representations, simultaneously. [8] is
the first multi-view method that combines representation learning and subspace
clustering. The algorithm assumes that all the observations (views) originate
from one common latent representation, and performs the self-expressive
reconstruction on it. Besides, [14, 33] impose Laplacian constraint on
subspace representations to preserve local manifold structures of each view.
Most standard methods cannot deal with highly nonlinear data, therefore some
kernel based methods are employed [7, 9, 32, 13, 34], but they also encounter
the dilemma of kernel selection. Inspired by [35], researchers begin to
explore the network structure of deep multi-view subspace clustering [36, 37,
12], and [38] further introduces attention mechanism in network to derive
dynamic weights corresponding multiple views.

###  2.2 Notations

In this paper, we define matrices and column vectors using bold uppercase and
lowercase characters, respectively, and scalars are denoted by italics and not
bold characters. Let $\left[\bf{A},\bf{B}\right]$ and
$\left[\bf{A};\bf{B}\right]$ denote the horizontal and vertical concatenation
between matrices $\bf{A}$ and $\bf{B}$. Then let $diag({\bf{X}})$ denotes the
diagonal elements of matrix ${\bf{X}}\in\mathbb{R}^{I\times J}$, and the norms
we used throughout this paper are summarized as follows:
2-norm: ${\left\|{\bf{x}}\right\|_{2}}=\sqrt{\sum\limits_{i=1}^{I}{{x_{i}}}}$;
Frobenius norm:
${\left\|{\bf{X}}\right\|_{F}}=\sqrt{\sum\limits_{i=1}^{I}{\sum\limits_{j=1}^{J}{x_{i,j}^{2}}}}$;
$L_{2,1}$-norm:
${\left\|{\bf{X}}\right\|_{2,1}}=\sum\limits_{j=1}^{J}{\sqrt{\sum\limits_{i=1}^{I}{x_{i,j}^{2}}}}$;
Nuclear norm:
${\left\|{\bf{X}}\right\|_{*}}=\sum\limits_{i}^{r}{{\sigma_{i}}}$, where
${\sigma_{i}}$ is $i$-th non-zero singular value, and $r$ is the total number
of non-zero singular values ($r\leq\min({I,J})$).

Since a number of symbols need to be defined to help the description of our
proposed algorithms, we have summarized all commonly used symbols in Table 1
for the convenience of reference.

Table 1: Notations Notation | Size |  Description
---|---|---
${{\bf{X}}^{v}}$ | ${{{M^{v}}\times N}}$ |  ${{\bf{X}}^{v}}=\left[{{\bf{x}}_{1}^{v};\cdots;{\bf{x}}_{N}^{v}}\right]$, where ${\bf{x}}_{n}^{v}\in{R^{{}^{{M^{v}}}}}$ is n-th instance of v-th view.
${\bf{P}}_{s}^{v}$ | ${{{M^{v}}\times{K_{s}}}}$ |  Projection matrix that projects $v$-th view’s specific latent representation into the corresponding observation space.
${\bf{P}}_{c}^{v}$ | ${{{M^{v}}\times{K_{c}}}}$ |  Projection matrix that projects $v$-th view’s consistent latent representation into the corresponding observation space.
${\bf{H}}_{s}^{v}$ | ${{{K_{s}}\times N}}$ |  Specific latent representation of $v$-th view.
${{\bf{H}}_{c}}$ | ${{{K_{c}}\times N}}$ |  Consistent latent representation shared by all views.
${\bf{H}}$ | ${{({K_{s}}\times V+{K_{c}})\times N}}$ |  Joint latent representation concatenated by $\left\\{{{{\bf{H}}_{s}^{v}}}\right\\}_{v=1}^{V}$ and ${\bf{H}}_{c}$ following the rule: $\left[{{\bf{H}}_{s}^{1};\cdots;{\bf{H}}_{s}^{V};{{\bf{H}}_{c}}}\right]$.
${\bf{Z}}$ | ${{N\times N}}$ |  self-expressive matrix for MVSC-CSLF.
${{\bf{Z}}^{v}}$ | ${{N\times N}}$ |  View-specific self-expressive matrix of $v$-th view for MVSC-CSLFS.
${{\bf{Z}}_{c}}$ | ${{N\times N}}$ |  Consistent self-expressive matrix shared by all views for MVSC-CSLFS.
${\bf{D}}$, ${{\bf{D}}^{v}}$, ${{\bf{D}}_{c}}$ | ${{N\times N}}$ |  Auxiliary variable of ${\bf{Z}}$, ${{\bf{Z}}^{v}}$, and ${{\bf{Z}}_{c}}$.
${\bf{E}}_{r}^{v}$ | ${{{M^{v}}\times N}}$ |  Observation recovery error of $v$-th view.
${{\bf{E}}_{s}}$ | ${{({K_{s}}\times V+{K_{c}})\times N}}$ |  self-expressive error for MVSC-CSLF.
${\bf{E}}_{s}^{v}$ | ${{{K_{s}}\times N}}$ |  View-specific self-expressive error of $v$-th view for MVSC-CSLFS.
${\bf{E}}_{s}^{c}$ | ${{{K_{c}}\times N}}$ |  Consistent self-expressive error shared by all views for MVSC-CSLFS.
${\boldsymbol{\pi}}$ | $1\times V$ |  Dynamic weights collection corresponding to the observation recovery process for MVSC-CSLF.
${{\boldsymbol{\pi}}_{1}}$ | $1\times V$ |  Dynamic weights collection corresponding to the observation recovery process for MVSC-CSLFS.
${{\boldsymbol{\pi}}_{2}}$ | $1\times(V+1)$ |  Dynamic weights collection corresponding to the self-expressive reconstruction process for MVSC-CSLFS.

##  3 Proposed Methods

Suppose $\left\\{{{{\bf{X}}^{v}}\in{{\mathbb{R}}^{{M^{v}}\times
N}}}\right\\}_{v=1}^{V}$ is the collection of observational data comprising
$V$ views, where $M^{v}$ and $N$ represent instance dimension and number,
respectively. We assume that observable views mainly originate from two parts:
the consistent latent representation ${\bf{H}}_{c}\in{\mathbb{R}}^{K_{c}\times
N}$ shared by all views; the view-specific latent representation
${\bf{H}}_{s}^{v}\in{\mathbb{R}}^{K_{s}\times N}$ related only to $v$-th view.
Then suppose that ${\bf{X}}^{v}$ can be recovered by ${\bf{H}}_{s}^{v}$ and
${\bf{H}}_{c}$ according to the following rule:

${{\bf{X}}^{v}}={\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}+{\bf{P}}_{c}^{v}{{\bf{H}}_{c}},$

where ${\bf{P}}_{s}^{v}\in{\mathbb{R}}^{M^{v}\times K_{s}}$ and
${\bf{P}}_{c}^{v}\in{\mathbb{R}}^{M^{v}\times K_{c}}$ are projections
corresponding to view-specific and consistent latent representations,
respectively. Matrix factorization and fusion process of multiple views is
illustrated in Fig. 1.

Figure 1: Illustration of matrix factorization and fusion: there are three
views in this example, and each of them approximately equals to the product of
projection matrix and latent representation matrix. Both of the matrices can
be divided into view-specific and consistent parts. Take view 1 as an example,
we concatenate ${\bf{P}}_{s}^{1}$ and ${\bf{P}}_{c}^{1}$ horizontally to
construct projection matrix, and concatenate ${\bf{H}}_{s}^{1}$ and
${\bf{H}}_{c}$ vertically to construct latent representation. And we can
finally obtain a unified joint latent representation $\bf{H}$ through vertical
concatenation of ${\bf{H}}_{s}^{1}$, ${\bf{H}}_{s}^{2}$, ${\bf{H}}_{s}^{3}$,
and ${\bf{H}}_{c}$.

We construct a pipeline that combines latent representation learning process
with subspace clustering, so that the specific task could guide the algorithm
to extract feature, and the representation learning process also provides
subspace clustering with compact and comprehensive factors. Therefore, our
proposed two algorithms all mainly contain two learning goals, one is
observation information recovery, the other is self-expressive reconstruction:

$\mathcal{I}({{{\bf{X}}^{v}},{\bf{H}}_{s}^{v},{{\bf{H}}_{c}};{\bf{\Theta}}_{1}})+\lambda\mathcal{S}({{\bf{H}}_{s}^{v},{{\bf{H}}_{c}};{\bf{\Theta}}_{2}}),$
(1)

where ${\bf{\Theta}}_{1}$ and ${\bf{\Theta}}_{2}$ are the underlying
parameters related to the two terms, and $\lambda$ is the trade-off parameter
to balance multiple terms. The first term $\mathcal{I}({\cdot,\cdot})$ of
equation (1) is to measure the recovery ability from latent representations to
observations, and the specific form in this paper is the same for two
algorithms we proposed, which is formulated as follow:

$\begin{array}[]{l}\mathcal{I}({{{\bf{X}}^{v}},{\bf{H}}_{s}^{v},{{\bf{H}}_{c}};{\bf{\Theta}}_{1}})=\mathop{\min}\limits_{{\bf{\Theta}}_{1},{\bf{H}}_{s}^{v},{{\bf{H}}_{c}}}\sum\limits_{v=1}^{V}{{{\left\|{{\bf{E}}_{r}^{v}}\right\|}_{{\rm{2,1}}}}}\\\
s.t.\;{({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v}={\bf{I}},\;{({{\bf{P}}_{c}^{v}})^{\rm
T}}{\bf{P}}_{c}^{v}={\bf{I}},\;{{\bf{X}}^{v}}={\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}+{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}+{\bf{E}}_{r}^{v},\\\
\end{array}$ (2)

where ${\bf{E}}_{r}^{v}\in\mathbb{R}^{M^{v}\times N}$ is the recovery error,
and
${\bf{\Theta}}_{1}=\\{{{\bf{P}}_{s}^{v}},{{\bf{P}}_{c}^{v}},{\bf{E}}_{r}^{v}\\}$.
${{\bf{P}}_{s}^{v}}$ and ${{\bf{P}}_{c}^{v}}$ are constrained to prevent the
value of ${\bf{H}}_{s}^{v}$ and ${{\bf{H}}_{c}}$ from being imposed to 0.

The second term of equation (1) measures the self-expressive ability. The
strategies employed by our proposed two methods are different, and we will
discuss the details in the next two subsections.

###  3.1 MVSC-CSLF

In this subsection, we propose a novel framework called Multi-View Subspace
Clustering with Consistent and view-Specific Latent Factors (MVSC-CSLF).
Firstly, refer to equation (2) and Fig. 1, a joint latent representation can
be defined as the vertical concatenation of view-specific and consistent
representations: ${\bf{H}}\in\mathbb{R}^{K\times
N}=\left[{{\bf{H}}_{s}^{1};\cdots;{\bf{H}}_{s}^{V};{{\bf{H}}_{c}}}\right]$,
where $K=K_{s}\times V+K_{c}$. Obviously, the representation concludes
multiple views’ complementary and consistent information, simultaneously.
Finally, Self-expressive reconstruction is applied on $\bf{H}$. We state such
a learning process as feature-level fusion strategy, i.e. fusing different
factors as a joint latent representation, and make the downstream subspace
clustering task degenerate to a single-view problem. Consequently, the second
term of equation (1) is formulated as follow:

$\begin{array}[]{l}\mathcal{S}({{\bf{H}}_{s}^{v},{{\bf{H}}_{c}};{\bf{\Theta}}_{2}})=\mathop{\min}\limits_{{\bf{\Theta}}_{2},{\bf{H}}_{s}^{v},{{\bf{H}}_{c}}}{\lambda_{\rm{1}}}{\left\|{{{\bf{E}}_{s}}}\right\|_{{\rm{2,1}}}}+\frac{{{\lambda_{\rm{2}}}}}{{\rm{2}}}\left\|{\bf{Z}}\right\|_{F}^{2}\\\
s.t.\;{\bf{H}}={\bf{HZ}}+{{\bf{E}}_{s}},\;diag({\bf{Z}})=0,\end{array}$ (3)

where ${\bf{E}}_{s}$ is self-expressive error, $\bf{Z}$ is self-expressive
matrix, $\lambda_{1}$ and $\lambda_{2}$ are trade-off parameters, and
${\bf{\Theta}}_{2}=\left\\{{\bf{E}}_{s},{\bf{Z}}\right\\}$. Then formulas (2)
and (3) are combined to construct a joint optimization problem by introducing
dynamic weight ${\boldsymbol{\pi}}$ and auxiliary variable $\bf{D}$:

$\small\begin{array}[]{l}\mathop{\min}\limits_{\bf{\Theta}}\sum\limits_{v=1}^{V}{{\pi^{v}}{{\left\|{{\bf{E}}_{r}^{v}}\right\|}_{{\rm{2,1}}}}}+{\lambda_{\rm{1}}}{\left\|{{{\bf{E}}_{s}}}\right\|_{{\rm{2,1}}}}+\frac{{{\lambda_{\rm{2}}}}}{{\rm{2}}}\left\|{\bf{D}}\right\|_{F}^{2}+\frac{{{\lambda_{3}}}}{2}\left\|{\boldsymbol{\pi}}\right\|_{2}^{2}\\\
s.t.\;{({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v}={\bf{I}},\;{({{\bf{P}}_{c}^{v}})^{\rm
T}}{\bf{P}}_{c}^{v}={\bf{I}},\\\
\;\;\;\;\;\;\;{{\bf{X}}^{v}}={\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}+{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}+{\bf{E}}_{r}^{v},\\\
\;\;\;\;\;\;\;{\bf{H}}={\bf{HZ}}+{{\bf{E}}_{s}},\\\
\;\;\;\;\;\;\;{\bf{D}}={\bf{Z}}-diag({\bf{Z}}),\\\
\;\;\;\;\;\;\;{\boldsymbol{\pi}}\geq
0,\;\sum\limits_{v=1}^{V}{{\pi^{v}}}=1,\end{array}$ (4)

where
${\bf{\Theta}}{\rm{=}}\left\\{{{\bf{E}}_{r}^{v},{{\bf{E}}_{s}},{\bf{P}}_{s}^{v},{\bf{P}}_{c}^{v},{\bf{H}}_{s}^{v},{{\bf{H}}_{c}},{\bf{D}},{\bf{Z}},{\boldsymbol{\pi}}}\right\\}$,
and $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ are trade-off parameters,
${\boldsymbol{\pi}}={\left\\{\pi^{v}\right\\}}_{v=1}^{V}$ weights the
importance of each view. The regularization on ${\boldsymbol{\pi}}$ is to
prevent the model from setting the weight value of the best view to 1. The
optimization process will be introduced in subsection 3.3 together with MVSC-
CSLFS.

###  3.2 MVSC-CSLFS

In this subsection,we further propose a improved framework based on MVSC-CSLF
named Multi-View Subspace Clustering with Consistent and view-Specific Latent
Factors and Subspaces (MVSC-CSLFS). Compare with MVSC-CSLF, a subspace-level
hierarchical strategy is employed to preform self-expressive reconstruction on
all views’ specific and consistent latent representations, respectively. Such
strategy imposes more appropriate prior constraints on the corresponding
representation with different attributes. Therefore, the second term of
equation (1) can be formulated as:

$\begin{array}[]{l}\mathcal{S}({{\bf{H}}_{s}^{v},{{\bf{H}}_{c}};{\bf{\Theta}}_{2}})=\mathop{\min}\limits_{{\bf{\Theta}}_{2},{\bf{H}}_{s}^{v},{{\bf{H}}_{c}}}{\lambda_{\rm{1}}}({\sum\limits_{v=1}^{V}{{{\left\|{{\bf{E}}_{s}^{v}}\right\|}_{2,1}}+{{\left\|{{\bf{E}}_{s}^{c}}\right\|}_{2,1}}}})+{\lambda_{2}}({\sum\limits_{v=1}^{V}{\frac{1}{2}\left\|{{{\bf{Z}}^{v}}}\right\|_{F}^{2}}+{{\left\|{{{\bf{Z}}_{c}}}\right\|}_{*}}})\\\
s.t.\;{\bf{H}}_{s}^{v}={\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}+{\bf{E}}_{s}^{v},\;{{\bf{H}}_{c}}={{\bf{H}}_{c}}{{\bf{Z}}_{c}}+{\bf{E}}_{s}^{c},\\\
\;\;\;\;\;\;\;diag({{{\bf{Z}}^{v}}})=0,\;diag({{{\bf{Z}}_{c}}})=0,\end{array}$
(5)

where ${{\bf{E}}_{s}^{v}}$ and ${{\bf{E}}_{s}^{c}}$ are reconstruction errors
related to ${\bf{H}}_{s}^{v}$ and ${\bf{H}}_{c}$, ${\bf{Z}}^{v}$ and
${\bf{Z}}_{c}$ are subspace representations related to ${\bf{H}}_{s}^{v}$ and
${\bf{H}}_{c}$, $\lambda_{1}$ and $\lambda_{2}$ are trade-off parameters, and
${\bf{\Theta}}_{2}=\left\\{{{\bf{E}}_{s}^{v}},{{\bf{E}}_{s}^{c}},{\bf{Z}}^{v},{\bf{Z}}_{c}\right\\}$.
Then formulas (2) and (5) are combined to construct a joint optimization
problem by introducing dynamic weights ${{\boldsymbol{\pi}}_{1}}$,
${{\boldsymbol{\pi}}_{2}}$, and auxiliary variable ${\bf{D}}^{v}$,
${\bf{D}}_{c}$:

$\small\begin{array}[]{l}\mathop{\min}\limits_{\bf{\Theta}}\sum\limits_{v=1}^{V}{\pi_{1}^{v}{{\left\|{{\bf{E}}_{r}^{v}}\right\|}_{2,1}}}+{\lambda_{\rm{1}}}({\sum\limits_{v=1}^{V}{\pi_{2}^{v}{{\left\|{{\bf{E}}_{s}^{v}}\right\|}_{2,1}}+\pi_{2}^{V+1}{{\left\|{{\bf{E}}_{s}^{c}}\right\|}_{2,1}}}})+{\lambda_{2}}({\sum\limits_{v=1}^{V}{\frac{{\pi_{2}^{v}}}{2}\left\|{{{\bf{D}}^{v}}}\right\|_{F}^{2}}+\pi_{2}^{V+1}{{\left\|{{{\bf{D}}_{c}}}\right\|}_{*}}})+\frac{{{\lambda_{3}}}}{2}({\left\|{{{\boldsymbol{\pi}}_{1}}}\right\|_{2}^{2}+\left\|{{{\boldsymbol{\pi}}_{2}}}\right\|_{2}^{2}})\\\
s.t.\;{({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v}={\bf{I}},\;{({{\bf{P}}_{c}^{v}})^{\rm
T}}{\bf{P}}_{c}^{v}={\bf{I}},\\\
\;\;\;\;\;\;\;{{\bf{X}}^{v}}={\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}+{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}+{\bf{E}}_{r}^{v},\\\
\;\;\;\;\;\;\;{\bf{H}}_{s}^{v}={\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}+{\bf{E}}_{s}^{v},\;{{\bf{H}}_{c}}={{\bf{H}}_{c}}{{\bf{Z}}_{c}}+{\bf{E}}_{s}^{c},\\\
\;\;\;\;\;\;\;{\bf{D}}^{v}={{{\bf{Z}}^{v}}}-diag({{{\bf{Z}}^{v}}}),\;{\bf{D}}_{c}={{{\bf{Z}}_{c}}}-diag({{{\bf{Z}}_{c}}}),\\\
\;\;\;\;\;\;\;{{\boldsymbol{\pi}}_{1}},{{\boldsymbol{\pi}}_{2}}\geq
0,\;\sum\limits_{v=1}^{V}{\pi_{1}^{v}}=1,\;\sum\limits_{v=1}^{V+1}{\pi_{2}^{v}}=1,\end{array}$
(6)

where
${\bf{\Theta}}{\rm{=}}\\{{\bf{E}}_{r}^{v},{\bf{E}}_{s}^{v},{\bf{E}}_{s}^{c},{\bf{P}}_{s}^{v},{\bf{P}}_{c}^{v},{\bf{H}}_{s}^{v},{{\bf{H}}_{c}},{\bf{D}}^{v},{\bf{D}}_{c},{{\bf{Z}}^{v}},{{\bf{Z}}_{c}},{{\boldsymbol{\pi}}_{1}}$
$,{\boldsymbol{\pi}}_{2}\\}$; $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$ are
trade-off parameters;
${\boldsymbol{\pi}}_{1}={\left\\{\pi_{1}^{v}\right\\}}_{v=1}^{V}$ weights the
recovery ability of each view;
${\boldsymbol{\pi}}_{2}={\left\\{\pi_{2}^{v}\right\\}}_{v=1}^{V+1}$ weights
the self-expressive ability of view-specific and consistent latent
representations from multiple views. Specifically, the regularization terms on
${{\boldsymbol{\pi}}_{1}}$ and ${\boldsymbol{\pi}}_{2}$ are to prevent the
model from setting the weight value of the best view to one. The optimization
process will be introduced in subsection 3.3 together with MVSC-CSLF.

###  3.3 Optimization

In this subsection, we will derive the updating formulas of MVSC-CSLF and
MVSC-CSLFS by constructing augmented lagrangian dual functions and using
alternating direction minimization strategy. Firstly, the lagrangian dual
forms for equations (4) and (6) are formulated as:

$\small\begin{array}[]{l}\mathop{\min}\limits_{\bf{\Theta}}\sum\limits_{v=1}^{V}{{\pi^{v}}{{\left\|{{\bf{E}}_{r}^{v}}\right\|}_{{\rm{2,1}}}}}+{\lambda_{\rm{1}}}{\left\|{{{\bf{E}}_{s}}}\right\|_{{\rm{2,1}}}}+\frac{{{\lambda_{\rm{2}}}}}{{\rm{2}}}\left\|{\bf{D}}\right\|_{F}^{2}+\frac{{{\lambda_{3}}}}{2}\left\|{\boldsymbol{\pi}}\right\|_{2}^{2}\\\
\;\;\;\;\;\;\;+\sum\limits_{v=1}^{V}\frac{\mu}{2}\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|_{F}^{2}\\\
\;\;\;\;\;\;\;+\frac{\mu}{2}\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{2}}+{\bf{H}}-{\bf{HZ}}-{{\bf{E}}_{s}}}\right\|_{F}^{2}\\\
\;\;\;\;\;\;\;+\frac{\mu}{2}\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+{\bf{D}}-{\bf{Z}}+diag({\bf{Z}})}\right\|_{F}^{2}\\\
\;\;\;\;\;\;\;-\sum\limits_{v=1}^{V}\frac{1}{{2\mu}}\left\|{{\bf{\Lambda}}_{1}^{v}}\right\|_{F}^{2}-\frac{1}{{2\mu}}\left\|{{{\bf{\Lambda}}_{2}}}\right\|_{F}^{2}-\frac{1}{{2\mu}}\left\|{{{\bf{\Lambda}}_{3}}}\right\|_{F}^{2}\\\
s.t.\;{({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v}={\bf{I}},\;{({{\bf{P}}_{c}^{v}})^{\rm
T}}{\bf{P}}_{c}^{v}={\bf{I}},\\\ \;\;\;\;\;\;\;{\boldsymbol{\pi}}\geq
0,\;\sum\limits_{v=1}^{V}{{\pi^{v}}}=1,\end{array}$ (7)

where ${\bf{\Lambda}}_{1}^{v}$, ${\bf{\Lambda}}_{2}$, and ${\bf{\Lambda}}_{3}$
are multipliers, $\mu$ is a positive penalty scalar.

$\small\begin{array}[]{l}\mathop{\min}\limits_{\bf{\Theta}}\sum\limits_{v=1}^{V}{\pi_{1}^{v}{{\left\|{{\bf{E}}_{r}^{v}}\right\|}_{2,1}}}+{\lambda_{\rm{1}}}({\sum\limits_{v=1}^{V}{\pi_{2}^{v}{{\left\|{{\bf{E}}_{s}^{v}}\right\|}_{2,1}}+\pi_{2}^{V+1}{{\left\|{{\bf{E}}_{s}^{c}}\right\|}_{2,1}}}})\\\
\;\;\;\;\;\;\;+{\lambda_{2}}({\sum\limits_{v=1}^{V}{\frac{{\pi_{2}^{v}}}{2}\left\|{{{\bf{D}}^{v}}}\right\|_{F}^{2}}+\pi_{2}^{V+1}{{\left\|{{{\bf{D}}_{c}}}\right\|}_{*}}})+\frac{{{\lambda_{3}}}}{2}({\left\|{{{\boldsymbol{\pi}}_{1}}}\right\|_{2}^{2}+\left\|{{{\boldsymbol{\pi}}_{2}}}\right\|_{2}^{2}})\\\
\;\;\;\;\;\;\;+\sum\limits_{v=1}^{V}{\frac{\mu}{2}\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|_{F}^{2}}\\\
\;\;\;\;\;\;\;+\sum\limits_{v=1}^{V}{\frac{\mu}{2}\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{2}^{v}+{\bf{H}}_{s}^{v}-{\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}-{\bf{E}}_{s}^{v}}\right\|_{F}^{2}}\\\
\;\;\;\;\;\;\;+\frac{\mu}{2}\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+{{\bf{H}}_{c}}-{{\bf{H}}_{c}}{{\bf{Z}}_{c}}-{\bf{E}}_{s}^{c}}\right\|_{F}^{2}\\\
\;\;\;\;\;\;\;+\sum\limits_{v=1}^{V}{\frac{\mu}{2}\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{4}^{v}+{{\bf{D}}^{v}}-{{\bf{Z}}^{v}}+diag({{{\bf{Z}}^{v}}})}\right\|_{F}^{2}}\\\
\;\;\;\;\;\;\;+\frac{\mu}{2}\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{5}}+{{\bf{D}}_{c}}-{{\bf{Z}}_{c}}+diag({{{\bf{Z}}_{c}}})}\right\|_{F}^{2}\\\
\;\;\;\;\;\;\;-\frac{1}{{2\mu}}(\sum\limits_{v=1}^{V}(\left\|{{\bf{\Lambda}}_{1}^{v}}\right\|_{F}^{2}+\left\|{{\bf{\Lambda}}_{2}^{v}}\right\|_{F}^{2}+\left\|{{\bf{\Lambda}}_{4}^{v}}\right\|_{F}^{2})+\left\|{{{\bf{\Lambda}}_{3}}}\right\|_{F}^{2}+\left\|{{{\bf{\Lambda}}_{5}}}\right\|_{F}^{2})\\\
s.t.\;{({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v}={\bf{I}},\;{({{\bf{P}}_{c}^{v}})^{\rm
T}}{\bf{P}}_{c}^{v}={\bf{I}},\\\
\;\;\;\;\;\;\;{{\boldsymbol{\pi}}_{1}},{{\boldsymbol{\pi}}_{2}}\geq
0,\;\sum\limits_{v=1}^{V}{\pi_{1}^{v}}=1,\;\sum\limits_{v=1}^{V+1}{\pi_{2}^{v}}=1,\end{array}$
(8)

where ${\bf{\Lambda}}_{1}^{v}$, ${\bf{\Lambda}}_{2}^{v}$,
${\bf{\Lambda}}_{3}$, ${\bf{\Lambda}}_{4}^{v}$ and ${\bf{\Lambda}}_{5}$ are
lagrangian multipliers, $\mu$ is a positive penalty scalar. We can decompose
the original problem into several sub-optimization problems, and derive their
updating rules, respectively.

####  3.3.1 ${\bf{P}}$ sub-problems

The sub-problems w.r.t ${\bf{P}}_{s}^{v}$ and ${\bf{P}}_{c}^{v}$ are the same
for MVSC-CSLF and MCSC-CSLFS, and they can be formulated as:

$\begin{array}[]{l}{\bf{P}}{{}_{s}^{v{\rm{*}}}}=\arg{\min_{{\bf{P}}_{s}^{v}}}\left\|{{{({\bf{Y}}_{1})}^{\rm
T}}-{{({{\bf{H}}_{s}^{v}})}^{\rm T}}{{({{\bf{P}}_{s}^{v}})}^{\rm
T}}}\right\|_{F}^{2}\\\ s.t.\;{({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v}={\bf{I}},\end{array}$ (9)

where
${\bf{Y}}_{1}=\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}$.

$\begin{array}[]{l}{\bf{P}}{{}_{c}^{v{\rm{*}}}}=\arg{\min_{{\bf{P}}_{c}^{v}}}\left\|{{{({{{\bf{Y}}_{2}}})}^{\rm
T}}-{{({{{\bf{H}}_{c}}})}^{\rm T}}{{({{\bf{P}}_{c}^{v}})}^{\rm
T}}}\right\|_{F}^{2}\\\ s.t.\;{({{\bf{P}}_{c}^{v}})^{\rm
T}}{\bf{P}}_{c}^{v}={\bf{I}},\end{array}$ (10)

where
${{\bf{Y}}_{2}}=\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{E}}_{r}^{v}$.
Equation (9) and (10) perform the Wahba problem [39], and according to [40],
such problem can be solved by introducing Theorem 1.
Theorem 1. Given the optimal problem as follow:

$\begin{array}[]{l}\arg{\min_{\bf{X}}}\left\|{{\bf{Y}}-{\bf{WX}}}\right\|_{F}^{2}\\\
s.t.\;{{\bf{X}}^{\rm T}}{\bf{X}}={\bf{X}}{{\bf{X}}^{\rm
T}}={\bf{I}}\end{array}$

The optimal solution is ${\bf{X}}={\bf{U}}{{\bf{V}}^{\rm T}}$, where
${\bf{U}}$ and ${\bf{V}}$ are left and right singular value matrices of
${{\bf{W}}^{\rm T}}{\bf{Y}}$.

Based on Theorem 1, the optimal solutions w.r.t equation (9) and (10) are:

${({{\bf{P}}{{{}_{s}^{v*}}}})^{\rm T}}={\bf{U}}_{s}{{\bf{V}}_{s}^{\rm T}},$
(11) ${({{\bf{P}}{{{}_{c}^{v*}}}})^{\rm T}}={\bf{U}}_{c}{{\bf{V}}_{c}^{\rm
T}},$ (12)

where ${\bf{U}}_{s}$ and ${\bf{V}}_{s}$ are the left and right singular value
matrices of ${{\bf{H}}_{s}^{v}}{({\bf{Y}}_{1})^{\rm T}}$, ${\bf{U}}_{c}$ and
${\bf{V}}_{c}$ are the left and right singular value matrices of
${{\bf{H}}_{c}}{({\bf{Y}}_{2})^{\rm T}}$.

####  3.3.2 ${\bf{H}}_{s}^{v}$ sub-problems

The sub-problems w.r.t ${\bf{H}}_{s}^{v}$ are different for MVSC-CSLF and
MVSC-CSLFS, which can be formulated as follows, respectively:

$\small\begin{array}[]{l}{\bf{H}}{{}_{s}^{v*}}=\arg{\min_{{\bf{H}}_{s}^{v}}}\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|_{F}^{2}+\left\|{\frac{1}{\mu}{{{\bf{\bar{\Lambda}}}}_{2}}+{\bf{H}}_{s}^{v}-{\bf{H}}_{s}^{v}{\bf{Z}}-{{{\bf{\bar{E}}}}_{s}}}\right\|_{F}^{2},\end{array}$
(13)

where
${{\bf{\bar{\Lambda}}}_{2}}={\bf{\Lambda}}_{2}^{({(v-1){K_{s}}:v{K_{s}},\cdot})}$,
and ${{\bf{\bar{E}}}_{s}}={\bf{E}}_{s}^{({(v-1){K_{s}}:v{K_{s}},\cdot})}$.

$\small\begin{array}[]{l}{\bf{H}}{{}_{s}^{v*}}=\arg{\min_{{\bf{H}}_{s}^{v}}}\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|_{F}^{2}+\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{2}^{v}+{\bf{H}}_{s}^{v}-{\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}-{\bf{E}}_{s}^{v}}\right\|_{F}^{2}.\end{array}$
(14)

Then let the derivatives of equations (13) and (14) w.r.t ${\bf{H}}_{s}^{v}$
equal to zero and we obtain the formulas as follows:

$\begin{array}[]{l}{\bf{AH}}_{s}^{v}+{\bf{H}}_{s}^{v}{\bf{B}}={\bf{C}}\\\
{\rm{with}}\;{\bf{A}}={({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v},\;{\bf{B}}=({{\bf{I}}-{\bf{Z}}}){({{\bf{I}}-{\bf{Z}}})^{\rm
T}}\\\ \;\;\;\;\;\;\;\;{\bf{C}}={({{\bf{P}}_{s}^{v}})^{\rm
T}}({\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}})\\\
\;\;\;\;\;\;\;\;-({\frac{1}{\mu}{{{\bf{\bar{\Lambda}}}}_{2}}-{{{\bf{\bar{E}}}}_{s}}}){({{\bf{I}}-{\bf{Z}}})^{\rm
T}},\end{array}$ (15)
$\begin{array}[]{l}{\bf{AH}}_{s}^{v}+{\bf{H}}_{s}^{v}{\bf{B}}={\bf{C}}\\\
{\rm{with}}\;{\bf{A}}={({{\bf{P}}_{s}^{v}})^{\rm
T}}{\bf{P}}_{s}^{v},\;{\bf{B}}=({{\bf{I}}-{{\bf{Z}}^{v}}}){({{\bf{I}}-{{\bf{Z}}^{v}}})^{\rm
T}}\\\ \;\;\;\;\;\;\;\;{\bf{C}}={({{\bf{P}}_{s}^{v}})^{\rm
T}}({\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}})\\\
\;\;\;\;\;\;\;\;-({\frac{1}{\mu}{\bf{\Lambda}}_{2}^{v}-{\bf{E}}_{s}^{v}}){({{\bf{I}}-{{\bf{Z}}^{v}}})^{\rm
T}}.\end{array}$ (16)

Equation (15) and (16) are sylvester equations [41], which can be solved using
lyap111github.com/python-control/python-control/blob/master/control/mateqn.py
function of a python package control 222github.com/python-control/python-
control.

####  3.3.3 ${\bf{H}}_{c}$ sub-problems

The sub-problems w.r.t ${\bf{H}}_{c}$ are different for MVSC-CSLF and MVSC-
CSLFS, which can be formulated as follows:

$\begin{array}[]{l}{{\bf{H}}_{c}}^{*}=\arg{\min_{{{\bf{H}}_{c}}}}\sum\limits_{v=1}^{V}{\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|_{F}^{2}}+\left\|{\frac{1}{\mu}{{{\bf{\bar{\Lambda}}}}_{2}}+{{\bf{H}}_{c}}-{{\bf{H}}_{c}}{\bf{Z}}-{{{\bf{\bar{E}}}}_{s}}}\right\|_{F}^{2},\end{array}$
(17)

where
${{\bf{\bar{\Lambda}}}_{2}}={\bf{\Lambda}}_{2}^{({V{K_{s}}{\rm{+1}}:\cdot,\cdot})}$,
${{\bf{\bar{E}}}_{s}}={\bf{E}}_{s}^{({V{K_{s}}{\rm{+1}}:\cdot,\cdot})}$.

$\begin{array}[]{l}{{\bf{H}}_{c}}^{*}=\arg{\min_{{{\bf{H}}_{c}}}}\sum\limits_{v=1}^{V}{\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|_{F}^{2}}+\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+{{\bf{H}}_{c}}-{{\bf{H}}_{c}}{{\bf{Z}}_{c}}-{\bf{E}}_{s}^{c}}\right\|_{F}^{2}.\end{array}$
(18)

Then let the derivatives of equations (17) and (18) w.r.t ${\bf{H}}_{c}$ equal
to zero and we obtain the formulas as follows:

$\begin{array}[]{l}{\bf{A}}{{\bf{H}}_{c}}+{{\bf{H}}_{c}}{\bf{B}}={\bf{C}}\\\
{\rm{with}}\;{\bf{A}}=\sum\limits_{v=1}^{V}{{{({{\bf{P}}_{c}^{v}})}^{\rm
T}}{\bf{P}}_{c}^{v}},\;{\bf{B}}=({{\bf{I}}-{\bf{Z}}}){({{\bf{I}}-{\bf{Z}}})^{\rm
T}}\\\
\;\;\;\;\;\;\;\;{\bf{C}}=\sum\limits_{v=1}^{V}{{{({{\bf{P}}_{c}^{v}})}^{\rm
T}}({\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{E}}_{r}^{v}})}\\\
\;\;\;\;\;\;\;\;-({\frac{1}{\mu}{{{\bf{\bar{\Lambda}}}}_{2}}-{{{\bf{\bar{E}}}}_{s}}}){({{\bf{I}}-{\bf{Z}}})^{\rm
T}},\end{array}$ (19)
$\begin{array}[]{l}{\bf{A}}{{\bf{H}}_{c}}+{{\bf{H}}_{c}}{\bf{B}}={\bf{C}}\\\
{\rm{with}}\;{\bf{A}}=\sum\limits_{v=1}^{V}{{{({{\bf{P}}_{c}^{v}})}^{\rm
T}}{\bf{P}}_{c}^{v}},\;{\bf{B}}=({{\bf{I}}-{{\bf{Z}}_{c}}}){({{\bf{I}}-{{\bf{Z}}_{c}}})^{\rm
T}}\\\
\;\;\;\;\;\;\;\;{\bf{C}}=\sum\limits_{v=1}^{V}{{{({{\bf{P}}_{c}^{v}})}^{\rm
T}}({\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{E}}_{r}^{v}})}\\\
\;\;\;\;\;\;\;\;-({\frac{1}{\mu}{{\bf{\Lambda}}_{3}}-{\bf{E}}_{s}^{c}}){({{\bf{I}}-{{\bf{Z}}_{c}}})^{\rm
T}}.\end{array}$ (20)

Equation (19) and (20) are sylvester equations [41], which can be solved using
lyap function of a python package control.

####  3.3.4 ${\bf{Z}}$ sub-problems

The sub-problems w.r.t ${\bf{Z}}$, ${\bf{Z}}^{v}$, and ${\bf{Z}}_{c}$ for
MVSC-CSLF and MVSC-CSLFS are formulated as:

$\begin{array}[]{l}{{\bf{Z}}^{*}}=\arg{\min_{\bf{Z}}}\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{2}}+{\bf{H}}-{\bf{HZ}}-{{\bf{E}}_{s}}}\right\|_{F}^{2}+\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+{\bf{D}}-{\bf{Z}}}\right\|_{F}^{2},\\\
{{\bf{Z}}^{*}}={{\bf{Z}}^{*}}-diag({{\bf{Z}}^{*}}),\end{array}$ (21)
$\begin{array}[]{l}{{\bf{Z}}^{v}}^{*}=\arg{\min_{{{\bf{Z}}^{v}}}}\sum\limits_{v=1}^{V}{\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{2}^{v}+{\bf{H}}_{s}^{v}-{\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}-{\bf{E}}_{s}^{v}}\right\|_{F}^{2}}+\sum\limits_{v=1}^{V}{\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{4}^{v}+{{\bf{D}}^{v}}-{{\bf{Z}}^{v}}}\right\|_{F}^{2}},\\\
{{\bf{Z}}^{v}}^{*}={{\bf{Z}}^{v}}^{*}-diag({{\bf{Z}}^{v}}^{*}),\end{array}$
(22)
$\begin{array}[]{l}{{\bf{Z}}_{c}}^{*}=\arg{\min_{{{\bf{Z}}_{c}}}}\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+{{\bf{H}}_{c}}-{{\bf{H}}_{c}}{{\bf{Z}}_{c}}-{\bf{E}}_{s}^{c}}\right\|_{F}^{2}+\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{5}}+{{\bf{D}}_{c}}-{{\bf{Z}}_{c}}}\right\|_{F}^{2},\\\
{{\bf{Z}}_{c}}^{*}={{\bf{Z}}_{c}}^{*}-diag({{\bf{Z}}_{c}}^{*}).\end{array}$
(23)

Then let the derivatives of equations (21), (22), and (23) w.r.t ${\bf{Z}}$,
${\bf{Z}}^{v}$, and ${\bf{Z}}_{c}$ equal to zero, and we obtain the
corresponding updating formulas:

$\begin{array}[]{l}{{\bf{Z}}^{*}}={({{\bf{I}}+{{\bf{H}}^{\rm
T}}{\bf{H}}})^{-1}}\cdot({\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+\frac{1}{\mu}{{\bf{H}}^{\rm
T}}{{\bf{\Lambda}}_{2}}+{\bf{D}}-{{\bf{H}}^{\rm
T}}{{\bf{E}}_{s}}+{{\bf{H}}^{\rm T}}{\bf{H}}})\\\
{{\bf{Z}}^{*}}={{\bf{Z}}^{*}}-diag({{\bf{Z}}^{*}})\end{array}$ (24)
$\begin{array}[]{l}{{\bf{Z}}^{v}}^{*}={\left[{{{({{\bf{H}}_{s}^{v}})}^{\rm
T}}{\bf{H}}_{s}^{v}+{\bf{I}}}\right]^{-1}}\cdot({\frac{1}{\mu}{\bf{\Lambda}}_{4}^{v}+\frac{1}{\mu}{{({{\bf{H}}_{s}^{v}})}^{\rm
T}}{\bf{\Lambda}}_{2}^{v}+{{\bf{D}}^{v}}-{{({{\bf{H}}_{s}^{v}})}^{\rm
T}}{\bf{E}}_{s}^{v}+{{({{\bf{H}}_{s}^{v}})}^{\rm T}}{\bf{H}}_{s}^{v}})\\\
{{\bf{Z}}^{v}}^{*}={{\bf{Z}}^{v}}^{*}-diag({{\bf{Z}}^{v}}^{*})\end{array}$
(25) $\begin{array}[]{l}{{\bf{Z}}_{c}}^{*}={({{\bf{H}}_{c}^{\rm
T}{{\bf{H}}_{c}}+{\bf{I}}})^{-1}}\cdot({\frac{1}{\mu}{{\bf{\Lambda}}_{5}}+\frac{1}{\mu}{\bf{H}}_{c}^{\rm
T}{{\bf{\Lambda}}_{3}}+{{\bf{D}}_{c}}-{\bf{H}}_{c}^{\rm
T}{\bf{E}}_{s}^{c}+{\bf{H}}_{c}^{\rm T}{{\bf{H}}_{c}}})\\\
{{\bf{Z}}_{c}}^{*}={{\bf{Z}}_{c}}^{*}-diag({{\bf{Z}}_{c}}^{*})\end{array}$
(26)

####  3.3.5 ${\bf{D}}$ sub-problems

We first give a brief account of the updating rules of ${\bf{D}}$ and
${\bf{D}}^{v}$ related to equations (7) and (8), respectively:

$\begin{array}[]{l}{{\bf{D}}^{*}}=\arg{\min_{\bf{D}}}{\lambda_{\rm{2}}}\left\|{\bf{D}}\right\|_{F}^{2}+\mu\left\|{\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+{\bf{D}}-{\bf{Z}}}\right\|_{F}^{2},\end{array}$
(27)
$\small\begin{array}[]{l}{{\bf{D}}^{v}}^{*}=\arg{{\min}_{{{\bf{D}}^{v}}}}{\lambda_{2}}\pi_{2}^{v}\left\|{{{\bf{D}}^{v}}}\right\|_{F}^{2}+\mu\left\|{\frac{1}{\mu}{\bf{\Lambda}}_{4}^{v}+{{\bf{D}}^{v}}-{{\bf{Z}}^{v}}}\right\|_{F}^{2}.\end{array}$
(28)

Then let the derivatives of equations (27) and (28) w.r.t ${\bf{D}}$,
${\bf{D}}^{v}$ equal to zero, and we obtain the corresponding updating
formulas:

${{\bf{D}}^{*}}=\frac{{\mu{\bf{Z}}-{{\bf{\Lambda}}_{3}}}}{{{\lambda_{\rm{2}}}+\mu}},$
(29)
${{\bf{D}}^{v}}^{*}=\frac{{\mu{{\bf{Z}}^{v}}-{\bf{\Lambda}}_{4}^{v}}}{{{\lambda_{2}}\pi_{2}^{v}{\rm{+}}\mu}}.$
(30)

And for ${\bf{D}}_{c}$, the optimal problem is:

$\begin{array}[]{l}{{\bf{D}}_{c}}^{*}=\arg{\min_{{{\bf{D}}_{c}}}}\bar{\mu}{\left\|{{{\bf{D}}_{c}}}\right\|_{*}}+\frac{1}{2}\left\|{{{\bf{D}}_{c}}-{\bf{G}}}\right\|_{F}^{2},\end{array}$

where $\bar{\mu}=\frac{{{\lambda_{2}}\pi_{2}^{V+1}}}{\mu}$. Obviously, this
kind of low-rank problem can be solved by singular value thresholding
algorithm [42]:

${{\bf{D}}_{c}}^{*}={\Pi_{\bar{\mu}}}({{{\bf{Z}}_{c}}-\frac{1}{\mu}{{\bf{\Lambda}}_{5}}})={\bf{U}}{\pi_{\bar{\mu}}}({\bf{\Sigma}}){{\bf{V}}^{\rm
T}},$ (31)

where ${\bf{U\Sigma}}{{\bf{V}}^{\rm T}}$ is the skinny SVD of
${{{\bf{Z}}_{c}}-\frac{1}{\mu}{{\bf{\Lambda}}_{5}}}$. Soft-thresholding
operation ${\pi_{\tau}}({\bf{\Sigma}})$ is defined as
${\pi_{\tau}}({\bf{\Sigma}})={({\left|{\bf{\Sigma}}\right|-\tau})_{+}}{\mathop{\rm
sgn}}({{\bf{\Sigma}}})$ and ${(t)_{+}}=\max({0,t})$.

####  3.3.6 $\bf{E}$ sub-problems

The updating rules of ${\bf{E}}_{r}^{v}$ are the same for MVSC-CSLF and MCSC-
CSLFS because ${\boldsymbol{\pi}}={\boldsymbol{\pi}}_{1}$, which are replaced
by ${\boldsymbol{\pi}}_{temp}={\left\\{{\pi}_{temp}^{v}\right\\}}_{v=1}^{V}$
for convenience when discuss ${\bf{E}}_{r}^{v}$. Therefore, the optimal sub-
problems w.r.t ${\bf{E}}_{r}^{v}$, ${\bf{E}}_{s}$, ${\bf{E}}_{s}^{v}$, and
${\bf{E}}_{s}^{c}$ for MVSC-CSLF and MCSC-CSLFS are listed as follows:

$\begin{array}[]{l}{\bf{E}}{{}_{r}^{v*}}=\arg{\min_{{\bf{E}}_{r}^{v}}}\bar{\mu}{\left\|{{\bf{E}}_{r}^{v}}\right\|_{{\rm{2,1}}}}+\frac{1}{2}\left\|{{\bf{E}}_{r}^{v}-{{\bf{G}}_{r}^{v}}}\right\|_{F}^{2},\end{array}$
(32)

where ${\bar{\mu}}_{r}^{v}=\frac{{\pi_{temp}^{v}}}{\mu}$, and
${{\bf{G}}_{r}^{v}}=\frac{1}{\mu}{\bf{\Lambda}}_{1}^{v}+{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}$.

$\begin{array}[]{l}{{\bf{E}}_{s}}^{*}=\arg{\min_{{{\bf{E}}_{s}}}}\bar{\mu}{\left\|{{{\bf{E}}_{s}}}\right\|_{{\rm{2,1}}}}+\frac{1}{2}\left\|{{{\bf{E}}_{s}}-{\bf{G}}_{s}}\right\|_{F}^{2},\end{array}$
(33)

where ${\bar{\mu}}_{s}=\frac{{{\lambda_{\rm{1}}}}}{\mu}$,
${\bf{G}}_{s}=\frac{1}{\mu}{{\bf{\Lambda}}_{2}}+{\bf{H}}-{\bf{HZ}}$.

$\begin{array}[]{l}{\bf{E}}{{}_{s}^{v*}}=\arg{\min_{{\bf{E}}_{s}^{v}}}\bar{\mu}{\left\|{{\bf{E}}_{s}^{v}}\right\|_{2,1}}+\frac{1}{2}\left\|{{\bf{E}}_{s}^{v}-{{\bf{G}}_{s}^{v}}}\right\|_{F}^{2},\end{array}$
(34)

where ${\bar{\mu}}_{s}^{v}=\frac{{{\lambda_{\rm{1}}}\pi_{2}^{v}}}{\mu}$,
${{\bf{G}}_{s}^{v}}=\frac{1}{\mu}{\bf{\Lambda}}_{2}^{v}+{\bf{H}}_{s}^{v}-{\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}$.

$\begin{array}[]{l}{\bf{E}}{{}_{s}^{c*}}=\arg{\min_{{\bf{E}}_{s}^{c}}}\bar{\mu}{\left\|{{\bf{E}}_{s}^{c}}\right\|_{2,1}}+\frac{1}{2}\left\|{{\bf{E}}_{s}^{c}-{\bf{G}}_{s}^{c}}\right\|_{F}^{2},\end{array}$
(35)

where ${\bar{\mu}}_{s}^{c}=\frac{{{\lambda_{\rm{1}}}\pi_{2}^{V+1}}}{\mu}$,
${\bf{G}}_{s}^{c}=\frac{1}{\mu}{{\bf{\Lambda}}_{3}}+{{\bf{H}}_{c}}-{{\bf{H}}_{c}}{{\bf{Z}}_{c}}$.

Equations (32) - (35) can be solved according to lemma 4.1 of [43], and their
solutions are:

${\bf{E}}_{r}^{v*({:,i})}=\left\\{\begin{array}[]{l}\frac{{{{\left\|{{\bf{G}}_{r}^{v({:,i})}}\right\|}_{2}}-{\bar{\mu}}_{r}^{v}}}{{{{\left\|{{\bf{G}}_{r}^{v({:,i})}}\right\|}_{2}}}}{\bf{G}}_{r}^{v({:,i})},\;{\rm{if}}\;{\left\|{{\bf{G}}_{r}^{v({:,i})}}\right\|_{2}}>{\bar{\mu}}_{r}^{v};\\\
0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{otherwise}}.\end{array}\right.$
(36)
${\bf{E}}_{s}^{*({:,i})}=\left\\{\begin{array}[]{l}\frac{{{{\left\|{{\bf{G}}_{s}^{({:,i})}}\right\|}_{2}}-{\bar{\mu}}_{s}}}{{{{\left\|{{\bf{G}}_{s}^{({:,i})}}\right\|}_{2}}}}{\bf{G}}_{s}^{({:,i})},\;{\rm{if}}\;{\left\|{{\bf{G}}_{s}^{({:,i})}}\right\|_{2}}>{\bar{\mu}}_{s};\\\
0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{otherwise}}.\end{array}\right.$
(37)
${\bf{E}}_{s}^{v*({:,i})}=\left\\{\begin{array}[]{l}\frac{{{{\left\|{{\bf{G}}_{s}^{v({:,i})}}\right\|}_{2}}-{\bar{\mu}}_{s}^{v}}}{{{{\left\|{{\bf{G}}_{s}^{v({:,i})}}\right\|}_{2}}}}{\bf{G}}_{s}^{v({:,i})},\;{\rm{if}}\;{\left\|{{\bf{G}}_{s}^{v({:,i})}}\right\|_{2}}>{\bar{\mu}}_{s}^{v};\\\
0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{otherwise}}.\end{array}\right.$
(38)
${\bf{E}}_{s}^{c*({:,i})}=\left\\{\begin{array}[]{l}\frac{{{{\left\|{{\bf{G}}_{s}^{c({:,i})}}\right\|}_{2}}-{\bar{\mu}}_{s}^{c}}}{{{{\left\|{{\bf{G}}_{s}^{c({:,i})}}\right\|}_{2}}}}{\bf{G}}_{s}^{c({:,i})},\;{\rm{if}}\;{\left\|{{\bf{G}}_{s}^{c({:,i})}}\right\|_{2}}>{\bar{\mu}}_{s}^{c};\\\
0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{otherwise}}.\end{array}\right.$
(39)

####  3.3.7 ${\boldsymbol{\pi}}$ sub-problems

For MVSC-CSLF and MVSC-CSLFS, the optimal sub-problems w.r.t
${\boldsymbol{\pi}}$, ${\boldsymbol{\pi}}_{1}$, and ${\boldsymbol{\pi}}_{2}$
are listed as follows:

$\begin{array}[]{l}\min\sum\nolimits_{v=1}^{V}{{\pi^{v}}{{\left\|{{\bf{E}}_{r}^{v}}\right\|}_{{\rm{2,1}}}}}+\frac{{{\lambda_{3}}}}{2}\left\|{\bf{\pi}}\right\|_{2}^{2}\\\
s.t.\;{\bf{\pi}}\geq 0,\;\sum\nolimits_{v=1}^{V}{{\pi^{v}}}=1,\end{array}$
(40)
$\begin{array}[]{l}\min\sum\nolimits_{v=1}^{V}{\pi_{1}^{v}{{\left\|{{\bf{E}}_{r}^{v}}\right\|}_{2,1}}}+\frac{{{\lambda_{4}}}}{2}\left\|{{{\bf{\pi}}_{1}}}\right\|_{2}^{2}\\\
s.t.\;{{\bf{\pi}}_{1}}\geq
0,\sum\nolimits_{v=1}^{V}{\pi_{1}^{v}}=1,\end{array}$ (41)
$\begin{array}[]{l}\min\sum\nolimits_{v=1}^{V}{\pi_{2}^{v}({{\lambda_{\rm{1}}}{{\left\|{{\bf{E}}_{s}^{v}}\right\|}_{2,1}}+\frac{{{\lambda_{2}}}}{2}\left\|{{{\bf{D}}^{v}}}\right\|_{F}^{2}})}+\pi_{2}^{V+1}({{\lambda_{\rm{1}}}{{\left\|{{\bf{E}}_{s}^{c}}\right\|}_{2,1}}+{\lambda_{2}}{{\left\|{{{\bf{D}}_{c}}}\right\|}_{*}}})+\frac{{{\lambda_{4}}}}{2}\left\|{{{\bf{\pi}}_{2}}}\right\|_{2}^{2}\\\
s.t.\;{{\bf{\pi}}_{2}}\geq
0,\;\sum\nolimits_{v=1}^{V+1}{\pi_{2}^{v}}=1.\end{array}$ (42)

They are standard convex optimization problems, which can be solved by the
python package CVXPY 333www.cvxpy.org/.

####  3.3.8 Lagrangian multipliers sub-problems

The lagrangian multipliers can be updated by the rules:

$\left\\{\begin{array}[]{l}{\bf{\Lambda}}_{1}^{v}={\bf{\Lambda}}_{1}^{v}+\mu({{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}})\\\
{{\bf{\Lambda}}_{2}}={{\bf{\Lambda}}_{2}}+\mu({{\bf{H}}-{\bf{HZ}}-{{\bf{E}}_{s}}})\\\
{{\bf{\Lambda}}_{3}}={{\bf{\Lambda}}_{3}}+\mu({{\bf{D}}-{\bf{Z}}})\end{array}\right.$
(43)
$\left\\{\begin{array}[]{l}{\bf{\Lambda}}_{1}^{v}={\bf{\Lambda}}_{1}^{v}+\mu({{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}})\\\
{\bf{\Lambda}}_{2}^{v}={\bf{\Lambda}}_{2}^{v}+\mu({{\bf{H}}_{s}^{v}-{\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}-{\bf{E}}_{s}^{v}})\\\
{{\bf{\Lambda}}_{3}}={{\bf{\Lambda}}_{3}}+\mu({{{\bf{H}}_{c}}-{{\bf{H}}_{c}}{{\bf{Z}}_{c}}-{\bf{E}}_{s}^{c}})\\\
{\bf{\Lambda}}_{4}^{v}={\bf{\Lambda}}_{4}^{v}+\mu({{{\bf{D}}^{v}}-{{\bf{Z}}^{v}}})\\\
{{\bf{\Lambda}}_{5}}={{\bf{\Lambda}}_{5}}+\mu({{{\bf{D}}_{c}}-{{\bf{Z}}_{c}}})\end{array}\right.$
(44)

The complete algorithms w.r.t MVSC-CSLF and MVSC-CSLFS are summarized in
Algorithms 1 and 2, respectively.

Algorithm 1 MVSC-CSLF

0: $\left\\{{{{\bf{X}}^{v}}}\right\\}_{v=1}^{V}$, $K_{s}$, $K_{c}$,
$\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\mu$, $\mu_{max}$, $\rho$,
$\epsilon$.

0: $\bf{Z}$ and $\bf{H}$.

1: Initialize: Set the initial values of
$\left\\{{{{\bf{P}}_{s}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{P}}_{c}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{H}}_{s}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{H}}_{c}$, ${\bf{Z}}$,
${\bf{D}}$, $\left\\{{{{\bf{E}}_{r}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{E}}_{s}$
equal to 0, and let $\left\\{\pi^{v}=\frac{1}{V}\right\\}_{v=1}^{V}$.

2: while not converged do

3: fix others and update $\left\\{{{{\bf{P}}_{s}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{P}}_{c}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{H}}_{s}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{H}}_{c}$, ${\bf{Z}}$,
${\bf{D}}$, $\left\\{{{{\bf{E}}_{r}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{E}}_{s}}}\right\\}_{v=1}^{V}$, $\boldsymbol{\pi}$, and
lagrangian multipliers by solving (11), (12), (15), (19), (24), (29), (36),
(37), (40), and (43) in turn;

4: update $\mu=\min(\rho\mu,\mu_{max})$

5: check convergence condition:
$\frac{1}{V}\sum\limits_{v=1}^{V}{{{\left\|{{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|}_{\infty}}}<\epsilon$,${\left\|{{\bf{H}}-{\bf{HZ}}-{{\bf{E}}_{s}}}\right\|_{\infty}}<\epsilon$,
${\left\|{{\bf{D}}-{\bf{Z}}}\right\|_{\infty}}<\epsilon$.

6: end while

Algorithm 2 MVSC-CSLFS

0: $\left\\{{{{\bf{X}}^{v}}}\right\\}_{v=1}^{V}$, $K_{s}$, $K_{c}$,
$\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, $\mu$, $\mu_{max}$, $\rho$,
$\epsilon$.

0: $\bf{Z}$ and $\bf{H}$.

1: Initialize: Set the initial values of
$\left\\{{{{\bf{P}}_{s}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{P}}_{c}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{H}}_{s}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{H}}_{c}$,
$\left\\{{{{\bf{Z}}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{Z}}_{c}$,
$\left\\{{{{\bf{D}}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{D}}_{c}$,
$\left\\{{{{\bf{E}}_{r}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{E}}_{s}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{E}}_{s}^{c}$ equal to
0, let $\left\\{\pi_{1}^{v}=\frac{1}{V}\right\\}_{v=1}^{V}$ and
$\left\\{\pi_{2}^{v}=\frac{1}{V+1}\right\\}_{v=1}^{V+1}$.

2: while not converged do

3: fix others and update $\left\\{{{{\bf{P}}_{s}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{P}}_{c}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{H}}_{s}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{H}}_{c}$,
$\left\\{{{{\bf{Z}}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{Z}}_{c}$,
$\left\\{{{{\bf{D}}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{D}}_{c}$,
$\left\\{{{{\bf{E}}_{r}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{E}}_{s}^{v}}}\right\\}_{v=1}^{V}$,
$\left\\{{{{\bf{E}}_{s}^{c}}}\right\\}_{v=1}^{V}$, ${\boldsymbol{\pi}}_{1}$,
${\boldsymbol{\pi}}_{2}$, and lagrangian multipliers by solving (11), (12),
(16), (20), (25), (26), (30), (31), (36), (38), (39), (41), (42), and (44) in
turn;

4: update $\mu=\min(\rho\mu,\mu_{max})$

5: check convergence condition:
$\frac{1}{V}\sum\limits_{v=1}^{V}{{{\left\|{{{\bf{X}}^{v}}-{\bf{P}}_{s}^{v}{\bf{H}}_{s}^{v}-{\bf{P}}_{c}^{v}{{\bf{H}}_{c}}-{\bf{E}}_{r}^{v}}\right\|}_{\infty}}}<\epsilon$,$\frac{1}{V}\sum\limits_{v=1}^{V}{{{\left\|{{\bf{H}}_{s}^{v}-{\bf{H}}_{s}^{v}{{\bf{Z}}^{v}}-{\bf{E}}_{s}^{v}}\right\|}_{\infty}}}<\epsilon$,${{{\left\|{{{\bf{H}}_{c}}-{{\bf{H}}_{c}}{{\bf{Z}}_{c}}-{\bf{E}}_{s}^{c}}\right\|}_{\infty}}}<\epsilon$,
$\frac{1}{V}\sum\limits_{v=1}^{V}{{{\left\|{{{\bf{D}}^{v}}-{{\bf{Z}}^{v}}}\right\|}_{\infty}}}<\epsilon$,${{{\left\|{{{\bf{D}}_{c}}-{{\bf{Z}}_{c}}}\right\|}_{\infty}}}<\epsilon$.

6: end while

###  3.4 Complexity and convergence analysis

Computational complexity of the algorithm mainly comes from some complex
computations, such as matrix inversion and svd decomposition, and therefore we
mainly consider these operations in this subsection.

####  3.4.1 MVSC-CSLF

The computational complexity mainly comes from five sources. For
$\left\\{{{{\bf{P}}_{s}^{v}}}\right\\}_{v=1}^{V}$ and
$\left\\{{{{\bf{P}}_{c}^{v}}}\right\\}_{v=1}^{V}$, the complexities are
$O(\sum\limits_{v=1}^{V}{{{({M^{v}})}^{3}}})$; for
$\left\\{{{{\bf{H}}_{s}^{v}}}\right\\}_{v=1}^{V}$ and ${\bf{H}}_{c}$, the
complexities are $O(VK_{s}^{3})$ and $O(K_{c}^{3})$; for $\bf{Z}$, the matrix
inversion’s complexity is $O({N^{3}})$. Therefore, the total computational
cost of this problem is
$O(\sum\limits_{v=1}^{V}{{{({M^{v}})}^{3}}}+VK_{s}^{3}+K_{c}^{3}+N^{3})$.

####  3.4.2 MVSC-CSLFS

The computational complexity mainly comes from seven sources. For
$\left\\{{{{\bf{P}}_{s}^{v}}}\right\\}_{v=1}^{V}$ and
$\left\\{{{{\bf{P}}_{c}^{v}}}\right\\}_{v=1}^{V}$, the complexities are
$O(\sum\limits_{v=1}^{V}{{{({M^{v}})}^{3}}})$; for
$\left\\{{{{\bf{H}}_{s}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{H}}_{c}$, the
complexities are $O(VK_{s}^{3})$ and $O(K_{c}^{3})$; for
$\left\\{{{{\bf{Z}}^{v}}}\right\\}_{v=1}^{V}$, ${\bf{Z}}_{c}$, the
complexities are $O({VN^{3}})$ and $O({N^{3}})$; for ${\bf{D}}_{c}$, the
complexity is $O({N^{3}})$. Therefore, the total computational cost of this
problem is
$O(\sum\limits_{v=1}^{V}{{{({M^{v}})}^{3}}}+VK_{s}^{3}+K_{c}^{3}+(V+2)N^{3})$.

Mostly, $K_{s}$ and $K_{c}$ should smaller than $M^{v}$, i.e.,
${K_{s}},{K_{c}}\ll{M^{v}}$, therefore the computational cost of our proposed
two algorithms can be rewritten as
$O(\sum\limits_{v=1}^{V}{{{({M^{v}})}^{3}}}+N^{3})$ and
$O(\sum\limits_{v=1}^{V}{{{({M^{v}})}^{3}}}+(V+2)N^{3})$. And for large scale
data, i.e. ${M^{v}}\ll N$, the computational cost are $O({N^{3}})$ and
$O((V+2){N^{3}})$, respectively.

####  3.4.3 Convergence analysis

MVSC-CSLF and MVSC-CSLFS both involve multiple sub-problems, which makes it
hard to give a solid proof of their convergence. However, we can give an
empirical demonstration by extensive experiments on multiple real-world
datasets referred to [12, 14, 13]. The convergence curves of MVSC-CSLF and
MVSC-CSLFS on six datasets are shown in Fig. 2 and Fig. 3, respectively. The
promising performance indicates that the proposed methods converge rate fast
and stable.

Figure 2: Convergence of MVSC-CSLFS. We display the curves about three stop
criteria in red, gray, and blue versus the iteration numbers on six datasets.
Figure 3: Convergence of MVSC-CSLFS. We display the curves about five stop
criteria in gray, red, blue, green, and purple versus the iteration numbers on
six datasets.

##  4 Experiment

###  4.1 datasets

Six real-world datasets are selected to test and validate the performance of
proposed algorithms. Among them,
ORL444http://www.cad.zju.edu.cn/home/dengcai/Data/FaceData.html  and
Yale${}^{\ref{f1}}$ are human face data, and we select intensity, LBP, and
Gabor features as three views; Prokaryotic[44] is a species data set described
from textual data, proteome composition, and genomic representations, each of
them is viewed as a view; Reuters[45] is document data that the contents are
written by 6 languages, each language is viewed as a view, and they are all in
the bag-of-words representations;
3-sources555http://mlg.ucd.ie/datasets/3sources.html is news data set with its
contents come from three sources (BBC, Reuters, and the Guardian), and they
are all in the bag-of-words representations; uci-
digit666http://archive.ics.uci.edu/ml/datasets/Multiple+Features concludes six
types of features of handwritten number, and 3 of them are utilized to
construct three views.Table 2 summarizes the above information.

Table 2: Description of datasets Data | View number ($V$) | Clusters ($C$) | Instance number ($N$) | Dimension of each view ($M^{v}$)
---|---|---|---|---
ORL | 3 | 40 | 400 | [4096,3304,6750]
Reuters | 5 | 6 | 600 |  | [21526,24892,34121,15487,11539]
---
3-sources | 3 | 6 | 169 | [3560,3631,3068]
Yale | 3 | 15 | 165 | [4096,3304,6750]
uci-digit | 3 | 10 | 2000 | [216,76,64]
Prokaryotic | 3 | 4 | 551 | [438,3,393]

###  4.2 Compared Methods

In comprising experiment, 9 algorithms are selected as baseline methods:
(1) LRSC [27]: the method impose a low-rank constraint on subspace clustering.
The algorithm has four variants depending on whether the data is corrupted and
whether the constraint is relaxed or exact. Since LRSC is a single-view
method, we perform each variant on each view’s data and select the best
result.
(2) LMSC [12]: the method combines representation learning with subspace
clustering. It learn a common latent representation first and then perform
self-expressive reconstruction on it with a low-rank constraint.
(3) CSMSC [10]: the method assumes that each view’s subspace representation
includes two parts: consistent and view-specific ones. It separates them from
each view, and constraints them with nuclear and Frobenius norm, respectively.
(4) MLRSSC [9]: the method performs self-expressive reconstruction on each
view, but impose sparse and low-rank constraints on each subspace
representation, simultaneously.
(5) MLRSSC-C [9]: the method is based on MLRSSC, but propose a centroid
alignment strategy.
(6) KMLRSSC [9]: the method is based on MLRSSC, and add kernel to enhance the
processing ability of nonlinear data.
(7) KMLRSSC-C [9]: the method is based on MLRSSC-C, and add kernel to enhance
the processing ability of nonlinear data.
(8) CSI [13]: the method is differ from most existing algorithms when
integrating multiple subspace representations. CSI propose a strategy that can
obtain a common subspace representation by keeping each view’s local manifold
preserving.
(9) FCMSC [14]: the method concatenates all views and performs self-expressive
reconstruction on it with low-rank and local manifold preserving
regularization.

###  4.3 Evaluation Metrics and Adjacency Matrix

We use ACC (accuracy), NMI (normalized mutual information), ARI (Adjusted Rand
Index), Precision, Recall, F-score as the evaluation metrics. Among them,
except for NMI and ARI, other metrics are not clustering metrics. Therefore
the predicted labels should be first mapped into the true categories by
applying Kuhn-Munkres algorithm. Note that a higher value indicates better
performance for all the above metrics.

Figure 4: Visualization for MVSC-CSLF’s adjacency matrices on six datasets.

Once model training is completed, the learned subspace representation can be
utilized to construct adjacency matrix, which can be applied to the spectral
clustering [46]. MVSC-CSLF constructs adjacency matrix according to the
following rule:

$\frac{{\left|{\bf{Z}}\right|+\left|{{{\bf{Z}}^{\rm T}}}\right|}}{2}.$ (45)

MVSC-CSLFS has $V+1$ coefficient matrix, and its construction rule is:

$\frac{1}{2}({\frac{{\left|{{{\bf{Z}}_{c}}}\right|+\left|{{\bf{Z}}_{c}^{\rm
T}}\right|}}{2}+\frac{1}{V}\sum\limits_{v=1}^{V}{\frac{{\left|{{{\bf{Z}}^{v}}}\right|+\left|{{{({{\bf{Z}}^{v}})}^{\rm
T}}}\right|}}{2}}}),$ (46)

where $\left|\cdot\right|$ denotes element-wise absolute operator.

Due to the higher correlation between samples belonging to the same category,
the adjacency matrix should present block diagonalization when we sort samples
by category. Therefore, we visualize the adjacency matrices of MVSC-CSLF and
MVSC-CSLFS computed thorough equations (45) and (46) on six datasets, which
are shown in Fig. 4 and Fig. 5, respectively. These figures show that the
adjacency matrices present obvious block diagonalization according to category
on all datasets. The property is more clearly expressed in MVSC-CSLFS than in
MVSC-CSLF, which is one of the reasons that MVSC-CSLFS performs better in most
cases.

Figure 5: Visualization for MVSC-CSLFS’s adjacency matrices on six datasets.

###  4.4 Clustering Performance on Real-World datasets

In this subsection, we compare our proposed methods with 9 baseline
approaches, and Table 3 tabulates the clustering results on six datasets in
terms of six metrics. Note that we conduct each method 10 times, and the
results are reported in form of the mean score, as well as the standard
deviation.

The results in Table 3 show that multi-view algorithms perform better than
single view algorithm on all datasets. As refereed in subsection 4.2, LRSC’s
four variants are conducted on all views of each datasets, and the best result
is selected as comparison. Even though, LRSC performs worse than almost all
multi-view algorithms, and the clustering results are difficult to find a
trade-off between Precision and Recall on Reuters and Prokaryotic.
Furthermore, it is not hard to see that our proposed MVSC-CSLF and MVSC-CSLFS
have satisfied performance on all datasets. In particular, MVSC-CSLFS
outperforms than others on ORL and Yale in terms of all metrics, and also has
best score on uci-digit and Prokaryotic in terms of three and four metrics,
respectively. On Reuters, MVSC-CSLF performs best in terms of ACC and F-score,
and since LRSC cannot find a good trade-off between Recall and F-score on this
data, MVSC-CSLF actually also has the best score in terms of Recall. As for
3-sources, FCSMC performs best, but expect for this method our proposed
approaches have a bigger improvement that other compared algorithms.

Table 3: Clustering results Dataset | Method | ACC | NMI | AR | Precision | Recall | F-score
---|---|---|---|---|---|---|---
ORL | LRSC | 0.769(0.015) | 0.883(0.015) | 0.688(0.037) | 0.668(0.037) | 0.726(0.036) | 0.695(0.036)
LMSC | 0.822(0.037) | 0.927(0.013) | 0.774(0.043) | 0.813(0.038) | 0.822(0.037) | 0.802(0.041)
CSMSC | 0.868(0.0012) | 0.942(0.005) | 0.827(0.002) | 0.860(0.002) | 0.804(0.003) | 0.831(0.001)
MLRSSC | 0.637(0.034) | 0.813(0.017) | 0.524(0.037) | 0.640(0.033) | 0.637(0.034) | 0.623(0.034)
MLRSSC-Centroid | 0.780(0.027) | 0.917(0.009) | 0.729(0.029) | 0.785(0.033) | 0.780(0.027) | 0.761(0.032)
KMLRSSC | 0.786(0.041) | 0.903(0.016) | 0.721(0.042) | 0.793(0.040) | 0.786(0.041) | 0.774(0.042)
KMLRSSC-Centroid | 0.783(0.031) | 0.907(0.008) | 0.721(0.026) | 0.793(0.039) | 0.783(0.031) | 0.772(0.035)
CSI | 0.863(0.152) | 0.930(0.095) | 0.805(0.158) | 0.782(0.170) | 0.840(0.141) | 0.810(0.158)
FCSMC | 0.847(0.019) | 0.928(0.008) | 0.782(0.021) | 0.746(0.025) | 0.833(0.020) | 0.787(0.020)
MVSC-CSLF | 0.849(0.009) | 0.931(0.006) | 0.800(0.012) | 0.854(0.009) | 0.849(0.009) | 0.837(0.010)
MVSC-CSLFS | 0.885(0.007) | 0.947(0.005) | 0.842(0.011) | 0.889(0.008) | 0.885(0.007) | 0.874(0.007)
Reuters | LRSC | 0.209(0.007) | 0.131(0.007) | 0.015(0.001) | 0.172(0.000) | 0.835(0.013) | 0.285(0.001)
LMSC | 0.495(0.001) | 0.355(0.001) | 0.223(0.010) | 0.312(0.009) | 0.497(0.008) | 0.383(0.006)
CSMSC | 0.513(0.001) | 0.358(0.001) | 0.247(0.001) | 0.343(0.001) | 0.444(0.001) | 0.387(0.001)
MLRSSC | 0.530(0.033) | 0.382(0.015) | 0.282(0.026) | 0.501(0.055) | 0.530(0.033) | 0.483(0.051)
MLRSSC-Centroid | 0.514(0.035) | 0.370(0.011) | 0.268(0.026) | 0.484(0.040) | 0.514(0.035) | 0.459(0.046)
KMLRSSC | 0.571(0.023) | 0.374(0.019) | 0.304(0.020) | 0.618(0.037) | 0.571(0.023) | 0.567(0.030)
KMLRSSC-Centroid | 0.551(0.024) | 0.357(0.016) | 0.294(0.016) | 0.591(0.046) | 0.551(0.024) | 0.540(0.027)
CSI | 0.527(0.009) | 0.394(0.009) | 0.288(0.007) | 0.364(0.006) | 0.511(0.008) | 0.425(0.007)
FCSMC | 0.518(0.001) | 0.369(0.001) | 0.268(0.000) | 0.357(0.000) | 0.470(0.001) | 0.405(0.000)
MVSC-CSLF | 0.582(0.010) | 0.352(0.004) | 0.290(0.012) | 0.604(0.005) | 0.582(0.005) | 0.581(0.012)
MVSC-CSLFS | 0.571(0.001) | 0.373(0.001) | 0.297(0.01) | 0.419(0.002) | 0.397(0.001) | 0.444(0.002)
Yale | LRSC | 0.675(0.027) | 0.706(0.019) | 0.491(0.032) | 0.501(0.030) | 0.547(0.030) | 0.523(0.030)
LMSC | 0.789(0.018) | 0.789(0.019) | 0.628(0.030) | 0.632(0.029) | 0.672(0.027) | 0.651(0.028)
CSMSC | 0.752(0.001) | 0.784(0.001) | 0.615(0.005) | 0.673(0.002) | 0.610(0.006) | 0.794(0.029)
MLRSSC | 0.660(0.050) | 0.697(0.033) | 0.477(0.054) | 0.482(0.056) | 0.544(0.045) | 0.511(0.050)
MLRSSC-Centroid | 0.627(0.034) | 0.702(0.021) | 0.458(0.024) | 0.711(0.050) | 0.627(0.034) | 0.648(0.041)
KMLRSSC | 0.649(0.053) | 0.689(0.032) | 0.485(0.046) | 0.679(0.070) | 0.649(0.053) | 0.653(0.059)
KMLRSSC-Centroid | 0.639(0.043) | 0.680(0.032) | 0.480(0.044) | 0.661(0.046) | 0.639(0.043) | 0.640(0.045)
CSI | 0.734(0.000) | 0.777(0.000) | 0.606(0.000) | 0.606(0.000) | 0.659(0.000) | 0.739(0.000)
FCSMC | 0.775(0.025) | 0.796(0.019) | 0.589(0.041) | 0.569(0.046) | 0.674(0.028) | 0.617(0.037)
MVSC-CSLF | 0.804(0.008) | 0.813(0.013) | 0.629(0.022) | 0.874(0.010) | 0.804(0.008) | 0.828(0.009)
MVSC-CSLFS | 0.859(0.017) | 0.861(0.016) | 0.727(0.024) | 0.902(0.018) | 0.859(0.017) | 0.853(0.006)
uci-digit | LRSC | 0.763(0.003) | 0.697(0.001) | 0.626(0.003) | 0.650(0.003) | 0.678(0.002) | 0.664(0.003)
LMSC | 0.905(0.004) | 0.833(0.006) | 0.812(0.008) | 0.835(0.007) | 0.840(0.007) | 0838(0.007)
CSMSC | 0.903(0.000) | 0.835(0.001) | 0.810(0.000) | 0.823(0.000) | 0.834(0.004) | 0.828(0.000)
MLRSSC | 0.881(0.061) | 0.852(0.022) | 0.814(0.050) | 0.877(0.070) | 0.881(0.061) | 0.873(0.070)
MLRSSC-Centroid | 0.893(0.055) | 0.853(0.023) | 0.818(0.050) | 0.891(0.063) | 0.893(0.055) | 0.887(0.063)
KMLRSSC | 0.894(0.049) | 0.861(0.018) | 0.826(0.044) | 0.888(0.060) | 0.894(0.049) | 0.886(0.058)
KMLRSSC-Centroid | 0.910(0.046) | 0.865(0.018) | 0.840(0.039) | 0.910(0.051) | 0.910(0.046) | 0.906(0.053)
CSI | 0.863(0.000) | 0.915(0.000) | 0.846(0.000) | 0.805(0.000) | 0.929(0.000) | 0.862(0.000)
FCSMC | 0.916(0.001) | 0.848(0.001) | 0.827(0.001) | 0.840(0.001) | 0.849(0.001) | 0.844(0.001)
MVSC-CSLF | 0.913(0.000) | 0.834(0.000) | 0.818(0.001) | 0.917(0.000) | 0.913(0.000) | 0.913(0.000)
MVSC-CSLFS | 0.917(0.002) | 0.852(0.001) | 0.828(0.004) | 0.921(0.000) | 0.917(0.000) | 0.917(0.000)
Prokaryotic | LRSC | 0.582(0.005) | 0.079(0.011) | 0.038(0.029) | 0.407(0.011) | 0.964(0.047) | 0.571(0.001)
LMSC | 0.709(0.032) | 0.418(0.034) | 0.394(0.052) | 0.669(0.067) | 0.585(0.068) | 0.618(0.026)
CSMSC | 0.658(0.006) | 0.352(0.004) | 0.367(0.005) | 0.665(0.005) | 0.527(0.007) | 0.588(0.004)
MLRSSC | 0.646(0.038) | 0.309(0.018) | 0.324(0.037) | 0.529(0.011) | 0.531(0.006) | 0.498(0.005)
MLRSSC-Centroid | 0.620(0.009) | 0.196(0.008) | 0.260(0.005) | 0.444(0.033) | 0.399(0.018) | 0.363(0.024)
KMLRSSC | 0.638(0.045) | 0.420(0.041) | 0.355(0.071) | 0.609(0.032) | 0.641(0.053) | 0.589(0.041)
KMLRSSC-Centroid | 0.655(0.045) | 0.405(0.030) | 0.350(0.072) | 0.606(0.031) | 0.657(0.046) | 0.597(0.042)
CSI | 0.619(0.000) | 0.413(0.000) | 0.291(0.000) | 0.634(0.000) | 0.440(0.000) | 0.520(0.000)
FCSMC | 0.674(0.001) | 0.443(0.000) | 0.394(0.001) | 0.688(0.001) | 0.537(0.001) | 0.603(0.001)
MVSC-CSLF | 0.763(0.001) | 0.449(0.004) | 0.521(0.004) | 0.614(0.001) | 0.596(0.007) | 0.600(0.002)
MVSC-CSLFS | 0.809(0.001) | 0.489(0.002) | 0.568(0.001) | 0.617(0.001) | 0.635(0.001) | 0.624(0.001)
3-sources | LRSC | 0.596(0.026) | 0.462(0.009) | 0.351(0.020) | 0.471(0.037) | 0.585(0.054) | 0.518(0.004)
LMSC | 0.734(0.020) | 0.676(0.018) | 0.579(0.017) | 0.729(0.024) | 0.621(0.047) | 0.669(0.018)
CSMSC | 0.798(0.002) | 0.747(0.005) | 0.673(0.004) | 0.730(0.005) | 0.773(0.001) | 0.751(0.003)
MLRSSC | 0.677(0.060) | 0.593(0.026) | 0.550(0.067) | 0.535(0.048) | 0.596(0.060) | 0.544(0.059)
MLRSSC-Centroid | 0.666(0.051) | 0.590(0.019) | 0.534(0.053) | 0.512(0.045) | 0.551(0.068) | 0.513(0.056)
KMLRSSC | 0.609(0.038) | 0.526(0.027) | 0.423(0.035) | 0.536(0.59) | 0.564(0.079) | 0.517(0.060)
KMLRSSC-Centroid | 0.621(0.025) | 0.531(0.020) | 0.457(0.031) | 0.544(0.057) | 0.562(0.051) | 0.519(0.046)
CSI | 0.763(0.000) | 0.747(0.000) | 0.622(0.000) | 0.717(0.000) | 0.702(0.000) | 0.709(0.000)
FCSMC | 0.908(0.003) | 0.795(0.006) | 0.793(0.008) | 0.887(0.004) | 0.795(0.007) | 0.839(0.006)
MVSC-CSLF | 0.843(0.016) | 0.696(0.014) | 0.659(0.034) | 0.810(0.013) | 0.873(0.007) | 0.833(0.012)
MVSC-CSLFS | 0.894(0.000) | 0.754(0.000) | 0.793(0.000) | 0.840(0.000) | 0.860(0.000) | 0.821(0.000)

It is worth noting that MVSC-CSLF, which only has a pretty simple constraint
strategy on coefficient matrix, performs better than many of compared methods
on most datasets. The promising performance indicates that well distributed
representation is significant for subspace clustering, and when obtained
suitable latent representation, even a simple form can derive satisfactory
clustering results. However, such strategy cannot make full use of the
obtained representations, therefore the proposed MVSC-CSLFS which has advanced
trick can further make a significant improvement on clustering performance.

###  4.5 Ablation Study: Is the latent representation good?

The core assumption in this study is that the proposed MVSC-CSLF and MVSC-
CSLFS perform well because they have good representations, and thus we design
an ablation experiment to demonstrate this. First of all, MVSC-CSLFS is
employed on ORL and Yale to calculate the latent representations:
${\bf{H}}_{s}^{1}$, ${\bf{H}}_{s}^{2}$, ${\bf{H}}_{s}^{3}$, ${\bf{H}}_{c}$,
and $\bf{H}$. Among them, the first three ones are view-specific
representations, ${\bf{H}}_{c}$ is consistent representation of all views, and
$\bf{H}$ is the joint latent representation concatenated by the first four
ones. Then LRSC [27] (a single view subspace clustering algorithm, and we use
its first variant: uncorrupted data and relaxed constriants) is employed to
test the performance on both raw data and latent representations in terms of
three major clustering metrics: ACC, NMI, and AR. As shown in Table 4, LRSC is
conducted on 3 views’ raw data, 3 views’ specific, consistent, and joint
representations, respectively. And the results obviously indicate that the
latent representations have made a significant improvement compared with raw
data. Note that joint representation may not bring a better improvement, and
this is why we further propose an advanced strategy to improve on the basis of
MVSC-CSLF. However, the ability of joint representation is also satisfied and
MVSC-CSLF only has one coefficient matrix that can reduce computing and
storage cost. Therefore, how to choose the two proposed algorithms should be
based on the actual situation and requirement.

Table 4: Comparison of the clustering performance on raw data and latent representation Data | Method | ACC | NMI | ARI
---|---|---|---|---
ORL | view1 | 0.691(0.021) | 0.834(0.009) | 0.567(0.022)
view2 | 0.769(0.015) | 0.883(0.015) | 0.688(0.037)
view3 | 0.560(0.018) | 0.747(0.010) | 0.418(0.021)
view1-specific | 0.855(0.015) | 0.933(0.005) | 0.809(0.013)
view2-specific | 0.847(0.020) | 0.928(0.005) | 0.787(0.013)
view3-specific | 0.815(0.016) | 0.910(0.009) | 0.744(0.021)
consistent | 0.847(0.017) | 0.928(0.007) | 0.788(0.017)
joint | 0.836(0.026) | 0.923(0.010) | 0.775(0.033)
Yale | view1 | 0.675(0.027) | 0.706(0.019) | 0.491(0.032)
view2 | 0.587(0.015) | 0.597(0.009) | 0.371(0.013)
view3 | 0.508(0.020) | 0.571(0.020) | 0.310(0.026)
view1-specific | 0.726(0.011) | 0.736(0.009) | 0.525(0.016)
view2-specific | 0.715(0.018) | 0.734(0.010) | 0.544(0.013)
view3-specific | 0.733(0.005) | 0.741(0.003) | 0.544(0.007)
consistent | 0.718(0.009) | 0.713(0.011) | 0.516(0.016)
joint | 0.718(0.018) | 0.733(0.013) | 0.537(0.019)

###  4.6 Hyper-parameters Tuning Strategy

Both MVSC-CSLF and MVSC-CSLFS have five hyperparameters, where $\lambda_{1}$,
$\lambda_{2}$, and $\lambda_{3}$ are trade-off parameters of different loss
and penalty terms, $K_{s}$ and $K_{c}$ are view-specific and consistent latent
representations’ dimensions, respectively. We design a greedy grid search
strategy in terms of the metric of ACC to find the best hyperparameters’
combination. Firstly, we fix $\lambda_{1}$, $\lambda_{2}$, $\lambda_{3}$, and
search the best combination of $K_{s}$ and $K_{c}$ range from 10 to 350, the
results on different datasets w.r.t MVSC-CSLF and MVSC-CSLFS are shown in Fig.
6 and Fig. 7, respectively. Secondly, fix $\lambda_{3}$, $K_{s}$ and $K_{c}$,
we search the best combination of $\lambda_{1}$ and $\lambda_{2}$, and the
search space is $\\{0.01,0.1,1,10,100\\}$. The results on different datasets
w.r.t MVSC-CSLF and MVSC-CSLFS are shown in Fig. 8 and Fig. 9, respectively.
Finally, we fix others and tune $\lambda_{3}$ from $\\{0.1,0.5,1,5,10\\}$, and
the results on different datasets w.r.t MVSC-CSLF and MVSC-CSLFS are shown in
Fig. 10.

Figure 6: Visualization for the hyperparameter selection procedure: $K_{s}$
and $K_{c}$. We conduct MVSC-CSLF 3 times with different hyperparameter
combinations on each data set, and the three axes of the coordinate system
represent $K_{s}$, $K_{c}$, and the mean value of clustering accuracy,
respectively. Figure 7: Visualization for the hyperparameter selection
procedure: $K_{s}$ and $K_{c}$. We conduct MVSC-CSLFS 3 times with different
hyperparameter combinations on each data set, and the three axes of the
coordinate system represent $K_{s}$, $K_{c}$, and the mean value of clustering
accuracy, respectively. Figure 8: Visualization for the hyperparameter
selection procedure: $\lambda_{1}$ and $\lambda_{2}$. We conduct MVSC-CSLF 3
times with different hyperparameter combinations on each data set, and the
three axes of the coordinate system represent $\lambda_{1}$, $\lambda_{2}$,
and the mean value of clustering accuracy, respectively. Figure 9:
Visualization for the hyperparameter selection procedure: $\lambda_{1}$ and
$\lambda_{2}$. We conduct MVSC-CSLFS 3 times with different hyperparameter
combinations on each data set, and the three axes of the coordinate system
represent $\lambda_{1}$, $\lambda_{2}$, and the mean value of clustering
accuracy, respectively.

Figure 10: Visualization for the hyperparameter selection procedure w.r.t
$\lambda_{3}$ of MVSC-CSLF and MVSC-CSLFS, respectively. We conduct MVSC-CSLF
and MVSC-CSLFS 3 times with different $\lambda_{3}$ on each data set,
therefore the two axes of the coordinate system represent $\lambda_{3}$ and
the mean value of clustering accuracy, respectively.

##  5 Conclusion and Future Work

In this study, we emphasize the importance of a good representation performing
in self-expressive subspace clustering, and introduce a matrix factorization
method to explore and separate the consistent and view-specific information
existing in multiple views. Based on the idea that representation learning
process should be guided by specific learning tasks, we propose two multi-view
subspace clustering algorithms MVSC-CSLF and MVSC-CSLFS. Among them, MVSC-CSLF
use a feature-level fusion strategy to fuse consistent and complementary
information into a joint latent representation, and then perform self-
expressive reconstruction on it. MVSC-CSLFS use a subspace-level hierarchical
strategy to impose the matching prior constraints on different subspace
representations. Experimental results indicate that the quality of data
representation is essential for subspace clustering. And our proposed MVSC-
CSLFS can further improve clustering results by designing suitable self-
expressive reconstruction strategies according to the properties of latent
representations.

Although this study has achieved good results, there are still many
challenges. One is to improve the capability of model to deal with non-linear
and corrupted data. Another is to transfer this paradigm to deep neural
network, which has more powerful representation ability. Finally, our proposed
algorithms are time-consuming, which is a common problem of subspace learning.
Therefore, some new techniques should be introduced to speed up the
computation, such as binary representation [47].

## References

  * Ojala et al. [2002] T. Ojala, M. Pietikäinen, T. Mäenpää,  Multiresolution gray-scale and rotation invariant texture classification with local binary patterns,  IEEE Trans. Pattern Anal. Mach. Intell. 24 (2002) 971–987.
  * Dalal and Triggs [2005] N. Dalal, B. Triggs,  Histograms of oriented gradients for human detection,  in: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, volume I, 2005, pp. 886–893.
  * Lowe [2004] D. G. Lowe,  Distinctive image features from scale-invariant keypoints,  Int. J. Comput. Vis. 60 (2004) 91–110.
  * Sun et al. [2019] S. Sun, L. Mao, Z. Dong, L. Wu, Multiview Machine Learning, Springer, 2019.
  * Liu et al. [2015] J. Liu, Y. Jiang, Z. Li, Z. H. Zhou, H. Lu,  Partially shared latent factor learning with multiview data,  IEEE Trans. Neural Netw. Learn. Syst. 26 (2015) 1233–1246.
  * Zhang et al. [2015] C. Zhang, H. Fu, S. Liu, G. Liu, X. Cao,  Low-rank tensor constrained multiview subspace clustering,  in: IEEE International Conference on Computer Vision, 2015, pp. 1582–1590.
  * Cao et al. [2015] X. Cao, C. Zhang, H. Fu, Si Liu, Hua Zhang,  Diversity-induced Multi-view Subspace Clustering,  in: IEEE Conference on Computer Vision and Pattern Recognition, 2015, pp. 586–594.
  * Zhang et al. [2017] C. Zhang, Q. Hu, H. Fu, P. Zhu, X. Cao,  Latent Multi-view Subspace Clustering,  in: IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 4333–4341.
  * Brbić and Kopriva [2018] M. Brbić, I. Kopriva,  Multi-view low-rank sparse subspace clustering,  Pattern Recognit. 73 (2018) 247–258.
  * Luo et al. [2018] S. Luo, C. Zhang, W. Zhang, X. Cao,  Consistent and specific multi-view subspace clustering,  in: AAAI Conference on Artificial Intelligence, 2018, pp. 3730–3737.
  * Zhou et al. [2019] T. Zhou, C. Zhang, X. Peng, H. Bhaskar, J. Yang,  Dual Shared-Specific Multiview Subspace Clustering,  IEEE T. Cybern. (2019) 1–14.
  * Zhang et al. [2020] C. Zhang, H. Fu, Q. Hu, X. Cao, Y. Xie, D. Tao, D. Xu,  Generalized latent multi-view subspace clustering,  IEEE Trans. Pattern Anal. Mach. Intell. 42 (2020) 86–99.
  * Weng et al. [2020] W. Weng, W. Zhou, J. Chen, H. Peng, H. Cai,  Enhancing multi-view clustering through common subspace integration by considering both global similarities and local structures,  Neurocomputing 378 (2020) 375–386.
  * Zheng et al. [2020] Q. Zheng, J. Zhu, Z. Li, S. Pang, J. Wang, Y. Li,  Feature concatenation multi-view subspace clustering,  Neurocomputing 379 (2020) 89–102.
  * Vidal et al. [2016] R. Vidal, Y. Ma, S. S. Sastry, Generalized Principal Component Analysis, Springer, 2016.
  * Chen et al. [2020] M. Chen, L. Huang, C.-D. Wang, D. Huang,  Multi-View Clustering in Latent Embedding Space,  in: AAAI Conference on Artificial Intelligence, 2020, pp. 3513–3520.
  * Lu et al. [2012] C.-Y. Lu, H. Min, Z.-Q. Zhao, L. Zhu, D.-S. Huang, S. Yan,  Robust and Efficient Subspace Segmentation via Least Squares Regression,  in: European Conference on Computer Vision, 2012, pp. 347–360.
  * Pan et al. [2014] J. Pan, M. Salzmann, H. Li,  Efficient dense subspace clustering,  in: IEEE Winter Conference on Applications of Computer Vision, 2014, pp. 461–468.
  * Cui et al. [2021] Z. Cui, X. Jing, P. Zhao, W. Zhang, J. Chen,  A new subspace clustering strategy for ai-based data analysis in iot system,  IEEE Internet of Things Journal 8 (2021) 12540–12549.
  * Xia et al. [2018] G. Xia, H. Sun, L. Feng, G. Zhang, Y. Liu,  Human motion segmentation via robust kernel sparse subspace clustering,  IEEE Trans. Image Process. 27 (2018) 135–150.
  * Wang et al. [2018] L. Wang, Z. Ding, Y. Fu,  Learning transferable subspace for human motion segmentation,  in: AAAI Conference on Artificial Intelligence, 2018, pp. 4195–4202.
  * Zheng et al. [2018] W. Zheng, Y. Zong, X. Zhou, M. Xin,  Cross-domain color facial expression recognition using transductive transfer subspace learning,  IEEE Trans. Affect. Comput. 9 (2018) 21–37.
  * Liao and Gu [2019] M. Liao, X. Gu,  Face recognition based on dictionary learning and subspace learning,  Digit. Signal Prog. 90 (2019) 110–124.
  * Qiu and Hao [2021] Y. Qiu, P. Hao,  Self-supervised deep subspace clustering network for faces in videos,  The Visual Computer 37 (2021) 2253–2261.
  * Elhamifar and Vidal [2013] E. Elhamifar, R. Vidal,  Sparse Subspace Clustering: Algorithm, Theory, and Applications,  IEEE Trans. Pattern Anal. Mach. Intell. 35 (2013) 2765–2781.
  * Patel et al. [2013] V. M. Patel, H. Van Nguyen, R. Vidal,  Latent Space Sparse Subspace Clustering,  in: IEEE International Conference on Computer Vision, 2013, pp. 225–232.
  * Vidal and Favaro [2014] R. Vidal, P. Favaro,  Low rank subspace clustering (LRSC),  Pattern Recognit. Lett. 43 (2014) 47–61.
  * Patel and Vidal [2014] V. M. Patel, R. Vidal,  Kernel sparse subspace clustering,  in: IEEE International Conference on Image Processing, 2014, pp. 2849–2853.
  * Yin et al. [2016] M. Yin, Y. Guo, J. Gao, Z. He, S. Xie,  Kernel sparse subspace clustering on symmetric positive definite manifolds,  in: IEEE Conference on Computer Vision and Pattern Recognition, 2016, pp. 5157–5164.
  * Xiao et al. [2016] S. Xiao, M. Tan, D. Xu, Z. Y. Dong,  Robust Kernel Low-Rank Representation,  IEEE Trans. Neural Netw. Learn. Syst. 27 (2016) 2268–2281.
  * Lu et al. [2019] C. Lu, J. Feng, Z. Lin, T. Mei, S. Yan,  Subspace Clustering by Block Diagonal Representation,  IEEE Trans. Pattern Anal. Mach. Intell. 41 (2019) 487–501.
  * Abavisani and Patel [2018] M. Abavisani, V. M. Patel,  Multimodal sparse and low-rank subspace clustering,  Inf. Fusion 39 (2018) 168–177.
  * Xie et al. [2020] D. Xie, W. Xia, Q. Wang, Q. Gao, S. Xiao,  Multi-view clustering by joint manifold learning and tensor nuclear norm,  Neurocomputing 380 (2020) 105–114.
  * Zhang et al. [2019] X. Zhang, H. Sun, Z. Liu, Z. Ren, Q. Cui, Y. Li,  Robust low-rank kernel multi-view subspace clustering based on the schatten _p_ -norm and correntropy,  Inf. Sci. 477 (2019) 430–447.
  * Ji et al. [2017] P. Ji, T. Zhang, H. Li, M. Salzmann, I. D. Reid, Deep Subspace Clustering Networks, 2017. arXiv:1709.02508.
  * Xue et al. [2019] Z. Xue, J. Du, D. Du, S. Lyu,  Deep low-rank subspace ensemble for multi-view clustering,  Inf. Sci. 482 (2019) 210--227.
  * Zhu et al. [2019] P. Zhu, B. Hui, C. Zhang, D. Du, L. Wen, Q. Hu, Multi-view deep subspace clustering networks, 2019\. arXiv:1908.01978.
  * Lu et al. [2021] R.-k. Lu, J.-w. Liu, X. Zuo,  Attentive multi-view deep subspace clustering net,  Neurocomputing 435 (2021) 186--196.
  * Wahba [1965] G. Wahba,  A Least Squares Estimate of Satellite Attitude,  SIAM Rev. 7 (1965) 409--409.
  * Farrell et al. [1966] J. L. Farrell, J. C. Stuelpnagel, R. H. Wessner, J. R. Velman, J. E. Brook,  A Least Squares Estimate of Satellite Attitude (Grace Wahba),  SIAM Rev. 8 (1966) 384--386.
  * Bartels and Stewart [1972] R. H. Bartels, G. W. Stewart,  Solution of the Matrix Equation AX + XB = C [F4],  Commun. ACM 15 (1972) 820--826.
  * Cai et al. [2010] J. Cai, E. Candès, Z. Shen,  A Singular Value Thresholding Algorithm for Matrix Completion,  SIAM J. Optim. 20 (2010) 1956--1982.
  * Liu et al. [2013] G. Liu, Z. Lin, S. Yan, J. Sun, Y. Yu, Y. Ma,  Robust recovery of subspace structures by low-rank representation,  IEEE Trans. Pattern Anal. Mach. Intell. 35 (2013) 171--184.
  * Brbić et al. [2016] M. Brbić, M. Piškorec, V. Vidulin, A. Kriško, T. Šmuc, F. Supek,  The landscape of microbial phenotypic traits and associated genes,  Nucleic Acids Res. 44 (2016) 10074--10090.
  * Lewis et al. [2004] D. D. Lewis, Y. Yang, T. G. Rose, F. Li,  Rcv1: A new benchmark collection for text categorization research,  J. Mach. Learn. Res. 5 (2004) 361--397.
  * Ng et al. [2002] A. Y. Ng, M. I. Jordan, Y. Weiss,  On Spectral Clustering: Analysis and an algorithm,  in: Advances in Neural Information Processing Systems 14, 2002, pp. 849--856.
  * Zhang et al. [2019] Z. Zhang, L. Liu, F. Shen, H. T. Shen, L. Shao,  Binary Multi-View Clustering,  IEEE Trans. Pattern Anal. Mach. Intell. 41 (2019) 1774--1782.


# Formatting Instructions For NeurIPS 2019

David S. Hippocampus
Department of Computer Science
Cranberry-Lemon University
Pittsburgh, PA 15213
<EMAIL_ADDRESS>
Use footnote for providing further information about author (webpage,
alternative address)— _not_ for acknowledging funding agencies.

###### Abstract

The abstract paragraph should be indented $\nicefrac{{1}}{{2}}$ inch (3 picas)
on both the left- and right-hand margins. Use 10 point type, with a vertical
spacing (leading) of 11 points. The word Abstract must be centered, bold, and
in point size 12. Two line spaces precede the abstract. The abstract must be
limited to one paragraph.

##  1 Submission of papers to NeurIPS 2019

NeurIPS requires electronic submissions. The electronic submission site is

https://cmt.research.microsoft.com/NeurIPS2019/

Please read the instructions below carefully and follow them faithfully.

###  1.1 Style

Papers to be submitted to NeurIPS 2019 must be prepared according to the
instructions presented here. Papers may only be up to eight pages long,
including figures. Additional pages _containing only acknowledgments and/or
cited references_ are allowed. Papers that exceed eight pages of content
(ignoring references) will not be reviewed, or in any other way considered for
presentation at the conference.

The margins in 2019 are the same as since 2007, which allow for $\sim$$15\%$
more words in the paper compared to earlier years.

Authors are required to use the NeurIPS LaTeX style files obtainable at the
NeurIPS website as indicated below. Please make sure you use the current files
and not previous versions. Tweaking the style files may be grounds for
rejection.

###  1.2 Retrieval of style files

The style files for NeurIPS and other conference information are available on
the World Wide Web at

http://www.neurips.cc/

The file `neurips_2019.pdf` contains these instructions and illustrates the
various formatting requirements your NeurIPS paper must satisfy.

The only supported style file for NeurIPS 2019 is `neurips_2019.sty`,
rewritten for LaTeX 2ε. Previous style files for LaTeX 2.09, Microsoft Word,
and RTF are no longer supported!

The LaTeX style file contains three optional arguments: `final`, which creates
a camera-ready copy, `preprint`, which creates a preprint for submission to,
e.g., arXiv, and `nonatbib`, which will not load the `natbib` package for you
in case of package clash.

##### Preprint option

If you wish to post a preprint of your work online, e.g., on arXiv, using the
NeurIPS style, please use the `preprint` option. This will create a
nonanonymized version of your work with the text “Preprint. Work in progress.”
in the footer. This version may be distributed as you see fit. Please do not
use the `final` option, which should only be used for papers accepted to
NeurIPS.

At submission time, please omit the `final` and `preprint` options. This will
anonymize your submission and add line numbers to aid review. Please do _not_
refer to these line numbers in your paper as they will be removed during
generation of camera-ready copies.

The file `neurips_2019.tex` may be used as a “shell” for writing your paper.
All you have to do is replace the author, title, abstract, and text of the
paper with your own.

The formatting instructions contained in these style files are summarized in
Sections 2, 3, and 4 below.

##  2 General formatting instructions

The text must be confined within a rectangle 5.5 inches (33 picas) wide and 9
inches (54 picas) long. The left margin is 1.5 inch (9 picas). Use 10 point
type with a vertical spacing (leading) of 11 points. Times New Roman is the
preferred typeface throughout, and will be selected for you by default.
Paragraphs are separated by $\nicefrac{{1}}{{2}}$ line space (5.5 points),
with no indentation.

The paper title should be 17 point, initial caps/lower case, bold, centered
between two horizontal rules. The top rule should be 4 points thick and the
bottom rule should be 1 point thick. Allow $\nicefrac{{1}}{{4}}$ inch space
above and below the title to rules. All pages should start at 1 inch (6 picas)
from the top of the page.

For the final version, authors’ names are set in boldface, and each name is
centered above the corresponding address. The lead author’s name is to be
listed first (left-most), and the co-authors’ names (if different address) are
set to follow. If there is only one co-author, list both author and co-author
side by side.

Please pay special attention to the instructions in Section 4 regarding
figures, tables, acknowledgments, and references.

##  3 Headings: first level

All headings should be lower case (except for first word and proper nouns),
flush left, and bold.

First-level headings should be in 12-point type.

###  3.1 Headings: second level

Second-level headings should be in 10-point type.

####  3.1.1 Headings: third level

Third-level headings should be in 10-point type.

##### Paragraphs

There is also a `\paragraph` command available, which sets the heading in
bold, flush left, and inline with the text, with the heading followed by 1 em
of space.

##  4 Citations, figures, tables, references

These instructions apply to everyone.

###  4.1 Citations within the text

The `natbib` package will be loaded for you by default. Citations may be
author/year or numeric, as long as you maintain internal consistency. As to
the format of the references themselves, any style is acceptable as long as it
is used consistently.

The documentation for `natbib` may be found at

http://mirrors.ctan.org/macros/latex/contrib/natbib/natnotes.pdf

Of note is the command `\citet`, which produces citations appropriate for use
in inline text. For example,

       \citet{hasselmo} investigated\dots

produces

> Hasselmo, et al. (1995) investigated…

If you wish to load the `natbib` package with options, you may add the
following before loading the `neurips_2019` package:

       \PassOptionsToPackage{options}{natbib}

If `natbib` clashes with another package you load, you can add the optional
argument `nonatbib` when loading the style file:

       \usepackage[nonatbib]{neurips_2019}

As submission is double blind, refer to your own published work in the third
person. That is, use “In the previous work of Jones et al. [4],” not “In our
previous work [4].” If you cite your other papers that are not widely
available (e.g., a journal paper under review), use anonymous author names in
the citation, e.g., an author of the form “A. Anonymous.”

###  4.2 Footnotes

Footnotes should be used sparingly. If you do require a footnote, indicate
footnotes with a number111Sample of the first footnote. in the text. Place the
footnotes at the bottom of the page on which they appear. Precede the footnote
with a horizontal rule of 2 inches (12 picas).

Note that footnotes are properly typeset _after_ punctuation marks.222As in
this example.

###  4.3 Figures

Figure 1: Sample figure caption.

All artwork must be neat, clean, and legible. Lines should be dark enough for
purposes of reproduction. The figure number and caption always appear after
the figure. Place one line space before the figure caption and one line space
after the figure. The figure caption should be lower case (except for first
word and proper nouns); figures are numbered consecutively.

You may use color figures. However, it is best for the figure captions and the
paper body to be legible if the paper is printed in either black/white or in
color.

###  4.4 Tables

All tables must be centered, neat, clean and legible. The table number and
title always appear before the table. See Table 1.

Place one line space before the table title, one line space after the table
title, and one line space after the table. The table title must be lower case
(except for first word and proper nouns); tables are numbered consecutively.

Note that publication-quality tables _do not contain vertical rules._ We
strongly suggest the use of the `booktabs` package, which allows for
typesetting high-quality, professional tables:

https://www.ctan.org/pkg/booktabs

This package was used to typeset Table 1.

Table 1: Sample table title Part |
---|---
Name | Description | Size ($\mu$m)
Dendrite | Input terminal |  $\sim$100
Axon | Output terminal |  $\sim$10
Soma | Cell body | up to $10^{6}$

##  5 Final instructions

Do not change any aspects of the formatting parameters in the style files. In
particular, do not modify the width or length of the rectangle the text should
fit into, and do not change font sizes (except perhaps in the References
section; see below). Please note that pages should be numbered.

##  6 Preparing PDF files

Please prepare submission files with paper size “US Letter,” and not, for
example, “A4.”

Fonts were the main cause of problems in the past years. Your PDF file must
only contain Type 1 or Embedded TrueType fonts. Here are a few instructions to
achieve this.

  * •

You should directly generate PDF files using `pdflatex`.

  * •

You can check which fonts a PDF files uses. In Acrobat Reader, select the menu
Files$>$Document Properties$>$Fonts and select Show All Fonts. You can also
use the program `pdffonts` which comes with `xpdf` and is available out-of-
the-box on most Linux machines.

  * •

The IEEE has recommendations for generating PDF files whose fonts are also
acceptable for NeurIPS. Please see
http://www.emfield.org/icuwb2010/downloads/IEEE-PDF-SpecV32.pdf

  * •

`xfig` "patterned" shapes are implemented with bitmap fonts. Use "solid"
shapes instead.

  * •

The `\bbold` package almost always uses bitmap fonts. You should use the
equivalent AMS Fonts:

           \usepackage{amsfonts}

followed by, e.g., `\mathbb{R}`, `\mathbb{N}`, or `\mathbb{C}` for
$\mathbb{R}$, $\mathbb{N}$ or $\mathbb{C}$. You can also use the following
workaround for reals, natural and complex:

           \newcommand{\RR}{I\!\!R} %real numbers
       \newcommand{\Nat}{I\!\!N} %natural numbers
       \newcommand{\CC}{I\!\!\!\!C} %complex numbers

Note that `amsfonts` is automatically loaded by the `amssymb` package.

If your file contains type 3 fonts or non embedded TrueType fonts, we will ask
you to fix it.

###  6.1 Margins in LaTeX

Most of the margin problems come from figures positioned by hand using
`\special` or other commands. We suggest using the command `\includegraphics`
from the `graphicx` package. Always specify the figure width as a multiple of
the line width as in the example below:

       \usepackage[pdftex]{graphicx} ...
       \includegraphics[width=0.8\linewidth]{myfile.pdf}

See Section 4.4 in the graphics bundle documentation
(http://mirrors.ctan.org/macros/latex/required/graphics/grfguide.pdf)

A number of width problems arise when LaTeX cannot properly hyphenate a line.
Please give LaTeX hyphenation hints using the `\-` command when necessary.

#### Acknowledgments

Use unnumbered third level headings for the acknowledgments. All
acknowledgments go at the end of the paper. Do not include acknowledgments in
the anonymized submission, only in the final paper.

## References

References follow the acknowledgments. Use unnumbered first-level heading for
the references. Any choice of citation style is acceptable as long as you are
consistent. It is permissible to reduce the font size to `small` (9 point)
when listing the references. Remember that you can use more than eight pages
as long as the additional pages contain _only_ cited references.

[1] Alexander, J.A. & Mozer, M.C. (1995) Template-based algorithms for
connectionist rule extraction. In G. Tesauro, D.S. Touretzky and T.K. Leen
(eds.), Advances in Neural Information Processing Systems 7, pp. 609–616.
Cambridge, MA: MIT Press.

[2] Bower, J.M. & Beeman, D. (1995) The Book of GENESIS: Exploring Realistic
Neural Models with the GEneral NEural SImulation System. New York:
TELOS/Springer–Verlag.

[3] Hasselmo, M.E., Schnell, E. & Barkai, E. (1995) Dynamics of learning and
recall at excitatory recurrent synapses and cholinergic modulation in rat
hippocampal region CA3. Journal of Neuroscience 15(7):5249-5262.