# On the phase diagram of a three-dimensional dipolar model.

V. Russier<EMAIL_ADDRESS>ICMPE, UMR 7182 CNRS and UPE 2-8 rue Henri
Dunant 94320 Thiais, France.  Juan J. Alonso<EMAIL_ADDRESS>Física Aplicada
I, Universidad de Málaga, 29071 Málaga, Spain  Instituto Carlos I de Física
Teórica y Computacional, Universidad de Málaga, 29071 Málaga, Spain

###### Abstract

The magnetic phase diagram at zero external field of an ensemble of dipoles
with uniaxial anisotropy on a FCC lattice has been investigated from tempered
Monte Carlo simulations. The uniaxial anisotropy is characterized by a random
distribution of easy axes and its magnitude $\lambda_{u}$ is the driving force
of disorder and consequently frustration. The phase diagram, separating the
paramagnetic, ferromagnetic and spin-glass regions, was thus considered in the
temperature, $\lambda_{u}$ plane. Here we interpret this phase diagram in
terms of the more convenient variables namely the bare dipolar interaction and
anisotropy energies $\epsilon_{d}$ and $\epsilon_{u}$ on the one hand and the
volume fraction $\Phi$ on the other hand and compare the result with that
corresponding to the random distribution of particles in the absence of
anisotropy. We also display the nature of the ordered phase reached at low
temperature by the ensemble of dipoles on the FCC lattice in terms of both the
dipolar coupling and the texturation of the easy axes distribution when the
latter is no more random. This system is aimed at modeling the magnetic phase
diagram of supracrystals of magnetic nanoparticles.

In Ref.Russier  _et al._ (2020) we investigated the phase diagram of an
ensemble of dipolar hard spheres located on the nodes of a perfect FCC
lattice. The phase diagram was investigated in the reduced temperature,
disorder control parameter, ($T^{*},\lambda_{u}$) plane. Given the $1/r^{3}$
dependence of the dipole-dipole interaction (DDI), it has been found
convenient to define the reduced temperature, $T^{*}=T/T_{r}$ through the
reference temperature $T_{r}=(\Phi/\Phi_{m})(\epsilon_{d}/k_{B})$ where
$\Phi$, $\Phi_{m}$ and $\epsilon_{d}$ are the volume fraction, its maximum
value of the FCC lattice and the dipole-dipole energy for particles at
contact. The disorder control parameter is nothing but the anisotropy energy
(MAE) over DDI ratio, namely
$\lambda_{u}=(\epsilon_{u}/\epsilon_{d})(\Phi_{m}/\Phi)$. Then the
($T^{*},\lambda_{u}$) phase diagram, displayed on figure (1), is qualitatively
similar to those corresponding to the $\pm{}J$ Ising model Hasenbusch  _et
al._ (2007); Papakonstantinou and Malakis (2013). The essential feature is the
succession at low temperature of long range order FM, quasi long range FM
(QLRO-FM) plus transverse SG and SG phases with increasing values of
$\lambda_{u}$. Now, when considering the system as a model for the magnetic
phase diagram of supracrystals of bare magnetic nanoparticles characterized by
given values of $\epsilon_{d}$ and $\epsilon_{u}$ where the volume fraction
$\phi$ can be tuned through the thickness of the coating layer, it is
convenient to translate the phase diagram in terms of $(k_{B}T/\epsilon_{u})$
and $(\Phi/\Phi_{m})(\epsilon_{d}/\epsilon_{u})$, the latter being understood
as the increasing dipolar coupling through the increase of concentration at
constant $\epsilon_{d}$ and $\epsilon_{u}$. This is shown on figure (2). In
this represention, the dipolar lattice phase diagram appears qualitatively
similar to the one deduced from experiments on the DMIM samples Petracic
(2010). Furthermore one must limit the reachable part of this phase diagram to
the region defined by $\Phi/\Phi_{m}\leq{}1$. Doing this we see, as diplayed
on figure (3), that according to the value of ($\epsilon_{d}/\epsilon_{u}$),
namely the relative importance of the dipolar coupling to the anisotropy
energy, the FM-QLRO an the FM phases may be not reachable. In the limit
$\epsilon_{d}/\epsilon_{u}\rightarrow\infty$, one gets only the PM/FM
transition of the pure dipolar system on the FCC lattice free of anisotropy
($\epsilon_{u}=0$) Bouchaud and Zerah (1993); Russier and Ngo (2017); Russier
_et al._ (2020) and all the non trivial features of the phase diagram are
shrinked on the $\Phi=0$ line. Moreover, the
($(k_{B}T/\epsilon_{u}),(\Phi/\Phi_{m})(\epsilon_{d}/\epsilon_{u})$) phase
diagram for particles located on the perfect FCC lattice can be directly
compared to the one we have got Alonso  _et al._ (2020) for the system of
dipolar spheres frozen in a random hard sphere like distribution, in the
absence of anisotropy, where the increasing disorder is quantified by
$1-(\phi/\phi_{RPC})$, the volume fraction being then limited to the so-called
RCP value, $\Phi_{RCP}\simeq{}0.64$, given on figure (4). It is also
qualitively close to the one of dipolar Ising model on the SC lattice in terms
of dilution Alonso and Fernández (2010), where however the ordered phase is
anti-FM due to the lattice symmetry. Finally, we present on figure (5) the
nature of the ordered phase at low temperature in terms of the dipolar
coupling and the texturation of the easy axes distribution as was studied in
Alonso  _et al._ (2019); Russier and Alonso (2020) in the limit of infinitely
strong anisotropy energy leading to the dipolar Ising model. As is the case
above, the condition $\Phi/\Phi_{m}\leq{1}$ restricts the diagram to the left
hand part defined by the abcissa $\leq(\epsilon_{d}/\epsilon_{u})$ and has
been displayed for two particular cases corresponding to a weak and a strong
dipolar coupling respectively.

Figure 1:  ($T_{c}/T_{r},\lambda_{u}$) phase diagram of the dipolar plus
uniaxial anisotropy with random distribution of easy axes on a FCC lattice.
$\lambda_{u}$ is the natural disorder control parameter.  Figure 2:  Phase
diagram of figure (1), shown as $T_{c}$ in terms of increasing dipolar
coupling for $\epsilon_{d}$ and $\epsilon_{u}$ considered as constant.  Figure
3:  Same as figure (2) where the constraint $\Phi/\Phi_{m}\leq{1}$ is
explicited for three characterisitc values of $\epsilon_{d}/\epsilon_{u}$ as
indicated.  Figure 4:  Phase diagram of the dipolar hard spheres distributed
according to a frozen hard-sphere like distribution with $\epsilon_{u}=0$. The
limiting value of the volumefraction, $\Phi_{RCP}$ is indicated.  Figure 5:
Nature of the ordered phase at low temperature for the FCC lattice in terms of
the texturation quantified by the variance $\sigma$ of the easy axes
distribution (see Ref. Alonso  _et al._ (2019); Russier and Alonso (2020)) and
the dipolar coupling $(\Phi/\Phi_{m})\epsilon_{d}/\epsilon_{u}$. Open circle
and open triangle are the FM/SG transition in the dipolar Ising limit Russier
and Alonso (2020), and $(\Phi/\Phi_{m})\epsilon_{d}/\epsilon_{u}=0.25$ Russier
(2020) respectively. The dotted line is only indicative. The upper horizontal
red line, $\sigma=\pi/2$ indicates the lower bound of random distribution of
easy axes. The SG region is separated in the true dipolar SG (3D) and the
limiting dipolar Ising SG (1D) obtained for $\lambda_{u}>30$ Russier  _et al._
(2020). The reachable region is located on the left hand side of the line
$(\Phi/\Phi_{m})\epsilon_{d}/\epsilon_{u}=\epsilon_{d}/\epsilon_{u}$ which is
shown for a weak and a strong dipolar couplings as particular cases.

## References

  * Russier  _et al._ (2020) V. Russier, J. J. Alonso, I. Lisiecki, A. T. Ngo, C. Salzemann, S. Nakamae, and C. Raepsaet, Phys. Rev. B 102, 174410 (2020).
  * Hasenbusch  _et al._ (2007) M. Hasenbusch, F. P. Toldin, A. Pelissetto, and E. Vicari, Phys. Rev. B 76, 184202 (2007).
  * Papakonstantinou and Malakis (2013) T. Papakonstantinou and A. Malakis, Phys. Rev. E 87, 012132 (2013).
  * Petracic (2010) O. Petracic, Superlattices and Microstructures 47, 569 (2010).
  * Bouchaud and Zerah (1993) J. P. Bouchaud and P. G. Zerah, Phys. Rev. B 47, 9095 (1993).
  * Russier and Ngo (2017) V. Russier and E. Ngo, Condensed Matter Physics 20, 33703 (2017).
  * Alonso  _et al._ (2020) J. J. Alonso, B. Allés, and V. Russier, Phys. Rev. B 102, 184423 (2020).
  * Alonso and Fernández (2010) J.-J. Alonso and J.-F. Fernández, Phys. Rev. B 81, 064408 (2010).
  * Alonso  _et al._ (2019) J.-J. Alonso, B. Allés, and V. Russier, Phys. Rev. B 100, 134409 (2019), arXiv:1909.13573.
  * Russier and Alonso (2020) V. Russier and J. J. Alonso, J. Phys.-Cond. Matt. 32, 135804 (2020).
  * Russier (2020) V. Russier, (2020), unpublished.


# Scalable Incomplete Multi-View Clustering with Structure Alignment

Yi Wen<EMAIL_ADDRESS>National University of Defense
TechnologyChangshaChina ,  Siwei Wang<EMAIL_ADDRESS>Intelligent
Game and Decision LabBeijingChina ,  Ke Liang<EMAIL_ADDRESS>National
University of Defense TechnologyChangshaChina ,  Weixuan Liang
<EMAIL_ADDRESS>National University of Defense
TechnologyChangshaChina ,  Xinhang Wan<EMAIL_ADDRESS>National
University of Defense TechnologyChangshaChina ,  Xinwang Liu
<EMAIL_ADDRESS>National University of Defense TechnologyChangshaChina
,  Suyuan Liu<EMAIL_ADDRESS>National University of Defense
TechnologyChangshaChina ,  Jiyuan Liu<EMAIL_ADDRESS>National
University of Defense TechnologyChangshaChina and  En Zhu<EMAIL_ADDRESS>National University of Defense TechnologyChangshaChina

(2023)

###### Abstract.

The success of existing multi-view clustering (MVC) relies on the assumption
that all views are complete. However, samples are usually partially available
due to data corruption or sensor malfunction, which raises the research of
incomplete multi-view clustering (IMVC). Although several anchor-based IMVC
methods have been proposed to process the large-scale incomplete data, they
still suffer from the following drawbacks: i) Most existing approaches neglect
the inter-view discrepancy and enforce cross-view representation to be
consistent, which would corrupt the representation capability of the model;
ii) Due to the samples disparity between different views, the learned anchor
might be misaligned, which we referred as the Anchor-Unaligned Problem for
Incomplete data (AUP-ID). Such the AUP-ID would cause inaccurate graph fusion
and degrades clustering performance. To tackle these issues, we propose a
novel incomplete anchor graph learning framework termed Scalable Incomplete
Multi-View Clustering with Structure Alignment (SIMVC-SA). Specially, we
construct the view-specific anchor graph to capture the complementary
information from different views. In order to solve the AUP-ID, we propose a
novel structure alignment module to refine the cross-view anchor
correspondence. Meanwhile, the anchor graph construction and alignment are
jointly optimized in our unified framework to enhance clustering quality.
Through anchor graph construction instead of full graphs, the time and space
complexity of the proposed SIMVC-SA is proven to be linearly correlated with
the number of samples. Extensive experiments on seven incomplete benchmark
datasets demonstrate the effectiveness and efficiency of our proposed method.
Our code is publicly available at https://github.com/wy1019/SIMVC-SA.

anchor graph, incomplete multi-view clustering, large-scale clustering, multi-
view clustering

††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
31st ACM International Conference on Multimedia; October 29–November 3, 2023;
Ottawa, ON, Canada.††booktitle: Proceedings of the 31st ACM International
Conference on Multimedia (MM ’23), October 29–November 3, 2023, Ottawa, ON,
Canada††price: 15.00††isbn: 979-8-4007-0108-5/23/10††doi:
10.1145/3581783.3611981††ccs: Computing methodologies Cluster analysis††ccs:
Theory of computation Unsupervised learning and clustering

##  1\. Introduction

Multi-view data, which consists of features extracted from objects by
different sensors, have been massively generated in recent years. Multi-modal
information (Wang, 2021; Li et al., 2023a; Liang et al., 2022; Li et al.,
2018; Liu et al., 2023c, 2022c, e) about the data can commonly be utilized to
enhance the expressive ability of the model. For instance, the same news can
be described from different views, i.e., textual reports (Liang et al., 2021)
and visual pictures (Nie et al., 2017; Xu et al., 2023; Zhao et al., 2022; Wan
et al., 2022). As an essential paradigm of multi-view learning, multi-view
clustering (MVC) has drawn substantial attention because of its promising
capability to reveal the intrinsic data structure (Cai et al., 2013; Li et
al., 2019). In general, multi-view clustering achieves remarkable performance
by learning a consensus representation by exploring consistency among diverse
views (Kumar et al., 2011; Zhao et al., 2017; Ren et al., 2019). For instance,
Zhan et al. (2018) optimizes the final consensus graph by imposing low-rank
constraints and minimizing the discrepancies of individual graphs. Zhang et
al. (2017) reconstruct samples in the latent space to achieve a more precise
and reliable subspace representation.

Although numerous methods have been proposed to enhance MVC in diverse ways,
most of them assume that all data are fully available (Gao et al., 2015; Qiang
et al., 2021; Zhan et al., 2017; Peng et al., 2019; Wan et al., 2023a, b; Li
et al., 2022d, 2023b; Zhang et al., 2022). However, samples are often
partially available in real scenarios due to data corruption or sensor
malfunction. For instance, in software traffic detection, people can’t use all
detected software, which leads to the incompleteness of the samples in the
corresponding view. The different sample absence between views destroys the
original cross-view alignment information and enlarges the difficulty of
exploring consensus and complementary information, causing incomplete multi-
view clustering (IMVC) a challenging problem. To tackle these issues, several
IMVC methods have been proposed in previous literature. For instance, Li et
al. (2014) learn a common potential representation from incomplete samples by
non-negative matrix decomposition and $\ell_{1}$ regularization terms. Wen et
al. (2020b) propose a new regularization term to preserve the local geometric
structure and fuse the individual incomplete graph. Although remarkable
success has been made, the high time complexity hinders their application in
large-scale scenarios(Liu et al., 2023a). One pioneer work, Liu et al. (2022a)
efficiently reduce the algorithm complexity by utilizing the anchor graph to
capture the clustering structure with incomplete views.

Although widely applied in large-scale applications, the existing anchor-based
IMVC methods still suffer from the following drawbacks: Firstly, most
approaches neglect the inter-view discrepancy and enforce cross-view
representation to be consistent, which would corrupt the representation
capability of the model. Secondly, as is shown in Fig. 1, the sample
distribution of different views might be biased due to the incomplete multi-
view data, which leads to the potential misalignment between cross-view
anchors, which we referred as the Anchor-Unaligned Problem for Incomplete Data
(AUP-ID). Such AUP-ID would result in inaccurate graph fusion and suboptimal
clustering performance. This issue for complete data has been demonstrated in
(Wang et al., 2022; Huang et al., 2020) and has more significant implications
for IMVC due to the incomplete cross-view alignment information. Besides, to
the best of our knowledge, no generalized framework for solving AUP-ID has
been proposed since generating the correct cross-view anchor correspondence
under incomplete scenarios would be more difficult due to the variance of the
feature dimension and available sample.

Figure 1. An example illustration of AUP-ID. Samples and representations of
different colors represent samples from different clusters and representations
of different anchors, respectively. With different missing samples and random
anchor initialization, the anchor learned may be unaligned and leads to
inaccurate correspondences.

To tackle these challenging issues, we propose a novel incomplete anchor graph
learning framework termed Scalable Incomplete Multi-View Clustering with
Structure Alignment (SIMVC-SA). Specifically, we construct the incomplete
anchor graph on each view to capture the complementary information from
different views. In order to address the AUP-ID, we adopt a novel structure
alignment module to refine the cross-view anchor correspondence mapping
adequately. Meanwhile, the anchor graph construction and alignment are jointly
optimized in our unified framework to enhance clustering quality. In addition,
through the anchor graph construction rather than a full pairwise graph, the
time complexity of the SIMVC-SA is effectively reduced from
$\mathcal{O}(n^{3})$ to $\mathcal{O}(nm)$. Meanwhile, a convergent five-step
alternative algorithm is designed in this paper to tackle the subsequent
optimization problem. We summarize the contributions as follows:

  * •

In order to solve the Anchor-Unaligned Problem for Incomplete Data, a novel
alignment module is proposed in this paper to capture the view-specific
structure. With the guidance of structure information, the cross-view anchor
correspondence mapping can be refined adequately.

  * •

We design a novel IMVC approach termed Scalable Incomplete Multi-View
Clustering with Structure Alignment (SIM VC-SA). Different from the existing
fixed anchor strategy, SIMVC-SA learns the anchor and constructs the
respective anchor graph to enhance the clustering performance.

  * •

Extensive experiments on seven incomplete benchmark datasets show the
effectiveness and efficiency of the proposed method.

##  2\. Related Work

###  2.1. Incomplete Multi-View Clustering (IMVC)

In real scenarios, samples are often partially available due to data dropout
and sensor corruption, which raises the incomplete multi-view clustering
(IMVC) study (Yang et al., 2022a; Hu and Chen, 2019; Lin et al., 2022c; Xu et
al., 2022; Jin et al., 2023; Wan et al., 2023c). The existing Incomplete Muti-
View Clustering (IMVC) approaches can be roughly divided into three types:
Non-negative matrix factorization (NMF) methods (Zhao et al., 2016), kernel or
graph-based methods (Nie et al., 2016; Han et al., 2022), and deep neural
networks (Wen et al., 2021; Zhu et al., 2019; Wang et al., 2020b).

In general, NMF jointly decomposes the raw matrix of each view into a
coefficient matrix and a basis matrix and learns a consensus matrix from the
coefficient matrices with a group of adaptive view weights. The graph or
kernel-based IMVC approach performs matrix complementation and achieves the
desired clustering performance by constructing consensus graphs or kernels
(Wang et al., 2020a; Liu et al., 2020; Ren et al., 2020). For example, Wang et
al. (2019c) propose a novel similarity matrix padding strategy based on matrix
perturbation theory. Because of the capacity to extract high-level
information, deep neural networks often achieve desirable performance for
solving IMVC problems (Wen et al., 2020a; Wang et al., 2018; Lin et al.,
2022b). Lin et al. (2021) design a deep IMVC model through the union of
representation and cross-view data recovery.

###  2.2. Graph-based IMVC Method

Graph structures (Kang et al., 2020a; Khan and Maji, 2019; Wang et al., 2019b;
Mo et al., 2022; Liang et al., 2023; Liu et al., 2023b; Yang et al., 2023,
2022c, 2022b; Liu et al., 2022b, 2023d), which can well describe the
relationships of pairwise data, are widely adopted in the field of IMVC.
Denoting the indicator vector $\mathbf{w}^{(v)}\in\mathbb{R}^{n_{v}}$
containing the index for $n_{v}$ available samples in the $v$-th view, we
define the index matrix $\mathbf{H}_{v}\in\mathbb{R}^{n\times n_{v}}$ for
$v$-th view as follows:

$\mathbf{h}_{ij}^{(v)}=\begin{cases}1,&\text{ if }w_{j}^{(v)}=i,\\\ 0,&\text{
otherwise. }\end{cases}$

where $\mathbf{h}_{ij}^{(v)}$ denotes the element in $i$-th row and $j$-th
column of $\mathbf{H}_{v}$. Then,
$\mathbf{X}_{v}\mathbf{H}_{v}\in\mathbb{R}^{d_{v}\times n_{v}}$ denotes the
existing data matrix of the $v$-th view.

As to the incomplete multi-view data, the subgraphs from each view may have a
few blanks in the respective rows and columns because of the incomplete
setting. Taking this into account, the classical graph-based IMVC paradigms
(Li et al., 2022a; Liu et al., 2010) could be mathematically represented in
two parts:

(1) $\displaystyle\min_{\mathbf{S}_{v},\mathbf{S}}$
$\displaystyle\left\|\mathbf{X}_{v}\mathbf{H}_{v}-\mathbf{X}_{v}\mathbf{H}_{v}\mathbf{S}_{v}\right\|_{\mathbf{F}}^{2}+\Psi\left(\mathbf{H}_{v}\mathbf{S}_{v}\mathbf{H}_{v}^{\top},\mathbf{S}\right),$
s.t. $\displaystyle\mathbf{S}_{v}\geq
0,\quad\mathbf{S}_{v}^{\top}\mathbf{1}=\mathbf{1},\quad\mathbf{S}\geq
0,\quad\mathbf{S}^{\top}\mathbf{1}=\mathbf{1}$

where $\mathbf{S}_{v}\in\mathbb{R}^{n_{v}\times n_{v}}$ is the view-specific
subgraph, $\mathbf{S}$ indicates the similarity among all the samples. The
$\Psi(\cdot)$ indicates the graph fusion process. However,
$\mathcal{O}\left(vn^{2}\right)$ space complexity and
$\mathcal{O}\left(n^{3}\right)$ time expenditure prevent this category of
algorithms from handling large-scale incomplete multiview tasks (Kang et al.,
2020b).

###  2.3. Anchor-based IMVC Method

As is shown in Eq.(1), the majority of the classical graph-based IMVC methods
involve the full graph construction, which makes them suffer from
$\mathcal{O}\left(n^{3}\right)$ time complexity. To tackle the issues, Liu et
al. (2022a); Li et al. (2022c) propose the anchor-based incomplete multi-view
clustering (AIMVC). The complexity of AIMVC is effectively reduced by merely
building the relationship between representative anchors and samples (Huang et
al., 2019). The classical AIMVC framework can be formulated as follows:

(2) $\displaystyle\min_{\mathbf{Z}}$
$\displaystyle\left\|\mathbf{X}_{v}\mathbf{H}_{v}-\mathbf{A}_{v}\mathbf{Z}\mathbf{H}_{v}\right\|_{\mathbf{F}}^{2}+\Omega\left(\mathbf{Z}\right),$
$\displaystyle\text{ s.t. }\mathbf{Z}\geq
0,\quad\mathbf{Z}^{\top}\mathbf{1}=\mathbf{1},$

where $\mathbf{A}_{p}\in\mathbb{R}^{d_{v}\times m}$ denotes the anchor matrix
from the $v$-th view, $\mathbf{Z}$ is the consistent anchor graph, $m$ is the
number of anchors, and $\Omega$ is the regularization term.

On the basis of the above paradigm, many methods adopt different
regularization terms to enhance clustering performance (Liu et al., 2022a; Guo
and Ye, 2019). However, most existing approaches overlook the inter-view
discrepancy and enforce cross-view representation to be consistent, which
would corrupt the representation capability of the model. Moreover, the
potential Anchor-Unaligned Problem for Incomplete Data has not been discussed
in previous research. Such AUP-ID would result in inaccurate graph fusion and
suboptimal clustering performance. In the next section, we will propose SIMVC-
SA to tackle these issues.

Figure 2. The framework of traditional anchor-based IMVC methods (left) and
the proposed SIMVC-SA (right). Different from traditional IMVC methods, the
proposed SIMVC-SA proposes a novel structure alignment module and adopts the
anchor learning strategy to efficiently enhance the clustering performance.

##  3\. Methods

###  3.1. Problem Formulation

As mentioned before, the main challenge for solving AUP-ID is the variation of
the feature dimension and available sample, which results in the anchors from
different views being under different metric spaces, and we can’t directly
measure the distance of cross-view anchors. As a result, a question worth
considering: how to effectively refine the cross-view anchor correspondence
under the incomplete scenario?

An intuitive method (Liu et al., 2022a) to implicitly avoid anchor
correspondence is to enforce cross-view anchor and the respective graph to be
consistent. While such a strategy overlooks the inter-view discrepancy and
corrupts the representation capability of the model. Inspired by (Wang et al.,
2022; Huang et al., 2020), we consider such principle: the correspondence
probability of the anchors should be high if their corresponding structure is
similar. Therefore, the original anchor correspondence problem can be
transferred to the structure alignment problem, as depicted in Fig. 1. In this
paper, we introduce the alignment matrix $\mathbf{P}_{v}$ that satisfies
$\mathbf{P}_{v}^{\top}\mathbf{P}_{v}=\mathbf{I}_{m}$ to efficiently tackle the
problem. Denoting the fusion representation as $\mathbf{F}$, and the anchor
graph alignment problem can be addressed as follow:

(3)
$\min_{\mathbf{P}_{v}}\left\|\mathbf{P}_{v}\mathbf{Z}_{v}-\mathbf{F}\right\|_{F}^{2},\quad\text{s.t.}\quad\mathbf{P}_{v}^{\top}\mathbf{P}_{v}=\mathbf{I}_{m},$

where $\mathbf{Z}_{v}\in$ $\mathbb{R}^{m\times n}$ is the view-specific anchor
graph.

Moreover, considering the traditional fixed anchor strategy relies on the
quality of anchors initialization and introduces unnecessary time overhead, we
adopt the anchor learning strategy to enhance our clustering performance in
this paper. In summary, the proposed Scalable Incomplete Multi-View Clustering
with Structure Alignment (SIMVC-SA) can be formulated as follows:

(4)
$\displaystyle\min_{\begin{subarray}{c}\boldsymbol{\gamma},\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V},\\\
\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V},\mathbf{P},\mathbf{F}\end{subarray}}\
\sum_{v=1}^{V}\gamma_{v}^{2}\left\|\mathbf{X}_{v}\mathbf{H}_{v}-\mathbf{A}_{v}\mathbf{Z}_{v}\mathbf{H}_{v}\right\|_{F}^{2}$
$\displaystyle\qquad\quad\
+\lambda\sum_{v=1}^{V}\left\|\mathbf{P}_{v}\mathbf{Z}_{v}-\mathbf{F}\right\|_{F}^{2}+\mu\sum_{v=1}^{V}\left\|\mathbf{Z}_{v}\right\|_{F}^{2}$
$\displaystyle\text{s.t.}\boldsymbol{\gamma}^{\top}1=1,\mathbf{A}_{v}^{\top}\mathbf{A}_{v}=\mathbf{I}_{m},\mathbf{P}_{v}^{\top}\mathbf{P}_{v}=\mathbf{I}_{m},\mathbf{Z}_{v}\geq
0,$
$\displaystyle\mathbf{Z}_{v}^{\top}\mathbf{1}_{m}=\mathbf{1}_{n},\mathbf{F}\mathbf{F}^{\top}=\mathbf{I}_{m},$

where $\mathbf{Z}_{v}\mathbf{H}_{v}$ can be considered as the similarities
between $m$ anchors and $n_{v}$ available samples of the $v$-th view. For the
sake of making the learned anchors $\mathbf{A}_{v}$ and consistent
representation $\mathbf{F}$ more discriminative, we impose orthogonal
constraints into them that
$\mathbf{A}^{\top}_{v}\mathbf{A}_{v}=\mathbf{I}_{m},\mathbf{F}\mathbf{F}^{\top}=\mathbf{I}_{m}$.
The learned bipartite graph $\mathbf{Z}_{v}$ should satisfy
$\mathbf{Z}_{v}\geq 0$ and $\mathbf{Z}_{v}^{\top}\mathbf{1}=\mathbf{1}$. The
$\boldsymbol{\gamma}\in\mathbb{R}^{V}$ captures the weight contribution of
every single view to all. $\lambda$ is the trade-off to balancing the
influence between anchor graph generation and alignment term. $\mu$ is the
hyperparameter of the regularization term. The framework of our SIMVC-SA is
shown in Fig. 2.

Although the Eq.(4) appears to be simple, we emphasize the superiority of
SIMVC-SA as follows:

  1. (1)

Joint Optimization Model. Unlike the existing two-stage ”aligning then
clustering” strategy (Wang et al., 2022), we propose a joint alignment-
clustering framework where the consistent representation $\mathbf{F}$ and the
alignment matrix $\mathbf{P}_{v}$ can be joint optimized to enhance the final
clustering performance in our model.

  2. (2)

Flexible Model with No Reference View. Different from the FMVACC (Wang et al.,
2022), which selects the first view for reference (all views align to the
first view) and MvCLN (Yang et al., 2021) iteratively selects the reference
view, we set a consistent representation $\mathbf{F}$ for alignment and
optimize it adaptively in our mode which avoids catastrophic performance
degradation when the reference view has poor quality(Lin et al., 2022a; Gong
et al., 2022).

  3. (3)

Soft Alignment Correspondence. The strict one-to-one mapping proposed in (Wang
et al., 2022) neglects the relationship between the different anchors and
brings higher time expenses. Besides, it is too harsh and unreasonable to
completely push away the different anchors. Recent work CLIP (Gao et al.,
2023) also noticed this problem. In the proposed method, we relax the original
strict constraint to an orthogonal constraint to achieve a soft assignment
while effectively reducing the time complexity of the alignment.

###  3.2. Optimization

The optimization problem in Eq.(4) is a non-convex problem when taking all
variables into account. To solve this problem in the section, we develop an
iterative optimization algorithm to address it. For the sake of simplifying
the optimization procedure, we have that
$\mathbf{X}_{v}\mathbf{H}_{v}\mathbf{H}_{v}^{\top}=\mathbf{X}_{v}\odot\mathbf{R}_{v}$,
where
$\mathbf{R}_{v}=\mathbf{1}_{d_{v}}\mathbf{r}^{(v)}\in\mathbb{R}^{d_{v}\times
n}$, $\mathbf{r}^{(v)}=[r_{1}^{(v)},\cdots,r_{n}^{(v)}]$, where
$r_{i}^{(v)}=\sum_{j=1}^{n_{v}}\mathbf{H}_{ij}^{(v)}$, $\odot$ represents the
Hadamard product. With this transformation, the space complexity drops from
$\mathcal{O}(vn^{2})$ to $\mathcal{O}(dn)$.

####  3.2.1. Optimization of Anchor Matrices
$\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$

When $\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$, $\boldsymbol{\gamma}$ and
$\mathbf{F}$ are fixed, the optimization for
$\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$ can be written as follows:

(5)
$\displaystyle\min_{\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}}\sum_{v=1}^{V}\gamma_{v}^{2}\left\|\mathbf{X}_{v}\mathbf{H}_{v}-\mathbf{A}_{v}\mathbf{Z}_{v}\mathbf{H}_{v}\right\|_{F}^{2},$
$\displaystyle\text{ s.t.
}\quad\mathbf{A}_{v}^{\top}\mathbf{A}_{v}=\mathbf{I}_{m}.$

Considering the optimization of each $\mathbf{A}_{v}$ is independent of the
corresponding view. Therefore, we extend the Frobenius norm with traces and
remove the irrelevant item, Eq.(LABEL:opt_A) can be formulated as:

(6)
$\max_{\mathbf{A}_{v}}\operatorname{Tr}\left(\mathbf{A}_{v}^{\top}\mathbf{M}_{v}\right),\text{
s.t. }\mathbf{A}_{v}^{\top}\mathbf{A}_{v}=\mathbf{I}_{m},$

where
$\mathbf{M}_{v}=\mathbf{X}_{v}\mathbf{H}_{v}\mathbf{H}_{v}^{\top}\mathbf{Z}_{v}^{\top}=\left(\mathbf{X}_{v}\odot\mathbf{R}_{v}\right)\mathbf{Z}_{v}^{\top}$.
According to (Wang et al., 2019a), the optimal solution for $\mathbf{A}_{v}$
is $\mathbf{U}_{m}\mathbf{V}_{m}^{\top}$, where $\mathbf{U}_{m}$ and
$\mathbf{V}_{m}$ represent the matrices which comprise the first $m$ left
singular vectors and right singular vectors of $\mathbf{M}_{v}$,
correspondingly. Their time overhead is $\mathcal{O}(nmd+m^{2}d)$ to obtain
all the optimal $\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$, where
$d=\sum_{v=1}^{V}d_{v}$.

####  3.2.2. Optimization of Anchor Graphs
$\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$

When $\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$, $\boldsymbol{\gamma}$ and
$\mathbf{F}$ are fixed, the optimization of
$\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$ can be written as follows:

(7)
$\displaystyle\min_{\left\\{\mathbf{\mathbf{Z}}_{v}\right\\}_{v=1}^{V}}\sum_{v=1}^{V}\gamma_{v}^{2}\left\|\mathbf{X}_{v}\mathbf{\mathbf{H}}_{v}-\mathbf{A}_{v}\mathbf{Z}_{v}\mathbf{H}_{v}\right\|_{F}^{2}$
$\displaystyle\qquad\ \
+\lambda\sum_{v=1}^{V}\left\|\mathbf{P}_{v}\mathbf{Z}_{v}-\mathbf{F}\right\|_{F}^{2}+\mu\sum_{v=1}^{V}\left\|\mathbf{Z}_{v}\right\|_{F}^{2},$
$\displaystyle\quad\text{s.t.}\quad\mathbf{Z}_{v}\geq
0,\mathbf{Z}_{v}^{\top}\mathbf{1}_{m}=\mathbf{1}_{n}.$

By removing the irrelevant items, the Eq.(7) can be rewrited as:

(8)
$\displaystyle\min_{\mathbf{Z}_{v}}\operatorname{Tr}\left(\mathbf{Z}_{v}^{\top}\mathbf{Z}_{v}\left(\gamma_{v}^{2}\mathbf{H}_{v}\mathbf{H}_{v}^{\top}+\left(\lambda+\mu\right)\mathbf{I}\right)\right)$
$\displaystyle\quad-2\operatorname{Tr}\left(\mathbf{Z}_{v}^{\top}\left(\gamma_{v}^{2}\mathbf{A}_{v}^{\top}\mathbf{X}_{v}\mathbf{H}_{v}\mathbf{H}_{v}^{\top}+\lambda\mathbf{P}_{v}^{\top}\mathbf{F}\right)\right)$
$\displaystyle\text{s.t.}\quad\mathbf{Z}_{v}\geq
0,\quad\mathbf{Z}_{v}^{\top}\mathbf{1}_{m}=\mathbf{1}_{n},$

with
$\mathbf{X}_{v}\mathbf{H}_{v}\mathbf{H}_{v}{}^{\top}=\mathbf{X}_{v}\odot\mathbf{R}_{v}$.
Denoting $\mathbf{z}_{j}^{(v)}$ as the $j$-th column vector of
$\mathbf{Z}_{v}$, we have

(9)
$\min_{\mathbf{z}_{j}^{(v)}}\frac{1}{2}\left\|\mathbf{z}_{j}^{(v)}-\mathbf{f}_{j}^{(v)}\right\|_{F}^{2},\quad\text{s.t.}\mathbf{z}_{j}^{(v)}\geq
0,\mathbf{z}_{j}^{(v)\top}\mathbf{1}_{m}=1,$

where
$\mathbf{f}_{ij}^{(v)}=\frac{\gamma_{v}^{2}\left[\mathbf{A}_{v}^{\top}\left(\mathbf{X}_{v}\odot\mathbf{R}_{v}\right)\right]_{ij}+\lambda\left[\mathbf{P}_{v}^{\top}\mathbf{F}\right]_{ij}}{\gamma_{v}^{2}r_{j}^{(v)}+\lambda+\mu}$,
$\left[\mathbf{P}_{v}^{\top}\mathbf{F}\right]_{ij}$ denotes the element of the
$i$-th row and $j$-th column of $\mathbf{P}_{v}^{\top}\mathbf{F}$.

We write the Lagrangian function of Eq.(9) as

$\small\mathcal{L}\left(\mathbf{z}_{j}^{(v)},\alpha_{j},\boldsymbol{\eta}_{j}\right)=\frac{1}{2}\left\|\mathbf{z}_{j}^{(v)}-\mathbf{f}_{j}^{(v)}\right\|_{F}^{2}-\alpha_{j}\left(\mathbf{z}_{j}^{(v)\top}\mathbf{1}_{m}-1\right)-\boldsymbol{\eta}_{j}^{\top}\mathbf{z}_{j}^{(v)},$

where $\alpha_{j}$ and $\boldsymbol{\eta}_{j}$ represent the respective
Lagrange multipliers. Their Kahn-Kuhn-Tucker (KKT) conditions can write as

$\left\\{\begin{array}[]{l}\mathbf{z}_{j}^{(v)}-\mathbf{f}_{j}^{(v)}-\alpha_{j}\mathbf{1}_{m}-\boldsymbol{\eta}_{j}=0,\\\
\boldsymbol{\eta}_{j}\odot\mathbf{z}_{j}^{(v)}=\mathbf{0}.\end{array}\right.$

Together with $\mathbf{z}_{j}^{(v)\top}\mathbf{1}_{m}=1$, we can derive the
equation below:

$\mathbf{z}_{j}^{(v)}=\max\left(\mathbf{f}_{j}^{(v)}+\alpha_{j}\mathbf{1}_{m},0\right),$

where $\alpha_{j}$ could be addressed by Newton’s method effectively. The time
complexity of optimizing $\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$ is
$\mathcal{O}(nmd)$.

####  3.2.3. Optimization of Consistent Representation $\mathbf{F}$

When $\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$ and $\boldsymbol{\gamma}$ are
fixed, the optimization for $\mathbf{Z}$ can be written as follows:

(10)
$\max_{\mathbf{F}}\operatorname{Tr}\left(\mathbf{F}\mathbf{Q}\right),\quad\text{s.t.}\mathbf{F}\mathbf{F}^{\top}=\mathbf{I}_{m},$

where $\mathbf{Q}=\sum_{v=1}^{V}\mathbf{Z}_{v}^{\top}\mathbf{P}_{v}^{\top}$,
the optimal solution of $\mathbf{F}$ is
$\boldsymbol{\Psi}_{m}\boldsymbol{\Sigma}_{m}^{\top}$, where
$\boldsymbol{\Sigma}_{m}$ and $\boldsymbol{\Psi}_{m}$ indicate the matrices
which comprise the first $m$ left singular vectors and the first $m$ right
singular vectors of $\mathbf{W}_{v}$, correspondingly. It costs
$\mathcal{O}(nm^{2}V)$ time.

Algorithm 1 Scalable Incomplete Multi-View Clustering with Structure Alignment
(SIMVC-SA)

0: $v$ views incomplete dataset $\left\\{\mathbf{X}_{v}\right\\}_{v=1}^{V}$,
the missing index $\left\\{\mathbf{H}_{v}\right\\}_{v=1}^{V}$, and the number
of cluster $k$.

1: Initialize $\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$, $\boldsymbol{\gamma}$.

2: repeat

3: Obtain $\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$ with Eq. (LABEL:opt_A).

4: Obtain $\mathbf{F}$ with Eq. (10).

5: Obtain $\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$ with Eq. (7).

6: Obtain $\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$ with Eq. (11).

7: Obtain $\boldsymbol{\gamma}$ with Eq. (12).

8: until converged

9: Obtain $\mathbf{U}$ by performing SVD on $\mathbf{F}$

9: Perform k-means on $\mathbf{U}$ to obtain discrete labels.

####  3.2.4. Optimization of Alignment Matrices
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$

When $\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$, $\boldsymbol{\gamma}$ and
$\mathbf{F}$ are fixed, the optimization for
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$ can be written as follows:

(11)
$\max_{\mathbf{P}_{v}}\operatorname{Tr}\left(\mathbf{P}_{v}^{\top}\mathbf{W}_{v}\right),\quad\text{s.t.}\mathbf{P}_{v}^{\top}\mathbf{P}_{v}=\mathbf{I}_{m},$

where $\mathbf{W}_{v}=\mathbf{F}\mathbf{Z}_{v}^{\top}$, Similar to Eq. (10),
this problem can be efficiently solved by rank-k truncated SVD.

####  3.2.5. Optimization of View Weight $\boldsymbol{\gamma}$

When $\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$,
$\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$ and $\mathbf{Z}$ are fixed, the
optimization for $\boldsymbol{\gamma}$ can be written as follows:

(12)
$\min_{\boldsymbol{\gamma}}\sum_{v=1}^{V}\gamma_{v}^{2}\tau_{v},\quad\text{s.t.}\boldsymbol{\gamma}^{\top}\mathbf{1}_{V}=1,\boldsymbol{\gamma}\geq
0,$

where
$\tau_{v}=\left\|\mathbf{X}_{v}\mathbf{H}_{v}-\mathbf{A}_{v}\mathbf{Z}_{v}\mathbf{H}_{v}\right\|_{F}^{2}$.
By Cauchy–Schwarz inequality, the view weight $\boldsymbol{\gamma}$ can be
acquired by

(13) $\gamma_{v}=\frac{1/\tau_{v}}{\sum_{v=1}^{V}1/\tau_{v}}.$

It consumes $\mathcal{O}(nmd)$ time. Algorithm 1 summarises the entire
optimization procedure for addressing Eq.(4).

###  3.3. Discussions

####  3.3.1. Convergence

As the iterations proceed, five variables of the above optimization procedure
will be separately addressed. As each sub-optimization problem reaches the
global optimum, the objective value monotonically decreases until the
convergence condition is attained (Bezdek and Hathaway, 2003). Furthermore,
because it is easy to prove that the lower boundary of the objective function
is zero, our proposed SIMVC-SA can converge to the local optimum.

####  3.3.2. Time Complexity

The time overhead of SIMVC-SA is composed of five optimization processes, as
previously mentioned. The time overhead of updating
$\left\\{\mathbf{A}_{v}\right\\}_{v=1}^{V}$ is
$\mathcal{O}\left(nmd+m^{2}d\right)$. When updating
$\left\\{\mathbf{Z}_{v}\right\\}_{v=1}^{V}$ and $\boldsymbol{\gamma}$ need
$\mathcal{O}\left(nmd\right)$. When analytically obtaining
$\left\\{\mathbf{P}_{v}\right\\}_{v=1}^{V}$, it costs
$\mathcal{O}((nm^{2}+m^{3})V)$ for all columns. The time overhead of
calculating $\mathbf{F}$ is $\mathcal{O}(nm^{2}V)$. As a result, the total
time overhead of the optimization procedure is
$\mathcal{O}\left(n\left(md+m^{2}V\right)+m^{3}V+m^{2}d\right)$. Consequently,
the computational complexity of SIMVC-SA is $\mathcal{O}(n)$, which is
linearly related to the number of samples.

Table 1. Incompete Multiview Datasets in our Experments Dataset | Size | Clusters | Views | Dimensionality
---|---|---|---|---
ORL | 400 | 40 | 3 | 4096/3304/6750
ProteinFold | 694 | 27 | 12 | 27/27/…/27/27/27
BDGP | 2500 | 5 | 3 | 1000/500/250
SUNRGBD | 10335 | 45 | 2 | 4096/4096
NUSWIDEOBJ | 30000 | 31 | 5 | 65/226/145/74/129
Cifar10 | 50000 | 10 | 3 | 512/2048/1024
MNIST | 60000 | 10 | 3 | 342/1024/64
Table 2. Empirical evaluation and comparison of SIMVC-SA with twelve baseline
methods on seven benchmark datasets.

Methods | BSV | MIC | MKKM-IK | AWP | DAIMC | APMC | UEAF | MKKM-IK-MKC | EEIMVC | FLSD | V3H | FIMVC-VIA | Proposed
---|---|---|---|---|---|---|---|---|---|---|---|---|---
ACC (%)
ORL | 24.32±0.89 | 37.56±1.66 | 59.80±2.44 | 68.69±0.00 | 68.03±2.32 | 65.58±1.91 | 60.25±2.50 | 64.95±2.62 | 73.24±2.54 | 48.09±1.85 | 67.03±1.45 | 76.36±2.79 | 76.06±2.36
ProteinFold | 22.25±0.53 | 15.99±0.78 | 26.03±1.06 | 28.00±0.00 | 28.65±1.65 | N/A | 28.72±1.53 | 17.99±0.83 | 27.75±1.67 | 25.98±1.33 | 17.33±0.48 | 28.15±1.26 | 30.17±1.23
BDGP | 34.96±1.06 | 25.37±0.61 | 32.17±0.24 | 23.62±0.00 | 28.12±0.01 | 28.12±0.01 | 44.88±0.02 | 40.77±0.20 | 44.00±0.05 | 42.96±0.03 | 43.63±0.75 | 39.84±0.16 | 48.11±0.21
SUNRGBD | 6.14±0.08 | 14.61±0.54 | 11.35±0.31 | 17.01±0.00 | 17.03±0.65 | 17.34±0.57 | 15.35±0.41 | 16.81±0.49 | 16.74±0.49 | 14.42±0.34 | N/A | 16.88±0.48 | 17.18±0.48
NUSWIDEOBJ | 12.05±0.03 | N/A | N/A | N/A | 13.79±0.37 | N/A | N/A | N/A | 12.73±0.16 | N/A | N/A | 12.96±0.12 | 15.40±0.32
Cifar10 | N/A | N/A | N/A | N/A | 95.81±0.45 | N/A | N/A | N/A | N/A | N/A | N/A | 96.16±0.00 | 96.32±0.22
MNIST | N/A | N/A | N/A | N/A | 97.57±0.31 | N/A | N/A | N/A | N/A | N/A | N/A | 98.16±0.01 | 98.40±0.04
NMI (%)
ORL | 48.49±0.90 | 56.44±1.00 | 75.95±1.33 | 83.79±0.00 | 82.89±1.06 | 80.20±0.82 | 76.16±1.25 | 79.76±1.41 | 85.37±1.32 | 67.91±1.28 | 81.05±0.61 | 88.08±1.31 | 87.53±1.25
ProteinFold | 27.60±0.59 | 16.64±1.02 | 33.70±0.84 | 36.17±0.00 | 37.67±1.08 | N/A | 38.18±0.88 | 24.88±0.84 | 36.03±0.98 | 35.66±0.79 | 22.75±0.53 | 36.22±0.96 | 37.72±0.98
BDGP | 12.88±0.94 | 4.47±0.70 | 7.41±0.16 | 4.68±0.00 | 8.68±0.01 | 8.68±0.01 | 23.55±0.04 | 16.35±0.13 | 19.91±0.09 | 18.95±0.06 | 24.15±0.40 | 15.11±0.10 | 23.67±0.14
SUNRGBD | 3.27±0.08 | 21.27±0.35 | 15.27±0.25 | 23.60±0.00 | 21.53±0.43 | 22.46±0.25 | 21.72±0.22 | 20.48±0.28 | 20.84±0.28 | 20.82±0.17 | N/A | 21.48±0.33 | 22.52±0.23
NUSWIDEOBJ | 2.68±0.03 | N/A | N/A | N/A | 11.70±0.36 | N/A | N/A | N/A | 10.31±0.16 | N/A | N/A | 10.27±0.07 | 11.78±0.11
Cifar10 | N/A | N/A | N/A | N/A | 90.47±0.35 | N/A | N/A | N/A | N/A | N/A | N/A | 91.18±0.00 | 91.21±0.18
MNIST | N/A | N/A | N/A | N/A | 93.89±0.53 | N/A | N/A | N/A | N/A | N/A | N/A | 95.76±0.02 | 95.58±0.00
Purity (%)
ORL | 26.80±0.92 | 40.81±1.40 | 62.79±2.11 | 70.42±0.00 | 71.82±1.79 | 69.24±1.48 | 63.90±1.90 | 67.68±2.34 | 76.09±2.19 | 50.88±1.72 | 70.22±1.09 | 78.59±2.29 | 78.79±2.08
ProteinFold | 25.89±0.60 | 19.78±0.84 | 30.91±1.04 | 31.97±0.00 | 34.99±1.54 | N/A | 35.47±1.16 | 22.73±0.87 | 33.13±1.30 | 32.82±0.97 | 22.24±0.51 | 33.67±1.12 | 35.97±1.14
BDGP | 36.75±0.89 | 25.67±0.59 | 33.42±0.18 | 24.02±0.00 | 28.46±0.01 | 28.46±0.01 | 45.92±0.01 | 41.10±0.13 | 46.40±0.05 | 44.41±0.03 | 45.38±0.50 | 40.12±0.15 | 49.05±0.19
SUNRGBD | 13.06±0.16 | 32.36±0.59 | 27.06±0.45 | 37.56±0.00 | 34.39±0.59 | 33.19±0.47 | 33.37±0.48 | 32.92±0.50 | 33.58±0.48 | 32.41±0.39 | N/A | 34.34±0.61 | 34.51±0.51
NUSWIDEOBJ | 13.72±0.04 | N/A | N/A | N/A | 23.41±0.63 | N/A | N/A | N/A | 21.83±0.21 | N/A | N/A | 21.97±0.12 | 23.71±0.27
Cifar10 | N/A | N/A | N/A | N/A | 95.81±0.45 | N/A | N/A | N/A | N/A | N/A | N/A | 96.23±0.00 | 96.29±0.25
MNIST | N/A | N/A | N/A | N/A | 97.57±0.31 | N/A | N/A | N/A | N/A | N/A | N/A | 98.24±0.02 | 98.42±0.05
Fscore (%)
ORL | 9.01±0.69 | 17.30±1.18 | 46.32±2.50 | 58.73±0.00 | 56.84±2.87 | 50.70±2.55 | 42.53±2.74 | 53.30±2.91 | 63.67±2.85 | 31.17±2.00 | 54.27±1.40 | 68.25±3.25 | 67.62±2.85
ProteinFold | 12.30±0.08 | 10.35±0.38 | 14.35±0.81 | 12.38±0.00 | 16.97±1.09 | N/A | 16.31±1.22 | 8.92±0.48 | 15.62±1.19 | 14.57±1.02 | 10.33±0.13 | 15.53±1.00 | 16.81±1.04
BDGP | 28.76±0.61 | 29.88±0.06 | 25.25±0.08 | 32.53±0.00 | 31.21±0.00 | 31.21±0.00 | 33.69±0.03 | 30.15±0.10 | 32.93±0.04 | 34.29±0.01 | 35.31±0.24 | 31.65±0.10 | 35.40±0.12
SUNRGBD | 6.89±0.01 | 9.45±0.24 | 7.10±0.14 | 11.58±0.00 | 10.71±0.35 | 10.96±0.25 | 10.20±0.16 | 10.06±0.21 | 10.20±0.20 | 12.09±0.00 | N/A | 10.33±0.24 | 11.74±0.10
NUSWIDEOBJ | 10.95±0.00 | N/A | N/A | N/A | 8.58±0.19 | N/A | N/A | N/A | 7.81±0.08 | N/A | N/A | 7.87±0.07 | 11.77±0.12
Cifar10 | N/A | N/A | N/A | N/A | 92.16±0.68 | N/A | N/A | N/A | N/A | N/A | N/A | 92.79±0.00 | 92.90±0.19
MNIST | N/A | N/A | N/A | N/A | 95.28±0.57 | N/A | N/A | N/A | N/A | N/A | N/A | 96.66±0.00 | 96.92±0.00

##  4\. Experiment

###  4.1. Datasets

Seven wide-used datasets are adopted to evaluate the effectiveness of the
proposed algorithm, including ORL, ProteinFold, BDGP, SUNRGBD, NUSWIDEOBJ,
Cifar10, and MNIST. The elaborate information of these datasets is listed in
Tab. 1. For the above datasets, we remove samples randomly on each view to
obtain its incomplete version. Specifically, according to (Li et al., 2022b),
with the principle that each sample appears in at least one view, we generate
incomplete datasets at 0.1 intervals from 0.1 to 0.9.

Figure 3. Clustering performance of SIMVC-SA on benchmark datasets with
different missing ratio. Figure 4. Time Comparison of Different IMVC Methods
on Seven Incomplete Datasets Figure 5. The ablation study of our structural
alignment strategy on five benchmark datasets. ”Unaligned” indicates without
our structural alignment strategy. Figure 6. The ablation study of our anchor
learning strategy.

###  4.2. Compared Methods and Setting

Along with our proposed SIMVC-SA, we run twelve state-of-the-art incomplete
multi-view clustering methods for comparison, including Best Single View (BSV)
(Ng et al., 2002), MIC Views Clustering via Weighted NMF With $\ell_{2,1}$
Regularization (MIC) (Shao et al., 2015), Multiple Kernel k-Means With
Incomplete Kernels (MKKM-IK) (Liu et al., 2017), Multiview Clustering via
Adaptively Weighted Procrustes (AWP) (Nie et al., 2018), Doubly Aligned
Incomplete Multi-view Clustering (DAIMC) (Hu and Chen, 2018), Anchor-based
partial multi-view clustering (APMC) (Guo and Ye, 2019), Unified Embedding
Alignment With Missing Views Inferring for Incomplete Multiview Clustering
(UEAF) (Wen et al., 2019), Multiple Kernel k k-Means With Incomplete Kernels
and Multiple Kernel Clustering (MKKM-IK-MKC) (Liu et al., 2020), Efficient and
Effective Regularized Incomplete Multiview Clustering (EEIMVC) (Liu et al.,
2020), Generalized IMVC With Flexible locality Structure Diffusion (FLSD) (Wen
et al., 2020b), View Variation and View Heredity for Incomplete Multiview
Clustering (V3H) (Fang et al., 2020) and Fast Incomplete Multi-View Clustering
With View Independent Anchors (FIMVC-VIA) (Liu et al., 2022a).

For all the algorithms mentioned above, we set their parameters as their
recommended range. In the proposed method, we adjusted $\lambda$ to
$[10^{-4},10^{-2},1,10^{2},10^{4}]$, $\mu$ to
$[0,10^{-4},10^{-2},1,10^{2},10^{4}]$, and the anchor numbers of [k, 2k, 5k]
using a mesh search scheme. In addition, we repeated each experiment 10 cycles
to calculate the average performance and standard bias. To assess the
clustering performance, we employ four well-used criteria consisting of
accuracy (ACC), normalized mutual information (NMI), Purity, and Fscore. All
experiments were conducted on a desktop computer with Intel core i9-10900X CPU
and 64G RAM, MATLAB 2020b (64-bit).

###  4.3. Experimental Results

Tab. 2 reports the clustering results on seven benchmark datasets. The best
results are marked in red, while the second-best results are marked in blue.
”N/A” indicates the unavailable results due to time-out or out-of-memory
errors. Besides, we compare the ACC of all methods with different missing
rates in Fig. 3. According to the results, we have the following conclusions:

Figure 7. Objectives of the proposed method.

  1. (1)

Compared with existing IMVC methods, our proposed algorithm demonstrates the
best performance in most datasets. The recently proposed FIMVC-VIA method
shows better performance than other methods, which demonstrates its
superiority in incomplete datasets. In terms of ACC, our SIMVC-SA achieves
better performance than FIMVC-VIA on the ProteinFold, BDGP, SUNRGBD,
NUSWIDEOBJ, and Cifar10 datasets, i.e., 2.02%, 8.27%, 0.3%, 2.44%, and 0.16%,
which demonstrates the effectiveness of view-specific representation and
cross-view alignment strategy.

  2. (2)

Compared to traditional subspace-based IMVC methods, our anchor-based method
achieves the best performance in most cases and is applicable to various
large-scale datasets.

  3. (3)

As shown in Fig. 3, we can observe that most IMVC methods show greater
fluctuations in performance with the missing rate rising, while our method is
more stable. We conjecture that this is because the alignment of the
representations well complements the missing information of different views.

###  4.4. Running Time Comparison

To validate the computational efficiency of the proposed SIMVC-SA, we plot the
average running time of each algorithm on seven benchmark datasets in Fig. 4.
The results of some compared algorithms on large-scale datasets are not
reported due to memory overflow errors. As shown in the Fig. 4, we can observe
that

  1. (1)

Compared to full graph-based clustering methods, the proposed SIMVC-SA
significantly reduces run time through the construction of anchor graphs.

  2. (2)

Compared to the anchor-based IMVC approach, i.e., FIMVC-VIA, the proposed
SIMVC-SA requires more time consumption, mainly due to our view-specific
representation and structure alignment strategy, the extra computational
complexity increases with the number of views, which is most obvious in
NUSWIDEOBJ (5 views). General, the extra time spent is worthwhile since our
proposed SIMVC-SA demonstrates its superiority over FIMVC-VIA in most
datasets.

###  4.5. Ablation Study

##### Structural Alignment Strategy

The structural alignment strategy is the main contribution of this paper. To
further demonstrate the effectiveness of this strategy, we present the
experimental results of the ablation study in Fig. 5, where ”Unaligned”
indicates not using our structural alignment strategy. In our experimental
setting, we fixed the alignment matrix $\mathbf{P}$ in the optimization
process to obtain the final clustering result. The effectiveness of the
proposed strategy can be clearly demonstrated in Fig. 5. In terms of ACC, the
proposed structural alignment strategy improves the algorithm performance on
the ORL, ProteinFold, BDGP, Cifar10, and MNIST datasets by 18.46%,11.16%,
6.43%, 23.38%, and 28.55% respectively, which demonstrates the effectiveness
of our strategy.

##### Anchor Learning Strategy

We conducted ablation experiments with the proposed anchor learning strategy,
as shown in Fig. 7. ”Fixed” indicates initializing anchors by k-means without
updating during the optimization process. Compared to the above methods, our
approach significantly improves the clustering performance and avoids the high
time expenditure of k-means.

Figure 8. Sensitivity analysis of anchor number m of out method on two
benchmark datasets. Figure 9. Sensitivity analysis of $\boldsymbol{\lambda}$
and $\boldsymbol{\mu}$ of out method on two benchmark datasets.

###  4.6. Convergence and Sensitivity

We conducted several experiments to exhibit the convergence of the proposed
SIMVC-SA. As shown in Fig. 6, the objective value of our algorithm is
monotonically decreasing in each iteration. These results clearly verify the
convergence of our proposed algorithm.

To investigate the sensitivity of SIMVC-SA to the number of anchors m, we
investigated how our performance shifts for different numbers of anchors. As
shown in Fig. 8, the number of anchors has little effect on the performance of
our algorithm. Moreover, two hyperparameters, $\lambda$, and $\mu$, are used
in our method, $\lambda$ is the structural alignment parameter, and $\mu$ is
the coefficient of the sparsity regularization term. As is shown in Fig. 9, we
conducted comparative experiments to indicate the effect of these two
parameters on performance.

##  5\. Conclusion

In this paper, we propose a novel incomplete anchor graph learning framework
termed Scalable Incomplete Multi-View Clustering with Structure Alignment
(SIMVC-SA). Specially, we construct the incomplete anchor graph on each view
in terms of the unaligned anchor. Besides, a novel structure alignment module
is proposed to refine the cross-view anchor correspondence. Meanwhile, the
anchor graph construction and alignment are jointly optimized in our unified
framework to enhance clustering quality. Through anchor graph construction
instead of full graphs, the time and space complexity of our proposed SIMVC-SA
is proven to be linearly related to the number of samples. Extensive
experiments on seven incomplete benchmark datasets demonstrate the
effectiveness and efficiency of our proposed method. In the future, we will
explore more flexible alignment strategies. For example, how to align the
anchor with different numbers.

##  6\. ACKNOWLEDGMENTS

This work was supported by the National Key R&D Program of China (no.
2020AAA0107100) and the National Natural Science Foundation of China (project
no. 62325604, 62276271).

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