11institutetext: Tongji University, Shanghai, China 11email:
<EMAIL_ADDRESS>22institutetext: Tianjin University, Tianjin, China
22email<EMAIL_ADDRESS>
# SLRL: Structured Latent Representation Learning for Multi-view Clustering

Zhangci Xiong 11 Meng Cao 22

###### Abstract

In recent years, Multi-View Clustering (MVC) has attracted increasing
attention for its potential to reduce the annotation burden associated with
large datasets. The aim of MVC is to exploit the inherent consistency and
complementarity among different views, thereby integrating information from
multiple perspectives to improve clustering outcomes.

Despite extensive research in MVC, most existing methods focus predominantly
on harnessing complementary information across views to enhance clustering
effectiveness, often neglecting the structural information among samples,
which is crucial for exploring sample correlations. To address this gap, we
introduce a novel framework, termed Structured Latent Representation Learning
based Multi-View Clustering method (SLRL). SLRL leverages both the
complementary and structural information. Initially, it learns a common latent
representation for all views. Subsequently, to exploit the structural
information among samples, a k-nearest neighbor graph is constructed from this
common latent representation. This graph facilitates enhanced sample
interaction through graph learning techniques, leading to a structured latent
representation optimized for clustering. Extensive experiments demonstrate
that SLRL not only competes well with existing methods but also sets new
benchmarks in various multi-view datasets.

###### Keywords:

Multi-view Clustering Latent Representation k-nearest Neighbor Graph Graph
Representation Learning.

##  1 Introduction

Multi-View Clustering (MVC) has attracted significant attention in recent
years, primarily because multi-view or multi-modal data offer shared semantics
that enhance learning effectiveness [1, 3, 4, 27]. This technique is
especially beneficial in scenarios lacking explicit labels.

There have been notable advancements in MVC methodologies, including the
development of subspace-based, non-negative matrix factorization-based, and
graph-based clustering algorithms. While these methods have yielded
satisfactory results, challenges remain. Specifically, most approaches
concentrate on harnessing the consistency and complementarity across views to
amalgamate information from diverse sources. However, they often neglect the
structural information among samples, which is crucial for uncovering latent
relationships essential for effective clustering. Although some graph-based
MVC methods have been introduced, they typically depend on constructing graphs
from original data matrices or shallow features and subsequently fusing these
graphs through various strategies. This method is not only computationally
demanding but also heavily reliant on the quality of the initial graph
construction, thus potentially limiting the effectiveness of MVC.

Figure 1: SLRL model framework.

In response to these issues, this paper proposes a Structured Latent
Representation Learning (SLRL) based multi-view clustering method. The model
structure of SLRL is illustrated in Figure 1. SLRL initially mines the
complementary information of different views flexibly by learning a common
latent representation. Subsequently, to leverage the structural information
among samples, a $k$-nearest neighbor graph is constructed based on this
latent representation. This graph is further enhanced through a graph
attention network, thereby learning a structured latent representation.
Moreover, a clustering loss further constrains the latent representation
learning, endowing the learned latent representation with a favorable cluster
structure. Compared with existing multi-view clustering methods, SLRL has the
following advantages:

(1) By considering both the complementary information between views and the
structural relationships among samples, the proposed method not only flexibly
integrates information from multiple views but also fully utilizes the
structural information among samples.

(2) The learned latent representations are not only well-structured but also
cohesive, making them particularly suitable for clustering tasks.

(3) Experiments conducted on six multi-view datasets demonstrate that the
clustering performance of SLRL surpasses that of several existing methods.

##  2 Related Work

###  2.1 Multi-view Clustering

Multi-view clustering aims to achieve enhanced clustering performance by
integrating information from different views. One straightforward approach
involves concatenating features from all views into a new feature vector,
followed by the application of traditional single-view clustering methods such
as K-means or spectral clustering on this vector. However, this strategy often
falls short of expectations as it overlooks the distinct characteristics of
each view and the complex relationships between them, and it lacks
interpretability. To better extract information from multi-view data, multi-
view clustering algorithms generally adhere to two principles: the principle
of consistency and the principle of complementarity. In recent years, numerous
multi-view clustering methods have been proposed by effectively considering
the consistency and complementarity of information from multiple views[6, 22,
23, 24, 25, 26, 19]. The existing multi-view clustering methods can be
categorized into four types: co-training based multi-view clustering, multi-
kernel learning based multi-view clustering, multi-view subspace clustering,
and graph-based multi-view clustering. Our work pertains to graph-based multi-
view clustering.

###  2.2 Graph-based Multi-view Clustering

Graph-based multi-view clustering firstly fuses all views into a single graph,
and then applies graph cutting algorithms or spectral clustering on the fused
graph to obtain the final clustering results.

In recent years, numerous graph learning-based multi-view clustering methods
have been proposed. For instance, Saha et al.[14] introduced a graph-based
multi-view clustering method that initially employs three different similarity
measures to construct nearest neighbor graphs projected onto individual
feature spaces. Subsequently, these graphs are clustered into feature
subspaces, followed by hierarchical agglomerative clustering on each view to
achieve the clustering outcome. Nie et al.[11] proposed a method that combines
multi-view clustering and semi-supervised classification. This approach
involves adaptively selecting neighbors of samples to construct graphs, and
then conducting multi-view learning based on these graphs, thereby realizing
more accurate clustering and classification. Tao et al.[16] introduced a
multi-view clustering method based on adaptive learning of graphs, which not
only considers the importance of each view at the view level but also
introduces sample-pair-specific weights within views, thereby adaptively
learning a common similarity matrix suitable for clustering. To enhance multi-
view clustering performance, Zhan et al.[23] proposed a graph learning based
clustering method, which learns a unified graph with rank constraints on its
Laplacian matrix from different perspectives. Then, clustering indicators are
directly obtained from the global graph, eliminating the need for any graph
cutting or K-means clustering. Tang et al.[15] employed a Cross-view Graph
Diffusion (CGD) approach, which takes predefined graph matrices from different
views as input and learns a unified graph for multi-view data clustering
through an iterative cross-diffusion process. Li et al.[8] presented a
consensus graph learning-based multi-view clustering method. This method
integrates graph structural information from different views to construct a
consensus graph, aiming to enhance clustering performance.

##  3 The Proposed Method

As shown in Figure 1, the SLRL method proposed in this paper is primarily
composed of three parts: common latent representation learning, structured
latent representation learning and clustering.

###  3.1 Common Latent Representation Learning

The essence of multi-view clustering is the effective use of consistency and
complementarity between views for a comprehensive information set. Therefore,
it requires integrating information from multiple views via specific
strategies. Addressing view complementarity, Zhang et al.[26] introduced a
latent multi-view subspace clustering approach. This approach generates a
common latent representation for all views and reconstructs each view from it.
Thanks to the multi-view data’s complementarity, this representation is more
complete, accurate, and robust than that from a single view.

Specifically, for the $v$ view $\bm{X}_{n}^{(v)}$, define the latent
representation of the $n^{th}$ sample as $h_{n}$; let
$f_{v}\left(\bm{h}_{n};\bm{\Theta}_{r}^{(v)}\right)$ be the reconstruction
network for the $v^{th}$ view, parameterized by $\Theta_{r}^{(v)}$. The
optimization objective for learning the common latent representation can then
be defined as follows:

$L_{r}=\sum_{v=1}^{V}\left\|f_{v}\left(\bm{h}_{n};\bm{\Theta}_{r}^{(v)}\right)-\bm{x}_{n}^{(v)}\right\|^{2},$
(1)

where $L_{r}$ represents the reconstruction loss for the common latent
representation, which aims to learn the mapping between the original data
space and the common latent space. In this paper, we utilize a multi-layer
perceptron as the encoder to achieve the mapping between the common latent
representation and the original features $x_{n}$ of data. By minimizing the
loss corresponding to the formula (1), a common latent representation
$\bm{H}=$ $\left\\{\bm{h}_{1},\bm{h}_{2},\ldots,\bm{h}_{N}\right\\}$ can be
learned.

###  3.2 Structured Latent Representation Learning

While the common latent representation can integrate complementary multi-view
information, it often lacks a well-defined cluster structure, making it
suboptimal for direct clustering use. Additionally, this perspective neglects
the structural information among samples, which is vital for clustering.
Noting that samples from the same class show similar features, we select the
$k$-nearest neighbors for each sample. Thus, we propose constructing an
undirected $k$-nearest neighbor graph on $\bm{H}$ to capture structural
information in the latent space, defining the neighbor graph as
$\bm{A}\in\bm{R}^{N*N}$. The distance between any two samples $i$ and $j$ with
representations $\bm{h}_{i}$ and $\bm{h}_{j}$ is calculated as follows.

For continuous data such as images, we employ a Gaussian kernel to measure the
similarity between sample $i$ and sample $j$. The neighbor relationship
between samples $i$ and $j$ can be calculated using the following formula:

$\centering\mathrm{A}_{ij}=\left\\{\begin{array}[]{l}\exp\left(-\frac{\left\|\bm{h}_{i}-\bm{h}_{j}\right\|^{2}}{2\sigma^{2}}\right),\bm{h}_{i}\in
N_{k}\left(\bm{h}_{j}\right)\text{ or }~{}\bm{h}_{j}\in
N_{k}\left(\bm{h}_{i}\right),\\\
0,\text{other},\end{array}\right.\@add@centering$ (2)

For discrete data, such as text, we utilize the dot product similarity for
calculating the similarity between entities. The neighbor relationship can
then be computed as follows:

$\centering\mathrm{A}_{ij}=\left\\{\begin{array}[]{l}\bm{h}_{j}^{T}\bm{h}_{i},\bm{h}_{i}\in
N_{k}\left(\bm{h}_{j}\right)\text{or}~{}\bm{h}_{j}\in
N_{k}\left(\bm{h}_{i}\right),\\\
0,\text{other},\end{array}\right.\@add@centering$ (3)

where $\bm{N}_{k}\left(\bm{h}_{i}\right)$ denotes the $k$-nearest neighbors of
the $i^{th}$ sample. Thus, we can obtain the neighbor graph $\bm{A}$ from non-
graph data, which effectively represents the neighbor relationships among
samples.

Given the effectiveness of nearest neighbor graphs in assessing distances
between diverse samples, we leverage Graph Attention Networks (GAT) to explore
the information among samples within the common latent representation, thereby
obtaining a structured latent representation.

The GAT takes $\bm{H}=$
$\left\\{\bm{h}_{1},\bm{h}_{2},\ldots,\bm{h}_{N}\right\\}$ as input, and the
output of the GAT node features is denoted as
$\tilde{\bm{h}_{i}}\in\mathbb{R}^{F^{\prime}}$, where $F$ represents the
dimensionality of the input node features, and $F^{\prime}$ denotes the
dimensionality of the output node features. The objective of GAT is to achieve
sufficient expressive capability by transforming the input features into
higher-dimensional features. Initially, it applies a learnable linear
transformation to each node, with the weight matrix represented as
$\bm{W}\in\mathbb{R}^{F^{\prime}\times F}$. Subsequently, it employs a single-
layer feed-forward neural network, denoted as
$\bm{a}\in\mathbb{R}^{2F^{\prime}}$, to calculate the attention coefficients
between nodes. The process of computation is illustrated in Equation (4).

${\alpha}_{ij}=\frac{\exp\left(\operatorname{LeakyReLU}\left(\bm{a}^{T}\left[\mathbf{W}\bm{h}_{i}\|\bm{W}\bm{h}_{j}\right]\right)\right)}{\sum_{j\in\mathcal{N}_{i}}\exp\left(\operatorname{LeakyReLU}\left(\bm{a}^{T}\left[\mathbf{W}\bm{h}_{i}\|\mathbf{W}\bm{h}_{j}\right]\right)\right)},$
(4)

where $T$ denotes the matrix transpose, and $\|$ represents the concatenation
operation. Node $j$ is a neighbor of node $i$, and $\alpha_{ij}$ signifies the
importance of the features of node $j$ relative to node $i$ among all the
neighbors of node $i$. The set of all neighbors of node $i$ is represented by
$\mathcal{N}_{i}$.

Utilizing the normalized attention coefficients $\alpha_{ij}$, GAT combines
the features of neighboring nodes to update the current node’s features, then
applying the nonlinear sigmoid function $\sigma$. The computation of the node
features is as follows:

$\tilde{\bm{h}_{i}}=\sigma\left(\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}\bm{W}\bm{h}_{j}\right).$
(5)

Finally, GAT stabilizes the learning process by employing multi-head
attention, and concatenates or averages the output features of the node’s
neighbors to form its final output feature, as illustrated in Equation (6):

$\tilde{\bm{h}_{i}}=\|_{k=1}^{K}\sigma\left(\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}^{k}\bm{W}\tilde{\bm{h}_{i}}\right),$
(6)

where $K$ denotes the number of attention heads, and $\alpha_{ij}^{k}$
represents the relative attention weight of the features of node $j$ with
respect to node $i$ for the $k^{th}$ attention head.

By applying attention mechanism learning to each node, we get a structured
latent representation
${\tilde{\bm{H}}}=\left\\{\tilde{\bm{h}_{1}},\tilde{\bm{h}_{2}},\ldots,\tilde{\bm{h}_{N}}\right\\}$.
For each sample in $\tilde{\bm{H}}$, similar samples feature closer
characteristics during the learning process, resulting in an enhanced
structural integrity.

###  3.3 Joint Training and Clustering

Through GAT, we obtain a structured latent representation $\tilde{\bm{H}}$
that preserves data’s inherent local structures but isn’t directly suitable
for clustering. Typically, existing works apply traditional clustering methods
like spectral clustering or K-means to these features, with clustering
effectiveness dependent on the feature quality. To address this, Xie et al.
[20] introduced an end-to-end clustering method that simultaneously optimizes
latent representation learning and clustering, enhancing the suitability of
the structured latent representation for clustering tasks.

In this paper, we use a two-step alternating unsupervised algorithm to improve
clustering. Initially, we compute the soft assignment of each sample in the
latent representation to cluster centroids. Next, we update the structured
latent representation and refine centroids based on high-confidence
assignments using an auxiliary target distribution. This cycle repeats until
convergence. Specifically, for each $i^{th}$ sample and $j^{th}$ cluster,
similarity between $\tilde{\bm{h}}{i}$ and $\bm{u}_{j}$ is measured using a
$t$-distribution kernel, calculated as follows:

$q_{ij}=\frac{\left(1+\left\|{\tilde{{\bm{h}_{i}}}}-\bm{\mu}_{j}\right\|^{2}\right)^{-1}}{\sum_{j}\left(1+\left\|{\tilde{\bm{h}}_{i}}-\bm{\mu}_{j}\right\|^{2}\right)^{-1}}.$
(7)

The cluster centers $u_{i}$ are initialized through K-means on
$\tilde{{\bm{h}_{i}}}$. $Q={[q_{ij}]}$ represents the probability of assigning
sample $i$ to cluster $j$, i.e. the soft assignment. Once the clustering
distribution $Q$ is obtained, our goal is to optimize the latent
representation through learning from high-confidence soft assignments.
Specifically, we aim to bring the feature representation of each sample closer
to its cluster center, thereby enhancing the cohesiveness within the cluster.
Therefore, the $p_{ij}$ in the target distribution $P$ can be calculated as
follows:

$p_{ij}=\frac{q_{ij}^{2}/\sum_{i}q_{ij}}{\sum_{j}\left(q_{ij}^{2}/\sum_{i}q_{ij}\right)}.$
(8)

Within the target distribution $P$, each assignment in $Q$ is squared and
normalized to confer a higher confidence level. Consequently, the clustering
loss function can be defined as follows:

$L_{c}=KL(P\|Q)=\sum_{i}\sum_{j}p_{ij}\log\frac{p_{ij}}{q_{ij}}.$ (9)

By minimizing the KL divergence loss between the distributions $Q$ and $P$,
the target distribution $P$ facilitates the network in learning a structured
latent representation that is more suitable for clustering tasks. This implies
that data representations around the cluster centers are brought closer
together, thereby yielding improved clustering performance.

1 Input: The multi-view data
$\bm{X}=\left\\{{\bm{X}}^{(1)},{\bm{X}}^{(2)},...,{\bm{X}}^{(V)}\right\\}$,
the number of nearest neighbors $k$, the hyperparameter $\gamma$, and the
maximum number of iterations $T$:

2 Initialization: Randomly initialize $\left\\{\bm{h}_{n}\right\\}_{n=1}^{N}$,
$\left\\{\bm{\Theta}_{r}^{(v)}\right\\}_{v=1}^{V}$ and
$\left\\{\bm{W}_{l}\right\\}_{l=1}^{L}$.

3for  _i $\in 0,1,\cdots,T$_ do

4 Update the common latent representation $H$ using Equation (1);

5 Construct the $k$-nearest neighbor graph using either Equation (2) or
Equation (3);

6 for  _$l=1$ to $L$_ do

7 Update the latent representation $\bm{H}^{(\ell)}$ for layer $l$ using
Equation (6);

8

9 end for

10 Calculate the reconstruction loss $L_{r}$ and the clustering loss $L_{c}$
separately, and compute the total loss using Equation (10);

11 Update network parameters through backpropagation.

12 end for

13

Output:  Network parameters
$\left\\{\bm{\Theta}_{r}^{(v)}\right\\}_{v=1}^{V}$,
$\left\\{\bm{W}_{l}\right\\}_{l=1}^{L}$, structured latent representation
${\tilde{\bm{H}}}$.

Algorithm 1 The Training Process of SLRL.

###  3.4 Loss Function

For SLRL, the loss function comprises reconstruction loss and clustering loss.
The reconstruction loss is utilized to constrain the error between the
features reconstructed from the common latent representation $\bm{H}$ and the
original features of the view, thus flexibly balancing consistency and
complementarity. Subsequently, by exploring the structural characteristics
among samples, a structured latent representation $\tilde{\bm{H}}$ is
obtained. The clustering loss is employed to disperse the latent
representation $\tilde{\bm{H}}$, while constraining the learning of both the
common latent representation and the graph representation. This approach
ensures that the learned latent representation possesses a better cluster
structure. Therefore, the overall loss function can be defined as:

$L=L_{r}+\gamma L_{c},$ (10)

where $L_{r}$ and $L_{c}$ represent the reconstruction loss and clustering
loss, respectively. $\gamma>0$ is a hyperparameter employed to balance the
optimization of clustering and the learning of the common latent
representation. Algorithm 1 succinctly summarizes the training process of
SLRL.

##  4 Experiments

In this section, we present comprehensive experiments to assess the efficacy
of our proposed SLRL algorithm. To this end, we compare its clustering
performance with other established methods, analyze the affinity matrix and
convergence behavior, investigate the parameter sensitivity, and visualize the
data.

###  4.1 Experimental Settings

In this paper, six commonly used multi-view datasets were selected for
comparison of all methods, including
100leaves111https://archive.ics.uci.edu/dataset/241/one+hundred+plant+
species+leaves+data+set, Scene-15, MSRCV1222http://research.microsoft.com/en-
us/projects/objectclassrecognition,
3Sources333http://erdos.ucd.ie/datasets/3sources.html,
BBCSport444http://erdos.ucd.ie/datasets/segment.html and
HW555https://archive.ics.uci.edu/dataset/72/multiple+features, and the
following ten methods were selected for comparison with the approach proposed:
SPCBestSV[10], LRRBestSV[9], DiMSC[2], AMGL[13], MLAN[12], LMSC[26], MVGL[23],
GMC[17], CGD[15] and DealMVC[21].

For each experiment, 80% of the dataset is used as the training set and 20% as
the test set. To ensure fairness, for all compared methods, adjustments are
made according to the source code and parameters provided by the authors of
the papers to achieve the best clustering results. For our model, the ReLU
function is used as the nonlinear activation function in the network. The
dimensionality of the latent representation $F$ for each dataset ranges from
$\left\\{16,32,64,128,256\right\\}$, with the default setting being 64. The
number of neighbors $k\in\\{3,15\\}$, with a default setting of $k=10$. As the
model is not sensitive to changes in the parameter $\gamma$, $\gamma$ is
uniformly set to 10 in the experiments of this paper. The loss function of the
model is optimized using the batch gradient descent algorithm, with a learning
rate of 0.01. To reduce random errors, we repeat each experiment 10 times and
calculate the average values and standard deviations of four clustering
performance metrics for comparison.

###  4.2 Experimental Analysis and Results

Tables 1 presents the average values and standard deviations of different
metrics across all methods on six datasets. For each clustering metric, we
utilize a red bold font to denote the best performance and an underlined font
to signify the second-best performance. Based on these experimental results,
the following conclusions can be drawn:

(1) It is evident that, across six commonly used multi-view datasets, the
method proposed in this paper has achieved excellent clustering performance.
Compared to the methods reviewed, the proposed method achieved the best
results on four clustering metrics across the datasets of 100Leaves, Scene-15,
MSRCV1, 3Sources and HW. Relative to the second-best method, SLRL improved by
9.05%, 8.84%, 1.54%, 6.48% and 5.13% on ACC, and by 2.34%, 25.21%, 7.57%,
2.18% and 0.37% on NMI, respectively. Although MLAN and CGD performed better
on the BBCSport dataset, their performance was lower on other datasets
compared to the proposed method. Overall, SLRL demonstrated desirable
clustering performance across different datasets, underscoring its
superiority.

(2) On some datasets, the best results of single-view clustering were superior
to those of some multi-view clustering algorithms, indicating that fully
utilizing the information from multiple views is challenging and requires more
rational exploration of the consistency and complementarity between views.

Table 1: Comparison of clustering performance on six datasets. Datasets | Method | ACC (%) | NMI (%) | F_score (%) | ARI (%)
---|---|---|---|---|---
100Leaves[5] | $\text{SPC}_{\text{BestSV}}$ | 48.03±1.11 | 76.61±2.18 | 40.76±1.84 | 40.19±1.51
$\text{LRR}_{\text{BestSV}}$ | 47.86±2.49 | 70.62±1.37 | 38.42±2.67 | 35.08±0.91
DiMSC | 85.58±3.13 | 72.34±2.31 | 57.97±3.25 | 60.89±2.96
AMGL | 76.76±1.93 | 87.57±2.86 | 57.66±1.30 | 56.12±1.94
MLAN | 84.02±0.72 | 94.11±1.30 | 80.21±1.07 | 79.56±2.36
LMSC | 74.37±2.06 | 86.41±2.73 | 64.31±2.84 | 63.30±2.95
MVGL | 76.59±1.26 | 85.87±0.78 | 51.32±2.74 | 51.37±0.70
GMC | 83.62±0.83 | 90.09±2.10 | 80.94±2.65 | 73.79±0.68
CGD | 86.34±1.21 | 94.92±1.93 | 83.74±0.93 | 81.86±1.02
DealMVC | 80.44±5.41 | 77.13±6.02 | 78.72±5.90 | 58.64±6.14
| Ours (SLRL) | 94.15±2.19 | 97.14±0.86 | 88.70±3.37 | 88.53±3.43
Scene-15[7] | $\text{SPC}_{\text{BestSV}}$ | 37.42±1.34 | 37.26±0.91 | 28.31±1.70 | 19.42±1.28
$\text{LRR}_{\text{BestSV}}$ | 36.29±0.92 | 36.13±1.47 | 25.31±0.90 | 18.95±0.91
DiMSC | 38.22±1.13 | 39.00±2.40 | 31.87±1.25 | 22.91±1.17
AMGL | 33.92±2.33 | 37.35±1.33 | 25.02±0.83 | 17.39±1.10
MLAN | 15.78±1.08 | 16.59±0.03 | 15.21±1.08 | 13.94±1.37
LMSC | 42.68±1.40 | 34.98±2.21 | 27.77±1.27 | 23.26±0.93
MVGL | 37.48±2.06 | 37.29±2.17 | 27.82±1.10 | 22.14±2.06
GMC | 41.55±1.30 | 37.09±1.81 | 30.04±0.92 | 22.18±1.98
CGD | 43.66±1.99 | 42.53±1.53 | 31.39±2.80 | 24.94±1.53
DealMVC | 29.87±6.56 | 25.16±4.98 | 29.05±6.62 | 22.66±4.49
| Ours (SLRL) | 47.52±1.87 | 53.25±1.02 | 42.31±0.97 | 31.88±1.24
MSRCV1[18] | $\text{SPC}_{\text{BestSV}}$ | 51.45±2.41 | 50.07±0.85 | 48.68±1.64 | 47.70±1.89
$\text{LRR}_{\text{BestSV}}$ | 66.75±3.06 | 59.68±1.55 | 60.44±3.03 | 58.24±1.17
DiMSC | 74.33±2.61 | 66.75±2.73 | 73.79±0.68 | 72.95±1.13
AMGL | 74.40±2.91 | 72.97±1.69 | 66.50±2.42 | 62.62±1.84
MLAN | 71.41±1.82 | 74.40±0.73 | 66.73±2.66 | 60.88±1.38
LMSC | 83.62±1.33 | 70.52±1.27 | 69.96±2.58 | 66.13±1.35
MVGL | 90.72±2.46 | 82.51±0.47 | 81.21±1.50 | 76.64±1.22
GMC | 90.55±2.29 | 82.76±2.03 | 78.48±1.94 | 76.95±2.27
CGD | 91.29±1.09 | 83.63±2.51 | 80.98±0.82 | 77.79±0.81
DealMVC | 76.52±6.85 | 65.01±7.60 | 74.64±6.32 | 59.25±7.84
| Ours (SLRL) | 92.70±2.12 | 89.96±1.52 | 84.35±3.09 | 82.26±1.84
BBCSport | $\text{SPC}_{\text{BestSV}}$ | 35.05±2.47 | 30.71±1.09 | 39.84±2.61 | 31.92±1.65
$\text{LRR}_{\text{BestSV}}$ | 89.68±1.91 | 77.34±2.90 | 80.18±1.39 | 73.75±2.26
DiMSC | 84.98±1.11 | 89.08±1.51 | 91.27±1.20 | 91.63±1.75
AMGL | 57.83±2.49 | 52.14±2.50 | 56.81±1.80 | 52.49±0.46
MLAN | 97.23±3.13 | 90.68±1.69 | 94.77±2.94 | 92.07±3.08
LMSC | 92.79±0.72 | 83.77±1.78 | 90.21±1.04 | 86.07±2.05
MVGL | 80.89±1.59 | 79.90±2.38 | 79.55±1.11 | 74.81±2.35
GMC | 85.56±2.30 | 81.01±0.97 | 80.85±1.26 | 81.15±0.37
CGD | 97.37±1.03 | 91.09±2.76 | 93.97±1.75 | 93.73±1.20
DealMVC | 66.06±11.94 | 49.87±12.67 | 63.34±12.55 | 43.30±14.93
| Ours (SLRL) | 93.02±2.52 | 86.44±0.93 | 91.20±1.73 | 90.75±1.10
3Sources | $\text{SPC}_{\text{BestSV}}$ | 64.08±2.44 | 56.25±1.36 | 60.26±0.87 | 53.15±0.70
$\text{LRR}_{\text{BestSV}}$ | 63.64±1.98 | 53.73±1.25 | 54.37±2.41 | 46.07±0.75
DiMSC | 73.75±2.22 | 68.84±0.82 | 70.23±1.25 | 60.43±1.51
AMGL | 63.88±1.08 | 60.89±1.91 | 62.75±1.27 | 54.91±1.03
MLAN | 68.72±2.41 | 54.38±1.21 | 55.73±1.33 | 35.55±1.77
LMSC | 72.22±2.08 | 68.40±0.91 | 70.61±1.15 | 57.76±0.55
MVGL | 70.54±1.66 | 65.69±2.31 | 60.63±1.59 | 43.10±1.08
GMC | 69.56±1.99 | 61.84±2.33 | 60.78±1.04 | 44.38±2.52
CGD | 76.65±1.53 | 70.79±0.58 | 70.18±3.23 | 60.64±1.20
DealMVC | 65.82±7.36 | 59.95±8.34 | 64.24±7.25 | 56.07±7.06
| Ours (SLRL) | 81.62±1.62 | 72.33±2.05 | 77.90±1.32 | 69.86±1.27
HW | $\text{SPC}_{\text{BestSV}}$ | 46.66±1.71 | 42.08±1.63 | 29.69±1.48 | 40.11±1.67
$\text{LRR}_{\text{BestSV}}$ | 45.42±0.31 | 40.64±0.17 | 27.05±0.23 | 34.76±0.20
DiMSC | 86.67±2.32 | 79.31±2.18 | 76.14±1.17 | 77.58±2.49
AMGL | 85.32±0.77 | 84.87±0.45 | 72.00±1.21 | 76.31±0.96
MLAN | 96.04±1.74 | 91.96±2.10 | 83.01±1.81 | 84.29±0.46
LMSC | 80.33±3.60 | 79.02±2.07 | 71.88±3.38 | 74.78±3.01
MVGL | 85.16±1.36 | 79.67±1.52 | 83.32±1.00 | 75.54±2.64
GMC | 88.12±2.41 | 91.79±1.43 | 80.10±0.88 | 86.62±1.17
CGD | 91.70±2.15 | 89.95±1.30 | 85.44±3.01 | 90.13±1.66
DealMVC | 82.67±4.84 | 77.29±5.93 | 80.37±5.56 | 75.82±6.27
| Ours (SLRL) | 96.40±2.06 | 95.11±2.21 | 92.75±3.77 | 92.02±4.16

###  4.3 Visualization of Clustering Results

To provide a more intuitive visualization, we present the t-SNE visualization
results of six methods applied to the MSRCV1 dataset in Fig.2. SLRL-$\bm{H}$
represents the visualization of the common latent representation $\bm{H}$,
while SLRL-$\tilde{\bm{H}}$ illustrates the visualization of the structured
latent representation $\tilde{\bm{H}}$. Observations from Fig.2 indicate that
the latent representations learned by our proposed method demonstrate superior
clustering performance compared to those of other methods. Additionally,
compared to SLRL-$\bm{H}$, SLRL-$\tilde{\bm{H}}$ shows samples from the same
class more tightly grouped and those from different classes more distinctly
separated, suggesting a more effective clustering structure. This underscores
the efficacy of incorporating graph representation learning and clustering
constraints into our model, enabling it to learn features that are more
conducive to clustering.

(a) $\text{LRR}_{\text{BestSV}}$

(b) $\text{SPC}_{\text{BestSV}}$

(c) MLAN

(d) LMSC

(e) SLRL-$\bm{H}$

(f) SLRL-$\tilde{\bm{H}}$

Figure 2: t-SNE visualization of various methods on MSRCV1.

###  4.4 Parameter Sensitivity Experiments

In our methodology, the number of nearest neighbors $k$ is crucial for
constructing the $k$-nearest neighbor graph and significantly influences most
graph-based algorithms. Meanwhile, $\gamma$ in the objective function balances
the learning of the common representation and clustering constraints. We
rigorously evaluated the impacts of $\gamma$ and $k$ on our method’s
performance, varying $\gamma$ from $10^{-5}$ to $10^{4}$ and adjusting $k$
between 3 and 15. Figure 3 visualizes these sensitivities across three
datasets, showing that the model is robust to $k$ variations within a certain
range and performs better when $\gamma>1$. Overall, our method’s performance
shows low sensitivity to parameter selection, ensuring reliability and
consistency in various scenarios.

(a) 100Leaves

(b) MSRCV1

(c) HW

Figure 3: Sensitivity experiments for parameters $k$ and $\gamma$.

###  4.5 Convergence Experiments

The convergence curves, depicted in Figure 4, illustrate the changes in the
values of the total loss function. Initially, there is a rapid decrease in the
loss function values within the first few iterations, followed by a slower
decline as the number of epochs increases, until convergence is achieved.
Typically, for these datasets, our model converges in fewer than 100 epochs,
which demonstrates the desirable convergence properties of our proposed
method.

(a) 100Leaves

(b) Scene-15

(c) MSRCV1

(d) BBCSport

(e) 3Sources

(f) HW

Figure 4: Convergent behavior of SLRL.

###  4.6 Ablation Study

To further validate our proposal, we conducted contrastive learning
experiments across various network structures, as documented in Table 2. The
following observations can be made:

(a) The common latent representation $\bm{H}$, learned directly from the input
features $\bm{X}$ and focusing solely on consistency across different views,
did not yield optimal performance.

(b) By incorporating the relationships between different nodes based on
$\bm{H}$, we derived a structured latent representation ${\tilde{\bm{H}}}$,
which significantly improved performance.

(c) By optimizes both the learning of latent representation and clustering
simultaneously, we can obtain a structured latent representation that is more
suitable for clustering tasks.

Table 2: Results across various network structures. |  | 100Leaves | Scene-15 | MSRCV1 | BBCSport | 3Sources | HW
---|---|---|---|---|---|---|---
(a) | $\mathbf{X}-\mathbf{H}_{\checkmark}$ | 80.30 | 35.91 | 71.43 | 89.16 | 72.50 | 76.19
(b) | $\mathbf{X}-\mathbf{H}_{\checkmark}-\mathbf{\tilde{\bm{H}}}_{\checkmark}$ | 87.44 | 45.23 | 85.71 | 86.49 | 77.52 | 95.23
(c) | $\mathbf{X}-\mathbf{H}_{\checkmark}-\mathbf{\tilde{\bm{H}}}_{\checkmark}-\mathbf{P}_{\checkmark}$ | 94.15 | 47.52 | 92.70 | 93.02 | 81.62 | 96.40

##  5 Conclusion

This paper introduces a multi-view clustering method based on SLRL, which
considers both the complementary relationships between views and the
structural relationships among samples, thereby learning a structured latent
representation more suitable for clustering tasks. Specifically, to mine the
structural information among samples, SLRL constructs a nearest neighbor graph
on the common latent representation of all views and employs a graph attention
network for graph representation learning, enhancing the structure of the
latent representation. Furthermore, a distribution-based clustering loss is
employed to further constrain both the common representation learning and
graph representation learning, resulting in a latent representation with an
improved cluster structure, making it more suitable for clustering tasks.
Experimental results on six multi-view datasets demonstrate that the
clustering method based on SLRL proposed in this paper achieves commendable
clustering performance.

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# Josephson oscillations of edge solitons in a photonic-topological coupler

N. V. Bazhan1, B. Malomed2,3 and A.I. Yakimenko1 1 Department of Physics,
Taras Shevchenko National University of Kyiv, 64/13, Volodymyrska Street, Kyiv
01601, Ukraine  2 Department of Physical Electronics, Faculty of Engineering,
and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978,
Israel  3 Instituto de Alta Investigación, Universidad de Tarapacá, Casilla
7D, Arica, Chile

###### Abstract

We introduce a scheme of a photonic coupler built of two parallel topological-
insulator slab waveguides with the intrinsic Kerr nonlinearity, separated by a
gap. Josephson oscillations (JO) of a single edge soliton created in one slab,
and of a pair of solitons created in two slabs, are considered. The single
soliton jumping between the slabs is subject to quick radiative decay. On the
other hand, the JO of the copropagating soliton pair may be essentially more
robust, as one soliton can absorb dispersive waves emitted by the other. The
most robust JO regime is featured by the pair of solitons with phase shift
$\pi$ between them.

##  I Introduction

Topological insulators are a class of materials that are insulating in a bulk
but conducting at their surfaces due to the existence of scattering-resistant
topological edge states. In recent years, it was discovered that the concepts
of topological phases are not restricted to fermionic states in solid-state
settings, but can also be realized in various optical and plasmonic media,
such as photonic crystals and metamaterials [1, 2, 3, 4, 5, 6]. The rapidly
growing interest in the study of topological effects in photonics is motivated
by the vision of guiding and routing light by optical circuits in a manner
that is robust against disturbances introduced by disorder, due to the
robustness of topological edge states.

Solitons, i.e., self-trapped nonlinear waves that propagate keeping their
shape, have been recently observed in the bulk of a photonic Floquet
topological insulator [7]. They exhibit dynamical behavior fundamentally
different from that demonstrated by solitons in ordinary bandgap settings [8],
viz., cyclotron-like orbits that are associated with the topology of the
lattice.

The concept of topological lasers, based on the photonic Floquet topological
insulator, was put forward in theoretical work [9], The system proposed in
that work was designed as a truncated array of lasing helical waveguides, with
the pseudo-magnetic field induced by the twist of the waveguides, and
propagation direction that opens up a topological gap by breaking the time-
reversal symmetry.

Further, the recent study [10] addressed topological edge states existing on
the domain wall between two helical honeycomb lattices with opposite
helicities. The intrinsic nonlinearity of this system helps to create more
robust edge states in the form of fundamental and multipole solitons,
including moving ones.

In this work, we address interaction between topological surface solitons
copropagating along opposite edges of two slab-shaped topological-insulator
waveguides separated by a free-space gap. The overall setup, displayed in Fig.
1, is identified as a coupler, i.e., a set of two parallel waveguides with an
empty space between them, which are coupled by tunneling of the field across
the empty gap. Oscillatory dynamics of composite wave modes in the coupler is
usually categorized as Josephson oscillations (JO) [11], as long Josephson
junctions [12] were actually the first well-studied example of nonlinear
couplers for wave modes (plasma waves or fluxons in bulk superconductors
separated by a narrow dielectric barrier). Various realizations of couplers
are also known in optics [13] and Bose-Einstein condensates [14, 15, 16, 17].
Thus, the present work aims to elaborate the scheme of the photonic
topological-insulator coupler and analyze the dynamics of composite solitons
in it.

##  II Model

The photonic structure of each waveguiding slab building the coupler is
similar to that proposed in Refs. [18, 19] for the creation of surface
solitons in the semi-infinite bulk waveguide. In the paraxial approximation,
the propagation of monochromatic optical-beam envelope $\psi(x,y,z)$ along
coordinate $z$ obeys the nonlinear Schrödinger equation (NLSE),

$i\partial_{z}\psi=-\frac{1}{2k_{0}}\nabla_{\perp}^{2}\psi-\frac{k_{0}}{n_{0}}\Big{(}n_{L}(x,y,z)+n_{2}|\psi|^{2}\Big{)}\psi,$
(1)

with the diffraction operator,
$\nabla_{\perp}^{2}\equiv\partial_{x}^{2}+\partial_{y}^{2}$, acting on
transverse coordinates $\left(x,y\right)$. Here $k_{0}=2\pi n_{0}/\lambda$ is
the wavenumber corresponding to the carrier wavelength ($\lambda$), $n_{0}$ is
the background refractive index, while $n_{L}$ and $n_{2}|\psi|^{2}$ are the
linear and nonlinear local corrections to $n_{0}$. Namely, $n_{L}(x,y,z)$
represents the helix-lattice background [20] with mean spacing $a$, helix
radius $R_{0}$, and modulation period $Z$. We adopt parameter values
consistent with fused silica glass at the standard wavelength,
$\lambda=1550\,$nm: $n_{0}=1.45$, $n_{L}=2.7\times 10^{-3}$ in the waveguides
(and $n_{L}=0$ outside), and $n_{2}=3\times 10^{-7}$ cm2/GW [21]. Individual
waveguiding helices have the circular cross section with radius
$4\,\mathrm{\mu}$m. The paraxial beam intensity $|\psi|^{2}$ is normalized by
a characteristic value $I_{0}=10^{3}$ GW/cm2, for which the nonlinear index
shift $n_{2}I_{0}$ is comparable to $n_{L}$. For the modal cross-section area
of the helix $w_{0}^{2}=(10$ $\mathrm{\mu}$m)2, this size of $I_{0}$ requires
peak powers $\sim 1$ MW, readily accessible with pulsed lasers [21].

Figure 1: A sketch of two photonic lattices composed of helical waveguides.
The lattices are separated by the gap of width $\Delta$. One lattice consists
of (green) waveguides twisted clockwise, while the other (orange) lattice is
the mirror reflection of the green one relative to the midplane of the gap.
Therefore, the orange lattice consists of waveguides twisted counterclockwise.
This setting allows two solitons launched on opposite edges of the gap to co-
propagate in the same direction. The waveguides in each lattice are shifted
relative to each other along the $z$-axis by half of the modulation period,
$Z=1$ cm. Shown are the lattices composed of $n_{y}^{s}=4$ shells ($n_{y}=2$).

In present work we consider a coupler composed by two mirror-symmetric
staggered helical lattices separated by the gap of width $\Delta$, as shown in
Fig. 1. Due to opposite signs of the twist of the waveguides in the lattices,
two edge solitons copropagate in the same direction along the edges of the
lattices, and interact with each other across the gap. For each lattice we use
the parameters of nonlinear photonic Floquet topological insulators introduced
in Refs. [18, 19]. The lattices consist of 2D square arrays of helical
waveguides, staggered so that adjacent waveguides have helix phase shifts of
$\pi$ relative to each other, as shown in the right panel of Fig. 1. Note
that, while $z$ varies within one period $Z$, each waveguide is approaching
its four neighbors sequentially. The respective modulation of the linear
refractive index in Eq. (1), which describes the coupler shown in Fig. 1, is
adopted as

$\displaystyle n_{L}(x,y,z)=\Delta
n_{1}\sum_{n=-[n_{x}/2]}^{[n_{x}/2]}\sum_{m=1}^{n_{y}}\Big{\\{}V_{0}(x-X_{n}^{-},y\mp
Y_{m-1}^{\pm})+$ $\displaystyle+V_{0}(x-X_{n-1/2}^{+},y\mp
Y_{m-1/2}^{\pm})\Big{\\}},$ (2)

where $n_{x}\,$ and $n_{y}$ are numbers of shells, equal in both lattices
(accordingly, total numbers of waveguiding rows and columns in each lattice
slab are $n_{y,x}^{s}=2n_{y,x}$). We build the coupler starting from its
center. Thus, we summation in Eq. (2) includes all waveguides along the $x$
axis in the range of $\left\\{-[n_{x}/2],[+[n_{x}/2]\right\\}$, where square
brackets stand for the integer part of $n_{x}/2$. Both slabs include $n_{y}$
shells. Factors $V_{0}(y-Y_{m}^{+})$ and $V_{0}(y+Y_{m}^{-})$ pertain to the
top and bottom mutually symmetric slabs. Further, functions introduced in Eq.
(2) are

$lX_{n}^{\pm}(z)=nd\pm x_{0}(z),\,\,\,\,Y_{m}^{\pm}(z)=\frac{\Delta}{2}+md\pm
y_{0}(z).$ (3)

Here, $\Delta n_{1}=2.7\times 10^{-3}$ is the modulation depth, $d=\sqrt{2}a$
and $\Delta$ are the lattice constant and the gap width shown in Fig. 1.

We assume a hypergaussian waveguide’s profiles Eq. (2), viz.,

$V_{0}(x,y)=\exp\left(-[(x^{2}+y^{2})/\sigma^{2}]^{3}\right),$ (4)

with $\sigma=4\,\mathrm{\mu}$m. The waveguides’ axes are shaped as the
helices,

$x_{0}(z)=R_{0}\cos(\Omega z),\quad y_{0}(z)=R_{0}\sin(\Omega z),$ (5)

with radius $R_{0}=4\,\mathrm{\mu}$m and pitch $Z=2\pi/\Omega=1$ cm, while
while value of the gap is defined by parameter $\Delta$, as shown below. These
parameters are chosen to provide a balance between the minimization of bending
losses, which inevitably accompany the strong nearest-neighbor couplings in
the setting under the consideration, and the necessity to fit sufficiently
many helix cycles to an experimentally feasible total array length, $\lesssim
10$ cm.

Our objective is to explore the interaction between two solitons copropagating
along edges of the slab-shaped lattice waveguides forming the coupler in Fig.
1. As shown by Fig. 2, the solitons move in the same direction due to the
mutual mirror symmetry of the parallel slabs. The interaction between the
solitons is mediated by the electromagnetic field traversing the gap between
the slabs.

Figure 2: Configurations of the copropagating solitons in the dual-waveguide
coupler in the $\left(x,y\right)$ cross-section at different values of
propagation distance $z$. The solitons are launched at $z=0$, and subsequent
configurations, produced by the numerical solution of Eq. (1), are displayed
at $z=1,2,3,4\,$cm. The distance between waveguides in each lattice (spacing)
is $d\approx 31$ $\mathrm{\mu}$m. The width of the gap between the lattices,
$\Delta$, is the same as in Eq. (3). Circles indicate cross-sections of
individual waveguides and arrows designate the direction of their twist.

First, we produced individual edge-soliton solutions of NLSE (1) with the
effective potential defined as per Eqs. (2)-(5), starting from input

$\Psi\left(x,y;z=0\right)=\sqrt{I}\exp(-(\mathbf{r}-\mathbf{r}_{0})^{2}/2a^{2}),$
(6)

where $\mathbf{r}_{0}$ is the initial position of the soliton, and $I$ is its
peak power. It is known that the edge soliton, while it can pass a
considerable distance, is not a completely stable object, being subject to
losses due to emission of small-amplitude dispersive waves [18], see Fig. 3.
The loss rate depends on number $n_{y}$ of layers in the lattice slab in the
direction across the lattice, see Fig. 1. To find the optimal number, we
computed the power of a single soliton propagating along the edge of one
lattice slab. The results, displayed in the top plot of Fig 3, demonstrate
that the edge soliton traveling in a thin slab, with $n_{y}\leq 3$, decays
faster than in the thicker one, with $n_{y}\geq 4$. Therefore, we ran the
systematic numerical analysis using the slabs with $n_{y}=4$. Then, Fig. 3
shows that the peak power of the soliton decreases by $\simeq 10$ times,
having passed $10$ cm in the $z$ direction. Because the soliton moves along
the $z$ axis in the lattice with pitch $Z=1$ cm, it shifts by $10$ helix
waveguides in the $x$ direction while passing $10$ cm along $z$. This means
that one needs to have the number of layers $n_{x}\geq 10$ in the slab along
$x$. In the simulations, we used the lattices with $n_{x}=18$.

Figure 3: The top plot: the peak power of the single edge soliton traveling
from right to left in lattice slabs built of $n_{y}$ layers. The bottom plot
shows color-coded consecutive intensity distributions in the lattice with
$n_{y}=3$. White circles designate cross-sections of individual helix
waveguides.

##  III  Edge solitons interacting across the gap

To address the interaction of two edge solitons carried by the parallel
lattice slabs in the coupler, as shown in Fig 2, we take the solitons moving
with equal velocities, which maximizes the impact of the interaction.
Generally, the interaction effects are transient ones because, as mentioned
above, each soliton by itself loses $90\%$ of its energy after passing the
distance $z=10$ cm. It is well known that the soliton-soliton interaction
depends on their peak powers, distance, and relative phase between them [22].
We monitored the evolution of relative phase $\Delta\phi(z)$ and power
difference $\Delta P(z)$ between two edge solitons maintained by the top and
bottom slab waveguides as

$\Delta
P(z)=\int\int_{\Omega_{w,\mathrm{top}}}\left|\psi_{\mathrm{top}}(x,y,z)\right|^{2}dxdy-\\\
-\int\int_{\Omega_{w,\mathrm{bott}}}\left|\psi_{\mathrm{bott}}(x,y,z)\right|^{2}dxdy,$
(7) $\Delta\phi=\phi_{\mathrm{top}}-\phi_{\mathrm{bott}},$ (8)

where $\Omega_{w}$ is the biggest area around the given helical waveguide
which does not touch adjacent ones, see the right panel of Fig. 2), while
$\phi_{\mathrm{top}}$ and $\phi_{\mathrm{bott}}$ are the phases measured at
centers of the waveguides.

Figure 4: The period of soliton’s JO (Josephson oscillations), $T_{1/2}$, vs.
width $\Delta$ of the gap separating the lattice slabs formimg the coupler.
Values of $\Delta$ are shown in units of the lattice spacing, $d$. (a) The
results for the JO of a single soliton, which was initially created in one
slab. (b) The results for the pair of solitons initially created in two slabs,
with phase difference $\Delta\phi(0)$ between them.

The power distribution in the interacting edge solitons depends on the gap’s
width $\Delta$, the initial value of $\Delta\phi$, and the initial power
(taken equal for both solitons). As the result of the interaction, a part of
the solitons’ energy oscillates between the slabs, realizing the JO. We define
the oscillation period, $T_{1/2}\equiv z_{2}-z_{1}$, as the distance between
the first two consecutive points, $z_{1}$ and $z_{2}$, at which the numerical
solution exhibits equal energies in both waveguiding slabs. The oscillations
are not strictly periodic because the period is affected by the continuing
energy loss. Only values $T_{1/2}>0.1$ cm were registered, as smaller ones may
be a result of fluctuations caused by the interaction of the solitons with the
dispersive waves.

First, we consider JO initiated by the single soliton, launched at $z=0$ in
one waveguiding slab. Depending on the gap’s width, the initial soliton
performs the oscillations periodically jumping across the gap, whose width is
smaller than a critical value $\Delta_{\mathrm{cr}}$, which takes values
$1.1d\leq\Delta_{\mathrm{cr}}\leq 1.2d$ (recall $d$ is the lattice spacing).
At $\Delta>\Delta_{\mathrm{cr}}$, the soliton stays in its initial slab. Thus,
JO performed by the single soliton are characterized by the dependence of the
JO period, $T_{1/2}$, on $\Delta$, which is presented in Fig. 4(a), in the
interval of $0.6d\leq\Delta\leq 1.1d$ (for $\Delta\leq 0.5d$ the two lattice
slabs start to touch each other, closing the gap between them, while at
$\Delta>1.1d$ the proximity to $\Delta_{\mathrm{cr}}$ makes $T_{1/2}$
extremely large).

Next, Fig. 4(b) summarizes the results produced by the simulations of JO of
two edge solitons which were initially created in the two slabs, with phase
difference $\Delta\phi(0)$. It is seen that the JO frequency, $2\pi/T_{1/2}$,
attains a maximum at $\Delta\phi(0)=\pi$ and a minimum at $\Delta\phi(0)=0$.
This observation is explained by the fact that the solitons repel and attract
each other in the former and latter cases, respectively [22], hence the
respective interaction force accelerates or decelerates the mutual motion of
the solitons. Indeed, a full picture of the simulated evolution shows that the
solitons periodically pass through each other, thus shuttling between the
opposite slabs, or bounce back from each other, thus oscillating between one
slab and the gap’s midplane, in the cases of $\Delta\phi(0)=0$ and $\pi$,
respectively. Case $\Delta\phi(0)=\pi/2$ corresponds to inelastic bouncing
with strong energy exchange between the edge solitons. Further, it is observed
that the period of two-soliton JO naturally increases approximately linearly
with the increase of the gap’s width $\Delta$.

Figure 5: (a) The peak power (normalized to $I_{0}$, see Eq. (6)) of the
single soliton and soliton pair, with different phase shifts, after having
passed distance $z=10$ cm, vs. the width of the gap between the parallel
waveguiding slabs. (b) The peak power of the single soliton, and largest
powers of the soliton pair in the top and bottom slabs, vs. the propagation
distance, for the coupler with gap $\Delta=0.8d$ and phase shift of the pair
$\Delta\phi(0)=\pi$ (which is the the optimal set of parameters for the pair,
as per panel (a)).

A significant result produced by systematic simulations of the JO is that the
two-soliton configuration is essentially _less vulnerable_ to the decay than
the single edge soliton. This trend is explained by the fact that dispersive
waves emitted by one soliton may be absorbed by the other. Accordingly, in the
two-soliton dynamical regimes the energy stays concentrated in the solitons,
making them brighter. In Fig. 5(a), the peak power of the single- and two-
soliton configurations is displayed as a function of the gap’s width,
$\Delta$, in the state produced by the simulations at $z=10$ cm. It is seen
that the strong interaction across the narrow gap, with $\Delta\leq 0.75d$,
leads to quick decay of the solitons, faster than the single soliton loses its
energy. However, at $\Delta=0.80d$, the interaction gives a significant boost
to the soliton’s brightness for all initial phase shifts between the solitons,
the largest boost being observed for $\Delta\phi(0)=\pi$ (the latter finding
is natural, as the repulsion between the solitons helps to hinder their
decay).

Further, in Fig. 5(b) we compare the results produced by the propagation of
the single edge soliton and the pair of two solitons with $\Delta\phi(0)=\pi$,
in the coupler with $\Delta=0.80d$ (the best configuration for the pair
produced by Fig. 5(b)). The results, displayed in the form of the dependence
of the peak power on the propagation distance, $z$, clearly demonstrate that
the interaction indeed leads to the increase of the peak power which stays in
the solitons. In particular, at $z=18$ cm it is three times higher in
comparison with the single soliton. Thus, the interaction between the solitons
makes their JO essentially more robust.

##  IV Conclusions

In conclusion, we have analyzed the dynamics of the single edge soliton and a
pair of solitons copropagating in the coupler built of parallel photonic
topological insulators separated by a gap. The single soliton performs JO
(Josephson oscillations) between the parallel waveguides, quickly losing its
energy through the emission of radiation. The evolution of the soliton pair
demonstrates essentially more robust dynamics, as dispersive waves emitted by
one soliton may be absorbed by the other, in the course of their collective
JO. The two-soliton dynamics depends on the phase shift $\Delta\phi(0)$
between the solitons, the most robust regime corresponding to
$\Delta\phi(0)=\pi$, in which case the two solitons periodically bounce back
from each other. As an extension of the analysis, it may be interesting, in
particular, to consider a ring cavity, in the form of the photonic
topological-insulator coupler closed into a ring. The two-soliton JO of the
solitons running along the ring may be additionally stabilized by external
gain.

The authors thank O.G. Chelpanova and V.R. Tuz for fruiful discussions. The
work of B.A.M. as supported, in part, by the Israel Science Foundation,
through grant No. 1286/17. NB and AY acknowledge support from National
Research Foundation of Ukraine, through grant No. 2020.02/0032.

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