# Preventing Dimensional Collapse of Incomplete Multi-View Clustering via
Direct Contrastive Learning

Kaiwu Zhang, Shiqiang Du, Baokai Liu, and Shengxia Gao

###### Abstract

Incomplete multi-view clustering (IMVC) is an unsupervised approach, among
which IMVC via contrastive learning has received attention due to its
excellent performance. The previous methods have the following problems: 1)
Over-reliance on additional projection heads when solving the dimensional
collapse problem in which latent features are only valid in lower-dimensional
subspaces during clustering. However, many parameters in the projection heads
are unnecessary. 2) The recovered view contain inconsistent private
information and useless private information will mislead the learning of
common semantics due to consistent learning and reconstruction learning on the
same feature. To address the above issues, we propose a novel incomplete
multi-view contrastive clustering framework. This framework directly optimizes
the latent feature subspace, utilizes the learned feature vectors and their
sub-vectors for reconstruction learning and consistency learning, thereby
effectively avoiding dimensional collapse without relying on projection heads.
Since reconstruction loss and contrastive loss are performed on different
features, the adverse effect of useless private information is reduced. For
the incomplete data, the missing information is recovered by the cross-view
prediction mechanism and the inconsistent information from different views is
discarded by the minimum conditional entropy to further avoid the influence of
private information. Extensive experimental results of the method on 5 public
datasets show that the method achieves state-of-the-art clustering results.

###### Index Terms:

Multi-view learning, incomplete multi-view clustering, contrastive learning

##  I Introduction

In the current era of information explosion, a large amount of data floods
into our lives. How to mine the hidden structure of the data from the massive
large-scale data, so that the useful information of the data can be better
utilized has gradually attracted widespread attention. As one of the most
important unsupervised methods, clustering [1, 2, 3, 4, 5] has long been a key
technique in pattern recognition and machine learning, which groups data
samples according to a certain criterion so that the similar samples are
accumulated into the same cluster. With the development of modern
communication technologies, most of the available data are collected from
multiple sources or described by various feature collectors, thus generating
multi-view data. Since multi-view data can provide common semantics to improve
learning efficiency, multi-view clustering [6, 7, 8] has received increasing
attention.

However, in practical applications, the information of multi-view data may be
missing in some views due to the influence of machine damage or sensor
problems, which leads to the generation of incomplete multi-view data. The
appearance of incomplete multi-view data makes the existing multi-view
clustering methods extremely limited and inapplicable. Therefore, in order to
deal with the problem of incomplete multi-view data, people have proposed
incomplete multi-view clustering methods [9, 10, 11, 12, 13]. The existing
incomplete multi-view clustering methods can be roughly divided into for
traditional and deep methods.

Traditional incomplete multi-view clustering methods [14, 15, 16, 17] usually
utilize traditional machine learning methods to obtain a discriminative new
feature representation to replace the original data for clustering. Existing
traditional incomplete multi-view clustering methods focus on using zero or
mean imputation techniques to fill in incomplete information. In this way,
Zhou et al. [18] proposed to fill missing samples with mean eigenvalues, and
then perform clustering on the filled data. However, this padding method may
introduce some useless or even noisy information, resulting in poor quality of
the construction graph. In order to avoid the above problems, people propose
to directly learn the common representation or subspace without the process of
filling. Hu et al. [19] proposed a doubly aligned incomplete multi-view
clustering (DAIMC) method, which uses regression constraints to align the
basis matrices, so that more usable information between views can be obtained.
Although traditional incomplete multi-view clustering methods have been well
developed, they suffer from the disadvantage of limited representational
ability, which leads to their limited performance in complex scenes with real
data.

In recent years, deep incomplete multi-view clustering methods [20, 21, 22]
have gradually become a popular trend in the community due to their strong
generalization performance and scalability. Deep incomplete multi-view
clustering methods typically utilize an imputation strategy to infer possible
values for missing data prior to multi-view clustering. For instance, Xu et
al. [23] proposed using generative adversarial networks to generate desired
views for missing views and implemented a weighted adaptive fusion scheme,
which can better utilize the complementary information between different
views. Due to the rise of contrastive learning, many methods based on
contrastive learning have been proposed. As a typical contrastive learning-
based method, COMPLETER [24] believed that cross-view consistency learning and
data recovery were intrinsically linked within the framework of information
theory, and proposed to recover missing data through dual prediction. Both of
the above methods learned the consistency information of views by utilizing
latent features and applied reconstruction learning and consistency learning
to the same features. Since there are both common information of all views and
private information for individual view in multi-view data, reconstruction
learning hopes to preserve the private information of views as much as
possible and consistent learning is dedicated to mining the common information
of views. Therefore, these two methods cannot avoid the influence of useless
private information in consistency learning. DSIMVC proposed by Tang et al.
[25] effectively solved the above problems by using MLP to learn high-level
features after learning latent representations and applying reconstruction
loss and contrastive loss on these two features, respectively. In order to
prevent the problem that some dimensions of the obtained latent representation
do not contain usable information and the model collapses along some
dimensions, DSIMVC and other methods [26, 27, 28] via contrastive learning
adopt the way of adding projection heads to solve it. But many parameters in
the added projection heads are unnecessary, which will increase the complexity
of the model.

Figure 1: The framework of our model. The features $\mathbf{Z}^{v}$ are first
learned from the raw features $\mathbf{X}^{v}$ of each view through the
autoencoder. In order to prevent dimensional collapse of latent features when
applying contrastive learning, we directly send sub-vectors of latent features
to the contrastive loss for learning. The reconstruction objective
$\mathcal{L}_{\mathbf{Z}}$ and the consistency objective
$\mathcal{L}_{\mathbf{C}}$ are performed on the $\mathbf{Z}^{v}$ and their
sub-vectors, respectively. The $\mathcal{L}_{\mathrm{R}}$ loss aims to recover
one view from another through parametric models $G^{(1)}$ and $G^{(2)}$.

In summary, although existing incomplete multi-view clustering methods have
made important progress by filling missing data with imputation strategies,
they suffer from two problems. On the one hand, in order to prevent the
phenomenon of dimensional collapse in incomplete multi-view clustering methods
based on contrastive learning, existing methods all choose to add projection
heads after obtaining latent representations to solve the above problems.
However, this method will introduce many unnecessary parameters and reduce the
efficiency of the model. Therefore, how to effectively prevent dimensional
collapse without increasing the amount of model parameters is the main
challenge of incomplete multi-view clustering. On the other hand, some
existing incomplete multi-view clustering methods exploit the coherence
objective of latent features to explore common semantics across all views.
However, these methods usually recover view-private information that is
inconsistent across different views when performing data recovery, and further
preserve private information in features when the reconstruction objective is
required to avoid trivial solutions. As a consequence, how to effectively
utilize the consistent information of incomplete data and avoid the influence
of view-private information is an crucial challenge for incomplete multi-view
clustering.

This paper proposes a novel method for incomplete contrastive multi-view
clustering to solve the above problems. Specifically, our main contents
include: (1) Design a simple and direct incomplete multi-view clustering model
to better utilize the information of latent features of each dimension.
Specifically, the autoencoder is first used to learn the latent features of
each view from the raw features, and then the sub-vectors of the learned
feature vectors are sent to the contrastive loss function, which directly
optimizes the representation space and effectively prevents dimensional
collapse in the representation space. (2) In order to prevent the performance
degradation caused by recovering the inconsistent information of the views,
the model adopts the minimization of conditional entropy to recover the
missing views, and the inconsistent information will be discarded skillfully.
Since reconstruction and consistency learning directly on latent features can
cause the above conflicts, unlike most existing methods that require MLP, we
adopt a simpler approach where reconstruction learning is implemented on
latent features and consistency learning is implemented on their sub-vectors.
The approach can achieve better utilization of the useful information of the
view at the same level of feature representation. The main contributions of
this paper are as follows:

  * •

To the best of our knowledge, this is the first work that considers avoiding
the effects of dimensional collapse in a way that does not rely on projection
heads when applying contrastive learning to clustering. And we design a novel
model that optimizes directly in the latent feature subspace, which
effectively avoids dimensional collapse in clustering without increasing the
amount of unnecessary parameters.

  * •

We propose an elegant framework for incomplete multi-view contrastive
learning, which performs reconstruction learning and consistency learning on
the latent features and their sub-vectors respectively. In this way, the
conflict between the inconsistency of view-private information and the
consistency of common semantics is flexibly resolved. Furthermore, the
predictive mechanism of latent feature representation is used to achieve data
recoverability.

  * •

Due to the exquisite design of our model, it has certain robustness in the
setting of hyperparameters. Extensive experiments demonstrate that compared
with other existing methods, our method has achieved the most advanced
effectiveness on 5 real datasets.

The rest of this paper is structured as follows. Section II introduces some
work that is relevant to the content of this paper. Section III provides the
main framework and related components of the framework. Section IV describes
and analyzes a large number of numerical experiments on real data sets.
Finally, we summarize this work in section V.

##  II Related Work

In this section, we briefly present recent developments on topics relevant to
our work, including two parts: incomplete multi-view clustering and
contrastive learning.

###  II-A Incomplete Multi-view Clustering

As mentioned above, existing incomplete multi-view clustering methods can be
divided into traditional incomplete multi-view clustering and deep incomplete
multi-view clustering.

Traditional incomplete multi-view clustering methods can be divided into
matrix factorization-based methods [29, 30], kernel learning-based methods
[31, 32, 33], and graph-based methods [34, 35, 36]. Matrix factorization-based
methods attempt to project incomplete data into a low-dimensionalal common
subspace satisfying low-rank constraints by generating latent representations.
In this category, partial multi-view clustering (PVC) [11] is one of the
earliest works that used non-negative matrix factorization (NMF) [37] to
obtain a consistent latent representation of incomplete data. Furthermore, MIC
[30] used the average of the observed samples in the corresponding views to
complete unobserved samples, and then performs joint weighted NMF to learn the
shared subspace between views. Kernel learning-based methods attempt to
recover kernel matrices that are consistent across multiple views in various
ways. Liu et al. [38] proposed two multi-kernel k-means methods with
incomplete kernels. They perform joint imputation on incomplete kernel
matrices and combine them for clustering, which is a successful solution with
good results. Graph-based methods aim to explore the original geometric
structure of the data by exploiting different graphs of all sample
relationships. Based on this idea, Wen et al. [13] proposed to construct
graphs via subspace learning, which fused the partitions generated by each
view into a unified matrix.

As far as we know, deep learning has attracted much attention due to its
strong scalability and excellent ability to handle large-scale data, and many
incomplete multi-view clustering methods have been proposed through various
different deep learning frameworks. Based on generative adversarial networks,
Wang et al. [39] developed a view-specific generative adversarial network with
multi-view cycle consistency to generate missing data for one view based on
shared representations. Based on the contrastive learning, Yang et al. [40]
use a noise-robust contrastive loss, which mitigates the effect of false
negatives and overcomes the incomplete information challenge.

###  II-B Contrastive Learning

Contrastive learning [28, 41, 42] is a discriminative representation learning
framework based on the idea of contrast, which is mainly used for unsupervised
representation learning. The main purpose of contrastive learning is to
maximize the similarity between instances of the same class and minimize the
similarity between instances of different classes. Compared with generative
learning, contrastive learning does not need to pay attention to the tedious
details of instances, but only needs to learn to distinguish data in the
feature space at the abstract semantic level. As such, the generalization
ability of the model based on contrastive learning is stronger. Due to its
advantages of simple model and saving hardware resources, it has been widely
used in the fields of image generation [43], image classification [44] and
clustering [26].

Contrastive learning has currently achieved promising results in computer
vision. In terms of image generation, Liu et al. [45] employed contrastive
learning to encourage diverse image synthesisers. The proposed DivCo improves
the performance of diverse image synthesis without sacrificing the visual
quality of the generated images. In terms of representation learning, Kaveh et
al. [46] developed a multi-view representation learning method to solve the
problem of graph classification through contrastive learning. In terms of
clustering, Li et al. [27] proposed a one-stage online clustering method
called contrastive clustering, which explicitly performs instance-level and
cluster-level contrastive learning. Contrastive learning has become the main
research direction of unsupervised learning because of its excellent
achievements in various fields.

##  III Proposed Method

In this section, we introduce each implementation detail of the proposed
incomplete contrastive multi-view clustering model, including the within-view
reconstruction module, multi-view contrastive learning module, and missing-
view data recovery module.

###  III-A Within-view Reconstruction Module

We treat the dataset as the original feature, where the dataset has multiple
views with $K$ common clusters. For multi-view data, the existing methods
generally try to learn the salient features of the sample from the information
contained in the original data. In particular, deep autoencoders can
accomplish the above tasks and have been applied in many research fields due
to their excellent performance. In this paper, we use
$\left\\{\mathbf{X}^{v}\in\mathbb{R}^{N\times M_{v}}\right\\}_{v=1}^{V}$ to
represent a multi-view dataset with $V$ views and $N$ samples, and define
$\mathbf{x}_{i}^{v}\in\mathbb{R}^{M_{v}}$ to represent the $M_{v}$-dimensional
samples from the $v$-th view. In order to obtain the feature information of
each view, we use autoencoder to transform heterogeneous multi-view data into
clustering-friendly features. Specifically, the feature representation
$\left\\{\mathbf{Z}^{v}\right\\}_{v=1}^{V}$ of $v$-th view is learned by
encoder $f_{ec}^{v}\left(\mathbf{X}^{v};\theta^{v}\right)$ and decoder
$f_{dc}^{v}\left(\mathbf{Z}^{v};\phi^{v}\right)$, where $\theta^{v}$ and
$\phi^{v}$ are network parameters, and
$\mathbf{z}_{i}^{v}=f_{ec}^{v}\left(\mathbf{x}_{i}^{v}\right)\in\mathbb{R}^{D}$
as the $D$-dimensionalal latent feature of the $i$-th sample. Therefore, in
the $v$-th view, the reconstruction loss between the encoder input
$\mathbf{X}^{v}$ and the decoder output
${\mathbf{X}^{*}}^{v}\in\mathbb{R}^{N\times M_{v}}$ can be defined as
$\mathcal{L}_{\mathrm{Z}}^{v}$, so the reconstruction loss of all views can be
expressed as:

$\displaystyle\mathcal{L}_{\mathrm{Z}}=\sum_{v=1}^{V}\mathcal{L}_{\mathrm{Z}}^{v}=\sum_{v=1}^{V}\sum_{i=1}^{N}\left\|\mathbf{x}_{i}^{v}-f_{dc}^{v}\left(f_{ec}^{v}\left(\mathbf{x}_{i}^{v}\right)\right)\right\|_{2}^{2}$
(1)

The representation $\mathbf{Z}^{v}$ is defined by
$\mathbf{Z}^{v}=f_{ec}^{v}\left(\mathbf{X}^{v}\right)$. The deep multi-view
clustering methods based on autoencoders generally aimed to learn cluster-
friendly common representations by maximizing the consistency between latent
features of different views. However, the existing multi-view clustering
methods have the following two issues: (1) Most methods suffer from the
dimensional collapse problem that the latent feature vector learned in the
model collapses along some dimensions so that the dimension of the embedding
vector cannot be fully utilized. Although some methods based on contrastive
learning solve the above problems by adding additional projection heads, this
approach will result in increased computation and many parameters in the
projection heads are unnecessary. (2) Some multi-view clustering methods
perform reconstruction learning and consistency learning on the same latent
features. The purpose of reconstruction learning is to enable the learned
latent features to contain more view-private information of the original
features, while consistency learning hopes to mine useful common information
between views, which will lead to poor quality of the learned latent features
and affect the final clustering results.

To avoid the above problems, we design a multi-view contrastive learning
module. Inspired by [47], the projector in contrastive learning is of great
significance to prevent dimensional collapse, and a linear projector only need
to satisfy the diagonal and low rank to effectively prevent dimensional
collapse. The details can be found in [47]. Accordingly, we adopt a novel way
to feed the sub-vectors of the latent representation $\mathbf{Z}^{v}$ directly
into the spectral contrastive loss, which satisfies the above conditions and
optimizes the representation space, thus preventing the dimensional collapse
that occurs in the clustering model. At the same time, since we perform
consistent learning on carefully selected sub-vectors, the remaining
irrelevant information does not participate in the contrastive learning, which
helps to filter out the view-private information of features and obtain
higher-quality common representations. The specific content of this part is
elaborated in the next subsection.

###  III-B Multi-view Contrastive Learning Module

It is obvious that the latent feature
$\left\\{\mathbf{Z}^{v}\right\\}_{v=1}^{V}$ of each view obtained from the
original features does not separate the consistent information and private
information of each view. Most existing methods directly perform contrastive
learning on features $\left\\{\mathbf{Z}^{v}\right\\}_{v=1}^{V}$, which causes
private information in features to interfere with consistent learning. To
solve this problem, Xu et al. [26] avoid the above situation by adding MLP
after the feature $\left\\{\mathbf{Z}^{v}\right\\}_{v=1}^{V}$ to obtain
advanced features. Although this method can resolve the conflict between the
reconstruction objective and the consistency objective, the addition of MLP
increases the complexity of the model and reduces the efficiency of the model.
Therefore, we adopt a simple way to use sub-vectors representing features and
feed the sub-vectors directly into the contrastive loss to solve the above
problem, where we utilize the $\left[0:d_{0}\right]$-dimensional vectors of
latent features as sub-vectors for consistency learning.

Since the purpose of contrastive learning is to maximize the similarity
between positive sample pairs and minimize the similarity between negative
sample pairs, the sub-vector $\mathbf{z}_{i}^{*v}$ defined for each feature in
our model has $(VN-1)$ feature pairs, where
$\left\\{\mathbf{z}_{i}^{*v},\mathbf{z}_{i}^{*n}\right\\}_{n\neq v}$ are
$(V-1)$ positive pairs and the rest $V(N-1)$ pairs are negative pairs. Based
on the spectral contrastive loss [48], we adopt the following loss to achieve
the consistency objective, so that the features focus on learning the common
semantics of all views:

$\displaystyle\mathcal{L}_{\mathrm{C}}=$
$\displaystyle\sum_{v=1}^{V}\sum_{n=v+1}^{N}\left[-\frac{2}{N}\sum_{i=1}^{N}{\mathbf{z}_{i}^{*v}}^{\top}{\mathbf{z}_{i}^{*n}}\right.$
$\displaystyle\left.+\frac{1}{2C_{N}^{2}}\sum_{i=1}^{N}\sum_{j\neq
i}\left({\mathbf{z}_{i}^{*v}}^{\top}{\mathbf{z}_{j}^{*v}}\right)^{2}\right]$
(2)

where $C_{N}^{2}$ denotes the combinatorial operations.

The feature representation $\mathbf{Z}^{v}$ obtained by our designed network
avoids model collapse through direct contrast learning on its sub-vector and
retains more useful information by further optimizing the representation
space. In the case of complete data, we can get a cluster-friendly
representation through the above steps and run k-means on the feature
representation to get good results. Whereas, most of the data in practical
applications are incomplete. Such incomplete data will greatly affect the
quality of the learned latent representation and make multi-view clustering
methods limited. Accordingly, researchers have proposed incomplete multi-view
clustering methods to deal with incomplete data. Most of the current
incomplete multi-view clustering methods use the mean filling method to solve
the problem of missing information. But this method introduces noise
information, and the learned representations cannot fully utilize the
consistent information of views. To address this issue, we attempt to use a
missing view recovery module to predict and recover missing views from the
existing available views, so that our method can reduce the impact of missing
information on clustering performance.

###  III-C Missing View Data Recovery Module

In this module, for the latent feature structure of the original data obtained
by the autoencoder, we encourage that $\mathbf{Z}^{p}$ can be determined by
$\mathbf{Z}^{q}$ by minimizing the conditional entropy
$H\left(\mathbf{Z}^{p}\mid\mathbf{Z}^{q}\right)$. This mechanism enables the
missing views to be determined by the available views, which can not only
achieve the representation recovery of the missing views but also cleverly
discard the inconsistent information between different views to obtain a
better consistent representation. We represent the conditional entropy by the
following equivalent log conditional likelihood transformation:

$\displaystyle\begin{aligned}
H\left(\mathbf{Z}^{p}\mid\mathbf{Z}^{q}\right)=-\mathbb{E}_{\mathcal{Z}_{\mathbf{z}^{p},\mathbf{z}^{q}}}\left[\log\mathcal{P}\left(\mathbf{Z}^{p}\mid\mathbf{Z}^{q}\right)\right]\end{aligned}$
(3)

However, it is difficult to directly estimate the expected value in equation
(3). A general approach to approximate this objective is by introducing a
variational distribution
$\mathcal{Q}\left(\mathbf{Z}^{p}\mid\mathbf{Z}^{q}\right)$ and maximize the
lower bound of the expectations, i.e.,
$\mathbb{E}_{\mathcal{P}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}}\left[\log\mathcal{Q}\left(\mathbf{Z}^{p}\mid\mathbf{Z}^{q}\right)\right]$.

The above $\mathcal{Q}$ has no strong restrictions and can be implemented in
various ways, such as Gaussian distribution or through neural networks. For
simplicity and directness, we use Gaussian distribution and utilize $G(\cdot)$
as a parameter function to map $\mathbf{Z}^{p}$ to $\mathbf{Z}^{q}$, i.e.,
$\mathcal{N}\left(\mathbf{Z}^{p}\mid
G^{(q)}\left(\mathbf{Z}^{q}\right),\sigma\mathbf{I}\right)$, the
$\sigma\mathbf{I}$ appearing here is a diagonal matrix. By ignoring the
constants derived from the Gaussian distribution, the objective function in
our data recovery can be defined as:

$\displaystyle\mathcal{L}_{\mathrm{R}}=\max-\mathbb{E}_{\mathcal{P}_{\mathbf{Z}^{p},\mathbf{Z}^{q}}}\left\|G^{(q)}\left(\mathbf{Z}^{q}\right)-\mathbf{Z}^{p}\right\|_{2}^{2}$
(4)

In order to visually observe the objective function of the data recovery part,
for the data of the two views, the loss function has the following form:

$\displaystyle\mathcal{L}_{\mathrm{R}}$
$\displaystyle=\left\|\left(f_{ec}^{2}\left(\mathbf{X}^{2}\right)\right)-G^{(1)}\left(f_{ec}^{1}\left(\mathbf{X}^{1}\right)\right)\right\|_{2}^{2}$
$\displaystyle+\left\|\left(f_{ec}^{1}\left(\mathbf{X}^{1}\right)\right)-G^{(2)}\left(f_{ec}^{2}\left(\mathbf{X}^{2}\right)\right)\right\|_{2}^{2}$
(5)

Through the above loss optimization model, the missing view data recovery
module can better recover complete information from another view, further
reducing the impact of missing data on the model. The missing information can
be recovered from such a way, i.e.,
$\mathbf{Z}^{2}=G^{(1)}\left(\mathbf{Z}^{1}\right)=G^{(1)}\left(f_{ec}^{1}\left(\mathbf{X}^{1}\right)\right)$
and
$\mathbf{Z}^{1}=G^{(2)}\left(\mathbf{Z}^{2}\right)=G^{(2)}\left(f_{ec}^{2}\left(\mathbf{X}^{2}\right)\right)$.

TABLE I: Details of widely used benchmark data sets. Data | BDGP | MNIST-USPS | Caltech | Fashion | Noisy MNIST
---|---|---|---|---|---
view1 | 1750 | 784 | 40 | 784 | 784
view2 | 75 | 784 | 254 | 784 | 784
size | 2500 | 5000 | 1400 | 10000 | 20000
classes | 5 | 10 | 7 | 10 | 10

##  IV Experiments

In this section, we compare the proposed method with 7 existing state-of-the-
art methods on 5 widely used multi-view datasets, while using different
missing rates on different datasets. To demonstrate that our method can
achieve missing view recovery, a visualization of missing view recovery is
added in the experimental section to further illustrate the effectiveness of
our method. All simulations are implemented using PyTorch 1.7.1, and the
original code of this method will be provided later.

###  IV-A Experimental Setup

In our experiments, we employ randomly generated incomplete multi-view
datasets to validate our algorithm. For each dataset, we generate incomplete
samples by randomly removing partial views across all samples. The missing
rate $\eta$ is defined as $\eta=(n-m)/n$, where $m$ is the number of complete
examples, and $n$ is the number of the whole dataset, which ranges from 0.1 to
0.7 with an interval of 0.2.

####  IV-A1 Datasets Descriptions

TABLE II: Clustering results ($\%$) on BDGP. | ACC | NMI | ARI
---|---|---|---
Method | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7
AE2-NET (CVPR’19) | 57.38 | 54.03 | 49.91 | 46.88 | 46.84 | 42.00 | 38.44 | 35.21 | 18.12 | 12.74 | 9.86 | 7.39
MFLVC (CVPR’22) | 97.18 | 84.33 | 78.07 | 57.14 | 91.84 | 73.64 | 62.91 | 38.25 | 93.12 | 70.55 | 57.04 | 26.26
UEAF (AAAI’19) | 92.44 | 89.48 | 86.92 | 79.92 | 79.26 | 72.66 | 67.56 | 55.70 | 82.08 | 75.59 | 70.21 | 56.85
FLSD (TCYB’20) | 90.88 | 80.64 | 79.08 | 75.60 | 75.03 | 58.56 | 56.44 | 56.01 | 78.60 | 59.29 | 57.18 | 51.80
CDIMC-net (IJCAI’20) | 82.52 | 73.20 | 61.72 | 52.76 | 70.15 | 66.52 | 41.78 | 46.13 | 62.95 | 57.26 | 37.15 | 29.60
COMPLETER (CVPR’21) | 52.65 | 51.10 | 43.86 | 41.66 | 44.00 | 42.52 | 37.09 | 31.76 | 23.66 | 16.83 | 12.25 | 7.61
DIMVC (AAAI’22) | 95.90 | 94.75 | 93.24 | 89.01 | 88.00 | 85.32 | 81.37 | 73.61 | 90.11 | 87.39 | 83.94 | 75.00
OURS | 97.97 | 96.86 | 95.43 | 91.36 | 93.90 | 90.49 | 86.60 | 78.12 | 95.02 | 92.33 | 88.98 | 79.83
TABLE III: Clustering results ($\%$) on MNIST-USPS. | ACC | NMI | ARI
---|---|---|---
Method | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7
AE2-NET (CVPR’19) | 74.42 | 64.63 | 50.51 | 40.19 | 72.53 | 61.58 | 46.65 | 37.47 | 63.76 | 48.70 | 29.35 | 19.49
MFLVC (CVPR’22) | 92.28 | 71.28 | 61.26 | 23.62 | 85.72 | 60.14 | 40.44 | 20.74 | 82.69 | 40.60 | 31.24 | 3.88
UEAF (AAAI’19) | 74.76 | 71.20 | 69.62 | 60.12 | 69.58 | 64.65 | 59.99 | 51.73 | 61.53 | 56.10 | 51.72 | 41.70
FLSD (TCYB’20) | 79.16 | 70.77 | 68.28 | 65.96 | 75.83 | 68.03 | 63.64 | 60.15 | 68.70 | 58.60 | 54.11 | 50.61
CDIMC-net (IJCAI’20) | 67.10 | 62.96 | 62.60 | 52.04 | 72.18 | 65.09 | 68.43 | 54.08 | 56.28 | 53.82 | 49.50 | 32.29
COMPLETER (CVPR’21) | 88.60 | 87.94 | 82.96 | 84.64 | 92.83 | 90.94 | 87.11 | 82.65 | 85.81 | 85.00 | 77.86 | 75.99
DIMVC (AAAI’22) | 85.44 | 79.37 | 78.64 | 78.22 | 78.14 | 77.90 | 77.93 | 76.14 | 73.57 | 70.83 | 70.17 | 70.33
OURS | 99.40 | 98.73 | 97.48 | 95.94 | 98.23 | 96.52 | 93.54 | 90.47 | 98.66 | 97.20 | 94.49 | 91.17

The experiments in this paper selected 5 widely-used data sets for testings,
the statistics of which are shown in Table I. The specific details of the
dataset are as follows:

  1. 1.

BDGP [49] dataset: BDGP is a dataset of images of drosophila embryos. It
contains 2,500 samples of 5 classes, where each sample has two views of
1750-dimensional visual features and 79-dimensional textual features.

  2. 2.

MNIST-USPS [50] dataset: MNIST-USPS contains 5,000 samples. It has a total of
10 categories and two views, where the first view is obtained from the MNIST
dataset and the second view is obtained from the USPS dataset.

  3. 3.

Caltech [51] dataset: Caltech is an image dataset. We construct a dual-view
dataset based on its multi-view to evaluate our method and compare algorithms.

  4. 4.

Fashion [52] dataset: Fashion is a 10,000-sample image dataset about products,
and we treat two different styles in the dataset as two views of a product by
using.

  5. 5.

Noisy MNIST [53] dataset: This dataset is the MNIST dataset with multi-views.
It uses randomly selected intra-class images with Gaussian noise and the
original MNIST image as its two views. In order to verify that the proposed
method can handle large-scale image datasets, this paper uses a Noisy MNIST
subset with 20,000 samples for experiments.

####  IV-A2 Comparison Algorithms

We verify the performance of the proposed method by comparing our algorithm
with state-of-the-art multi-view clustering methods. The specific information
of these seven algorithms is as follows:

  1. 1.

UEAF [54] uses graph learning to mine the local structure of the data, and
simultaneously considers the reconstruction of the hidden information of
missing views and the adaptive importance assessment of different views in one
framework.

  2. 2.

FLSD [55] introduces semantic consistency constraints and proposes a graph
regularized matrix factorization model, which simultaneously considers the
latent and shared representations of different views.

  3. 3.

AE2-NET [56] is to encode the internal information from heterogeneous views
into a comprehensive representation, and this model can automatically balance
the complementarity and consistency of different views.

  4. 4.

MFLVC [26] proposes a multi-level feature learning framework for contrastive
multi-view clustering. The model can learn different levels of features and
realizes reconstruction learning and consistency learning in a fusion-free
manner.

  5. 5.

CDIMC-net [57] captures latent features of views by combining deep encoders
and graph embedding strategies into a model framework.

  6. 6.

COMPLETER [24] is the first work that provides a theoretical framework to
unify representation learning and data recovery.

  7. 7.

DIMVC [58] proposes a deep incomplete multi-view model without interpolation
and fusion. DIMVC provides an implementation of a high-dimensionalal mapping
and transforms this complementary information into supervised information with
high confidence to achieve multi-view clustering consistency for complete and
incomplete data.

TABLE IV: Clustering results ($\%$) on Caltech. | ACC | NMI | ARI
---|---|---|---
Method | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7
AE2-NET (CVPR’19) | 46.54 | 36.98 | 18.81 | 17.44 | 37.51 | 23.36 | 1.71 | 1.09 | 28.20 | 13.29 | 5.3 | 1.6
MFLVC (CVPR’22) | 60.99 | 46.27 | 38.86 | 36.36 | 52.06 | 35.56 | 28.39 | 25.31 | 43.16 | 25.19 | 16.70 | 12.64
UEAF (AAAI’19) | 41.01 | 35.57 | 35.00 | 34.14 | 26.36 | 22.83 | 22.03 | 21.07 | 19.09 | 16.70 | 11.67 | 13.82
FLSD (TCYB’20) | 46.07 | 42.57 | 41.74 | 40.30 | 31.57 | 27.37 | 26.09 | 25.49 | 24.88 | 20.02 | 19.41 | 19.24
CDIMC-net (IJCAI’20) | 50.79 | 49.07 | 44.29 | 42.93 | 49.29 | 42.75 | 41.65 | 38.48 | 35.15 | 33.97 | 28.55 | 26.78
COMPLETER (CVPR’21) | 47.24 | 46.73 | 47.53 | 44.99 | 45.93 | 44.22 | 44.72 | 43.12 | 27.49 | 27.28 | 24.59 | 24.26
DIMVC (AAAI’22) | 46.88 | 43.96 | 37.25 | 36.30 | 32.90 | 26.72 | 23.58 | 20.03 | 25.40 | 19.25 | 14.73 | 12.99
OURS | 67.51 | 56.47 | 55.93 | 53.66 | 56.98 | 51.26 | 47.55 | 44.15 | 47.36 | 42.29 | 39.83 | 37.54
TABLE V: Clustering results ($\%$) on Fashion. | ACC | NMI | ARI
---|---|---|---
Method | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7
AE2-NET (CVPR’19) | 65.52 | 59.37 | 50.52 | 43.97 | 65.09 | 59.40 | 51.13 | 44.04 | 52.99 | 44.37 | 31.71 | 23.53
MFLVC (CVPR’22) | 88.75 | 80.35 | 52.72 | 39.93 | 79.84 | 69.57 | 50.59 | 36.85 | 75.34 | 59.29 | 26.92 | 10.66
UEAF (AAAI’19) | 60.70 | 51.89 | 40.59 | 31.07 | 62.73 | 52.81 | 40.86 | 29.34 | 47.68 | 35.80 | 22.35 | 14.61
FLSD (TCYB’20) | 66.57 | 65.06 | 63.66 | 56.02 | 69.43 | 66.87 | 64.65 | 62.15 | 54.81 | 52.06 | 49.70 | 45.41
CDIMC-net (IJCAI’20) | 65.42 | 51.05 | 46.32 | 43.47 | 71.67 | 69.08 | 50.01 | 48.72 | 55.17 | 40.19 | 30.81 | 29.45
COMPLETER (CVPR’21) | 78.55 | 75.99 | 72.97 | 71.91 | 84.76 | 80.64 | 78.14 | 74.05 | 71.33 | 75.34 | 67.32 | 62.99
DIMVC (AAAI’22) | 64.49 | 62.60 | 62.65 | 61.43 | 75.81 | 70.51 | 69.70 | 64.43 | 58.49 | 54.93 | 53.78 | 49.58
OURS | 96.53 | 94.14 | 90.98 | 86.94 | 92.86 | 88.82 | 84.23 | 79.19 | 92.67 | 87.93 | 82.09 | 74.98

####  IV-A3 Evaluation Metrics

The clustering performance of all methods was measured by the following widely
used evaluation metrics: Accuracy (ACC), Normalized Mutual Information (NMI)
and ARI. The larger the value of all indicators, the better the clustering
performance. The specific information of these three evaluation indicators is
as follows:

  1. 1.

Accuracy (ACC). ACC is used to compare the obtained labels with the true
labels provided by the data. ACC is defined as follows:

$\displaystyle\mathrm{ACC}=\frac{\sum_{i=1}^{n}\delta(t_{i},map(h_{i}))}{n},$
(6)

where $t_{i}$ represents the true label of the data. $h_{i}$ represents the
label of the $i$-th sample obtained by clustering, and $map(h_{i})$ represents
the reproduction allocation of the best clustering index to ensure the
correctness of statistics. $\delta(\cdot,\cdot)$ is the indication function,
which is defined as:

$\displaystyle\delta(u,v)=\left\\{\begin{array}[]{ll}1,&u=v,\\\
0,&otherwise.\end{array}\right.$ (9)

  2. 2.

Normalized Mutual Information (NMI). NMI is defined as follows:

$\operatorname{NMI}=\frac{2\sum_{i=1}^{k}\sum_{j=1}^{\hat{k}}\frac{m_{ij}}{m}\log\frac{m_{ij}m}{\sum_{i=1}^{k}q_{i}\sum_{j=1}^{k}p_{j}}}{-\sum_{i=1}^{k}\frac{q_{i}}{m}\log\frac{q_{i}}{m}-\sum_{j=1}^{\hat{k}}\frac{p_{j}}{m}\log\frac{p_{j}}{m}}$
(10)

where $m$ is the total number of samples and $k$ is the number of clusters.
$q_{i}$ and $p_{j}$ are the number of the obtained clustering results and true
labels respectively.

  3. 3.

ARI. ARI is defined as follows:

$\displaystyle\mathrm{ARI}=\frac{\sum_{i,j=1}^{c}C_{n_{ij}}^{2}-\frac{\sum_{i=1}^{c}C_{s_{i}}^{2}\sum_{i=1}^{c}C_{t_{i}}^{2}}{C_{n}^{2}}}{\frac{1}{2}\left(\sum_{i=1}^{c}C_{s_{i}}^{2}+\sum_{i=1}^{c}C_{t_{i}}^{2}\right)-\frac{\sum_{i=1}^{c}C_{s_{i}}^{2}\sum_{i=1}^{c}C_{t_{i}}^{2}}{C_{n}^{2}}}.$
(11)

$C_{n}^{m}$ represents the number of $n$ samples selected from $m$ samples.

###  IV-B Experimental Results and Analysis

For all compared algorithms, we use default parameters or iteratively search
in a given parameter set to find the best parameters for these methods and
report their best clustering results. For the clustering method that can only
be applied to the complete multi-view data (AE2-NET and MFLVC) adopt the same
way as [24], we use the average value of the feature values of the
corresponding views to fill in the missing views, and cluster on the completed
data set obtained after filling. For each dataset, all methods were performed
on the same randomly formed incomplete cases and their best results were
reported for fair comparison. The clustering metrics (ACC, NMI and ARI)
comparison is presented in Table II-VI. From these tables, we obtain the
following observations:

1) From the Table II-VI, it can be observed that the clustering metrics (ACC,
NMI and ARI) gradually decrease as the missing rate increases. The performance
of other incomplete multi-view clustering methods is better than the complete
multi-view clustering method when the missing rate is large, which indicates
that the imputed views inferred by other incomplete multi-view clustering
methods contain more semantic information than the average vectors and thus
alleviate the risk of degraded clustering performance due to the adverse
effects of noise introduced by mean value filling.

2) The clustering results of the proposed method on all data sets are
obviously better than other methods, especially when dealing with large-scale
data sets and high miss rate. For example, on Noisy MNIST with a missing rate
of 0.7, our method exceeds the second best value by approximately 7.5%, 5.2%
and 7.1% in terms of ACC, NMI and ARI, respectively. And since our method can
effectively prevent the effect of dimensional collapse by using direct
contrastive learning, it can more explicitly capture the useful information
contained in latent features, thus greatly improving the clustering
performance.

3) The clustering performance of all algorithms decreases with increasing
missing rate, but the least decrease is clearly observed for our proposed
method. Especially on the MNIST-USPS dataset, the performance drop of the
comparison algorithm is larger than ours, and the ACC of our method only drops
3.46% in the case of missing rate 0.1-0.7. These observations proved that our
algorithm in the case of a larger missing rate can still normal processing
data. This reflects that our recovered views are more conducive to clustering,
and our method discards inconsistent information between views so that it can
better utilize views to obtain consistent information. This advantage allows
our method to perform clustering tasks well even when many view data are
missing.

TABLE VI: Clustering results ($\%$) on Noisy MNIST. ’-’ indicates that the experimental results could not be obtained due to insufficient memory, and the best results of the first place are shown in bold. | ACC | NMI | ARI
---|---|---|---
Method | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7 | 0.1 | 0.3 | 0.5 | 0.7
AE2-NET (CVPR’19) | 36.25 | 28.07 | 27.91 | 26.07 | 30.48 | 21.22 | 19.93 | 16.08 | 18.25 | 10.50 | 7.49 | 4.61
MFLVC (CVPR’22) | 89.99 | 71.29 | 43.08 | 23.94 | 79.48 | 51.04 | 38.51 | 16.95 | 78.79 | 45.69 | 15.32 | 4.15
UEAF (AAAI’19) | - | - | - | - | - | - | - | - | - | - | - | -
FLSD (TCYB’20) | - | - | - | - | - | - | - | - | - | - | - | -
CDIMC-net (IJCAI’20) | 40.09 | 38.09 | 34.72 | 27.86 | 48.33 | 43.83 | 32.85 | 22.99 | 30.89 | 27.22 | 21.30 | 12.17
COMPLETER (CVPR’21) | 89.99 | 87.51 | 80.94 | 74.41 | 86.05 | 82.36 | 76.52 | 67.35 | 82.85 | 79.17 | 71.87 | 61.83
DIMVC (AAAI’22) | 64.93 | 58.56 | 51.57 | 48.88 | 61.57 | 60.46 | 48.97 | 47.17 | 49.68 | 45.43 | 37.82 | 33.55
OURS | 94.64 | 93.44 | 90.61 | 81.93 | 88.24 | 85.64 | 80.73 | 72.60 | 88.79 | 86.28 | 80.75 | 68.82

(a) Epoch 0 (NMI:63.78)

(b) Epoch 5 (NMI:91.15)

(c) Epoch 10 (NMI:97.47)

(d) Epoch 15 (NMI:98.23)

Figure 2: t-sne visualization and NMI(%) on the MNIST-USPS dataset for the
contrastive learning process.

(a) Epoch 5 (NMI:82.14)

(b) Epoch 20 (NMI:90.91)

(c) Epoch 50 (NMI:92.57)

(d) Epoch 100 (NMI:92.86)

Figure 3: t-sne visualization and NMI(%) on the Fashion dataset for the
contrastive learning process.

###  IV-C Ablation Studies

In this section, we will analyze the ablation studies of the proposed method
on the MNIST-USPS dataset. In the evaluation, the missing rate is fixed at
0.5.

TABLE VII: Ablation study of the loss part on the MNIST-USPS. $\mathcal{L}_{\mathrm{R}}$ | $\mathcal{L}_{\mathrm{C}}$ | $\mathcal{L}_{\mathrm{Z}}$ | ACC | NMI | ARI
---|---|---|---|---|---
|  | $\checkmark$ | 48.00 | 43.40 | 27.49
$\checkmark$ |  |  | 47.14 | 44.83 | 32.42
| $\checkmark$ |  | 53.06 | 62.04 | 41.07
$\checkmark$ |  | $\checkmark$ | 78.87 | 72.77 | 66.94
| $\checkmark$ | $\checkmark$ | 60.34 | 64.19 | 38.38
$\checkmark$ | $\checkmark$ |  | 83.22 | 80.58 | 74.21
$\checkmark$ | $\checkmark$ | $\checkmark$ | 97.48 | 93.54 | 94.49
TABLE VIII: Ablation study of the contrastive learning part on the MNIST-USPS $\mathbf{X}-\mathbf{Z}$ | $\mathbf{X}-\mathbf{Z}^{*}$ | ACC | NMI | ARI
---|---|---|---|---
$\checkmark$ |  | 81.40 | 72.48 | 65.94
$\checkmark$ | $\checkmark$ | 89.72 | 81.07 | 79.05
| $\checkmark$ | 97.48 | 93.54 | 94.49

Loss function component analysis. To further verify the importance of each
module of our method in the overall algorithm framework, we evaluate each
module of the algorithm through the following ablation studies. In detail, as
shown in Table VII, the following seven experiments are designed to isolate
the effect of the contrastive loss $\mathcal{L}_{\mathrm{C}}$, the
reconstruction loss $\mathcal{L}_{\mathrm{Z}}$, and the prediction loss
$\mathcal{L}_{\mathrm{R}}$. It can be observed from the table that the loss of
each part is very important for the model, no matter which part of the loss is
missing will greatly affect the performance of the algorithm.

Contrastive learning on different representations. To further illustrate the
effectiveness of our direct contrastive learning model, we target and perform
contrastive learning on different latent representations. As shown in the
Table VIII, we can find that the model performance is poor if both the
reconstruction loss and the contrastive loss are applied to $\mathbf{Z}$. The
main reason for the above situation is that the dataset contains a large
amount of private information, and this method cannot handle the conflict
between private information and common semantics, thus leading to performance
degradation. And the performance advantage of our structure can be clearly
observed, which demonstrates the effectiveness of our method on dealing with
incomplete multi-view clustering problems.

###  IV-D Visually Verify on Data Recovery

In this section, we verify whether our method has the ability to restore the
view through visual experiments. Experiments are carried out on the Noisy
MNIST and Fashion datasets. By visualizing the recovered views, we can observe
the effectiveness of the algorithm more intuitively. In the experiments, the
missing rate $\eta$ is fixed at 0.5.

Figure 4 demonstrates the data recovery ability of our method on the dataset
Noisy MNIST and Fashion, and our method has the ability to infer missing view
representations compared to existing state-of-the-art methods. As shown in the
figure, we can observe that in the first three rows of data, the recovered
images are very similar to the complete images in the first row, but the
background of our newly acquired data is consistent with the lost view, so it
can be concluded that our method can recover the important information of the
missing view through the information of other views, and can discard the
blurred noise in the original view.

Figure 4: Data recovery on Noisy MNIST and Fashion. Line 1 is the full view,
Line 2 is the missing view, and Line 3 is the result of restoring from the
full view.

###  IV-E T-sne Visualizations

In addition to the above visualizations, we also show t-sne visualizations of
common representations learned by the model. As shown in the Figures 2 and 3,
as the epoch increases, the learned representation becomes more compact and
discriminative.

###  IV-F Parameter and Convergence Analysis

In this section, we investigate the convergence of our algorithm by reporting
the loss value and the corresponding cluster performance over time. As shown
in Figure 5, the x-axis represents training epochs, and the left and right
y-axes represent clustering performance and corresponding loss values,
respectively. It can be observed from the figure that the clustering effect of
the model increases as the loss value decreases after several epochs,
indicating that the proposed method has good convergence performance.

In the parameter sensitivity analysis, we studied the loss components in two
hyper-parameters balance loss components, i.e.,
$\mathcal{L}_{\mathbf{Z}}+\lambda_{1}\mathcal{L}_{\mathbf{C}}+\lambda_{2}\mathcal{L}_{\mathbf{R}}$.
Figure 6 shows the average value of ACC and NMI for 5 independents runs, as
shown in the figure, our method is robust to the choice of $\lambda_{2}$. In
addition, a good choice of $\lambda_{1}$ will significantly improve the
performance of our model.

(a) Loss vs. performance on BDGP

(b) Loss vs. performance on MNIST-USPS

Figure 5: Figure (a) is the result of the BDGP dataset , and Figure (b) is the
result of the MNIST-USPS dataset.

(a) $\lambda_{1}$ vs. $\lambda_{2}$ (ACC)

(b) $\lambda_{1}$ vs. $\lambda_{2}$ (NMI)

Figure 6: Parameter analysis on MNIST-USPS.

##  V Conclusion

To solve the dimensional collapse problem in incomplete multi-view clustering
without relying on additional projection heads, this paper proposes a novel
framework for contrastive learning of incomplete multi-view clustering. For
each view, the framework performs reconstruction learning and consistency
learning on the latent features and their sub-vectors respectively, which
makes our model simpler to avoid dimensional collapse and guarantees that
latent features can obtain more useful information. And recover the view by
minimizing the conditional entropy while discarding inconsistent information.
Since our method does not achieve consistency learning and reconstruction
learning on the same feature, the conflict between view-private information
inconsistency and common semantic consistency is well reduced. Extensive
experiments on 5 public datasets demonstrate that our method achieves state-
of-the-art performance.

## Acknowledgment

This work was supported in part by the Fundamental Research Funds for the
Central Universities (No.31920220130, 31920220019), the Introduction of Talent
Research Project of Northwest Minzu University (No.xbmuyjrc201904), the
National Natural Science Foundation of China (No.61866033), the Leading Talent
of National Ethnic Affairs Commission (NEAC), the Young Talent of NEAC, and
the Innovative Research Team of NEAC (2018) 98.

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|  Kaiwu Zhang received the B.S. degree in Software Engineering from the
College of the Dalian Maritime University, Dalian, China, in 2020. He is
currently pursuing the M.S. degree with the College of China National
Institute of Information Technology, Northwest Minzu University, Lanzhou,
China. His current research interests include contrastive learning, and multi-
view clustering.
---|---
|  Shiqiang Du received the Ph.D. degree in radio physics from School of
Information Science and Engineering at Lanzhou University, Lanzhou, China, in
2017. He was a visiting scholar with the Department of Mathematics and
Statistics, Utah State University, Logan, USA, in 2013. From 2019 to 2020, he
was a visiting scholar with the Department of Engineer, University of Modena
and Reggio Emilia, Modena, Italy. He is currently the Professor with the
College of Mathematics and Computer Science, Northwest Minzu University,
Lanzhou, China. His current research interests include machine learning, image
processing, and pattern recognition.
---|---
|  Baokai Liu received the B.S. degree in communication engineering from the
Shanghai Second Polytechnic University, Shanghai, China, in 2016. He is
currently pursuing the M.S. degree with the College of China National
Institute of Information Technology, Northwest Minzu University, Lanzhou,
China. His current research interests include image processing, and deep
learning.
---|---
|  Shengxia Gao received the B.S. degree in software engineering from the
College of Mathematics and Computer Scinence, Northwest Minzu University,
Lanzhou, China, in 2020. She is currently pursuing the M.S. degree with the
College of China National Institute of Information Technology, Northwest Minzu
University, Lanzhou, China. Her current research interests include contrastive
learning.
---|---