# Multiview Clustering of Adaptive Sparse Representation Based on Coupled P Systems

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- A new coupled P system is proposed, which integrates the construction of a single view matrix and the formation of a unified graph into the P system to perform clustering tasks.
- (2)
- To construct the similarity matrix of each view, this paper introduces a natural neighbor search algorithm without parameters, which can automatically determine the number of neighbors in each view. After that, sparse representation and various learning methods are imported to construct the similarity matrix to preserve the internal geometry of the views.
- (3)
- In forming a unified graph, this paper adopts a soft thresholding operator to learn a consistent sparse structure affinity matrix from the similarity matrix of each view and then obtain the clustering result. Iterative optimization is not required, and better clustering results can be captured and obtained quickly.
- (4)
- Nine multiview data sets are employed to simulate and verify the clustering performances of MVCS-CP.

## 2. Related Work

#### 2.1. Notations

#### 2.2. Graph-Based Clustering and Graph Learning

**Theorem**

**1.**

**L**is equal to the number of connected components of the similarity matrix

**S**.

#### 2.3. Natural Neighbours

**Definition**

**1.**

**(The Natural Characteristic Value Ncv)**Ncv is equivalent to the number of natural neighbors (That is, k) of the data point$\mathit{x}$.

**Definition**

**2.**

**(**

**The Natural Neighbors**

**)**The natural neighbors of the object$\mathit{x}$in the data set are the

**k**nearest neighbors, expressed as NaN ($\mathit{x}$).

#### 2.4. P System

## 3. Multi-View Clustering of Adaptive Sparse Representation Based on Coupled P Systems

#### 3.1. The General Framework of the Proposed Coupled P System

**Definition**

**3.**

- $\Gamma =\left\{{\mathit{X}}^{1},{\mathit{X}}^{2},\dots ,{\mathit{X}}^{m},{\mathit{S}}^{1},{\mathit{S}}^{2},\dots ,{\mathit{S}}^{m},NaN\left(\mathit{x}\right),Ncv,Nb\left(\mathit{x}\right),\mathit{W},\mathit{D},\mathit{L},para,c,\right\}$. ${\mathit{X}}^{i}$, ${\mathit{S}}^{i}$represent the original data of$m$views and the similarity matrix corresponding to each view, respectively.$NaN\left(\mathit{x}\right)$is the natural neighbor of the data point$\mathit{x}$in the view.$Ncv$refers to the characteristic natural value, and the number of reverse neighbors of$\mathit{x}$is denoted as$Nb\left(\mathit{x}\right)$. $\mathit{W}$represents the learned uniform unified graph matrix.$\mathit{D}$and$\mathit{L}$indicate the degree matrix and Laplacian matrix, respectively. The parameters$para$and$c$respectively refer to the parameters required for the experiment and the number of clusters.
- $\epsilon =\left\{{\mathit{X}}^{1},{\mathit{X}}^{2},\dots ,{\mathit{X}}^{m},para,c\right\}\subseteq \Gamma $is the initial objects in the coupled systems.
- $syn=\left\{\left\{0,1\right\},\left\{0,3\right\},\left\{1,2\right\},\left\{2,3\right\},\left\{3,4\right\}\right\}$signifies the synapse between cells, whose main function is to connect cells and make them communicate with each other.
- ${\sigma}_{0},\cdots ,{\sigma}_{t}$denotes the cells (membrane) in the system.$t$depends on the number of views and the number of clusters in the data set, that is, the total number of cells in the system.
- $R$represents a collection of communication rules and evolution rules in the system. The role of evolution rules is to modify objects and communication rules are used to transfer objects between cells (membranes).
- $in$is cell 0, which is the input membrane.$out$is cell 5, output membrane, used to store the final clustering results.

#### 3.2. The Evolution Rules

- ${R}_{01}=\left\{{\mathit{X}}^{1},{\mathit{X}}^{2},\dots ,{\mathit{X}}^{m},para\to {\mathit{X}}^{1},{\mathit{X}}^{2},\dots {,}_{go{[]}_{1}}\right\}$
- ${R}_{02}=\left\{c,para\to c,para{,}_{go{[]}_{2}}\right\}$

#### 3.2.1. The Evolution Rules of Determining Ncv and Constructing Similarity Matrix in Cell 1

_{1}:

- R
_{11}(Iterative search rules): At the $k$th iteration, for each data point ${\mathit{x}}_{i}$ in the single view ${X}^{v}$, we search for its $r$th neighbor ${x}_{j}$ using a KD tree. After that, $Nb\left({x}_{j}\right)=Nb\left({\mathit{x}}_{i}\right)+1,\mathit{N}\mathit{a}{\mathit{N}}_{\mathit{k}}\left({x}_{i}\right)=\mathit{N}\mathit{a}{\mathit{N}}_{\mathit{k}-\mathbf{1}}\left({x}_{i}\right){\displaystyle \cup}{x}_{j}$ correspond to the concepts in Section 2.3. $NaN\left(\mathit{x}\right)$ will be transported to the related subcell to construct the similarity matrix ${\mathit{S}}^{\nu}$. - R
_{12}(Iterative stop rules): If the number of reverse neighbors $Nb\left(\mathit{x}\right)$ of data point $\mathit{x}$ does not change or $Nb\left(\mathit{x}\right)==0$, the evolution rules stop. - R
_{13}(Determine the $Ncv$ rule): The natural characteristic value $Ncv$ is calculated by Equation (2), which is equivalent to the number of neighbors $k$, and then $k$ is transmitted to the relevant subunits to prepare for the construction of ${\mathit{S}}^{\nu}$.

- R
_{14}(Lagrange function rule): The Lagrange function of Equation (17) is $\mathcal{L}\left({\mathit{s}}_{i}^{v},\u03f5,\zeta \right)=\left|\right|{\mathit{s}}_{i}^{v}+{\frac{{d}_{i}}{2\beta}\left|\right|}_{2}^{2}-\u03f5\left({\mathbf{1}}^{T}{\mathit{s}}_{i}^{v}-1\right)-{\zeta}^{T}{\mathit{s}}_{i}^{v}$ - R
_{15}(Constraint rule): Based on the Karush–Kuhn–Tucker constraint, the optimal solution ${\widehat{s}}_{ij}^{v}={(-\frac{{d}_{ij}}{2\alpha}+\u03f5)}_{+}$ can be acquired, where ${\left(a\right)}_{+}=max\left(a,0\right)$. As a result of the constraints ${\mathbf{1}}^{T}{\mathit{s}}_{i}^{v}=1$, it has $\u03f5=\frac{1}{k}+\frac{1}{2k\beta}{\displaystyle \sum}_{j=1}^{k}{\mathit{h}}_{ij}$. - R
_{16}(Determining $\beta $ rule): Since there are only $k$ non-zero values in ${\mathit{s}}_{i}^{v}$, $\beta $ has a maximum value, which is conveyed as $\beta =\frac{k}{2}{d}_{i,k+1}-\frac{k}{2}\Sigma j={1}^{k}{d}_{ij}$. - R
_{17}(Getting the ${\mathit{s}}_{i}^{v}$ rule): The $j$-th element of ${\mathit{s}}_{i}^{v}$ is as follows:$${\mathit{s}}_{ij}^{v}=\{\begin{array}{cc}\frac{{d}_{i,k-1}-{d}_{ij}}{k{d}_{i,k+1}-{{\displaystyle \sum}}_{h=1}^{k}{b}_{ih}}& j\le k\\ 0& j>k\end{array}.$$

#### 3.2.2. The Evolution Rules of Constructing the Unified Graph Matrix, Degree Matrix, Laplacian Matrix in Cell 2

- (1).
- The unified graph matrix $\mathit{W}$ and the similarity matrix ${\mathit{S}}^{v}$ of each view tend to be as consistent as possible.
- (2).
- The unified graph matrix $\mathit{W}$ is sparse, which can further alleviate the noises generated by different views.

_{2}:

- R
_{21}(Removing the $cons$ rule): Removing the cons, then the problem (11) is redefined as $\underset{\mathit{W}}{\mathrm{min}}||\mathit{T}-{\mathit{W}\left|\right|}_{F}^{2}+\frac{para}{m}{||\mathit{W}||}_{1}$, where $T={\Sigma}_{v=1}^{m}{\mathit{S}}^{v}/m$. - R
_{22}(Soft-thresholding operator rule): Based on the above, when $\mu >0,$ the soft-thresholding operator is introduced here:$${\mathcal{S}}_{\mu}\left(x\right)=\{\begin{array}{cc}x-\mu ,& x>\mu \\ x+\mu ,& x<-\mu \\ 0,& otherwise\end{array}.$$ - R
_{23}(Obtaining $\mathit{W}$ rule): By conducting the ${\mathcal{S}}_{\mu}$ element-wise, it can be extended to the matrix. In addition, as shown in [52], the approximate solution to problem (13) is ${\mathit{W}}^{*}={\mathcal{S}}_{\frac{para}{2m}}\left(\mathit{T}\right)$. - R
_{24}(Constructing the Degree Matrix rule): According to ${D}_{ii}={{\displaystyle \sum}}_{j=1}^{n}{W}_{ij}$, the degree matrix $\mathit{D}$ is gained. - R
_{25}(Constructing the Laplacian Matrix rule): In terms of the Laplacian Matrix, it is based upon $\mathit{W}$ and $\mathit{D}$, $\mathit{L}=\mathit{I}-{\mathit{D}}^{-1/2}\mathit{W}{\mathit{D}}^{-1/2}$.

#### 3.2.3. The Evolution Rules of K-Means in Cell 3

_{3}.

- R
_{31}(Building new cluster instances rule): The formation of clustering new instances is conducive to K-means clustering. We select the eigenvectors $U=\left\{{u}_{1},{u}_{2},\cdots {u}_{c}\right\},U\in {R}^{n\ast c}$ corresponding to the first $c$ eigenvalues of $L$, and standardize it to obtain ${Y}_{ij}={U}_{ij}/{({{\displaystyle \sum}}_{j}{U}_{ij}^{2})}^{1/2}$. - R
_{32}(Randomly selecting clustering centers rule): Among the $n$ points of $Y$, it randomly selects $c$ points as the initial clustering centers and stores them in the subcells. - R
_{33}(Clustering rule): After that, the distance from each instance to each cluster center is computed in the subcells simultaneously and transported to cell 3. Finally, the instances are allocated based on the principle of minimum distance to form $c$ different clusters in cell 3. - R
_{34}(Outputting result rules): For clusters divided in accordance with rule R_{35}, it takes the current average distance of each cluster as the new cluster center. Comparing the current cluster center with the previous cluster center, if there is a change, it repeats rule R_{35}. Conversely, the result of clustering is outputted to cell 4.

#### 3.3. The Communication Rules between Different Cells

- (1)
- Unidirectional transport between cells. $u$ is a string containing the object. $\lambda $ is the empty string.
- $Rule1:\left(0,u/\lambda ,1\right)$: It feeds $u$ containing the original data $\mathit{X}$ of $m$ views into cell 2 for the determination of the similarity matrix for each view.
- $Rule2:\left(0,u/\lambda ,2\right)$: The $u$ including the parameter $para$ and the number of clusters $c$ are transferred to cell 3 to format the unified graph matrix and construct the degree matrix and the Laplacian matrix.
- $Rule4:\left(1,u/\lambda ,2\right)$: The string $u$ of similarity matrix $\mathit{S}$ for each view produced by cell 1 is transported to cell 2 for the construction of the unified graph matrix.
- $Rule5:\left(2,u/\lambda ,3\right)$: It conveys the string $u$ containing the Laplacian matrix and the number of clusters $c$ to cell 3 for K-means clustering.
- $Rule6:\left(2,u/\lambda ,3\right)$: The string $u$ of clustering results generated by K-means is transmitted to cell 4 for storage.

- (2)
- Unidirectional transport between cells and the environment.
- $Rule3:\left(0,u/\lambda ,2\right)$: It transports the string $u$ of the resulting reverse neighbor $Nb\left(\mathit{x}\right)$ into the environment to release.

## 4. Experiments

#### 4.1. Datasets

- Caltech101 [59]: Coltech101-07 and Coltech101-20 are selected from the Caltech101 dataset, which includes 2386 and 1474 images, respectively. Each image contains six feature vectors of GABOR, WM (wavelet moment), CENT (Centrist features), HOG, GIST and LBP.
- NUS [60]: It contains 2400 images in 12 categories. The six features of colour histogram, CM, edge direction histogram, wavelet texture, block-wise colour moment and SIFT description are included for each image.
- ORL [61]: This dataset contains 400 images with four feature vectors of GIST, HOG, LBP, and CENT.
- 3sources: This dataset contains 169 news documents reported by three online news organizations, BBC, The Guardian and Reuters.
- BBC [62]: It is a collection of 685 documents from the BBC News website, each divided into four feature vectors.
- BBC_Sport [62]: This dataset consists of 544 documents collected from the BBC Sports website; each document has two feature vectors.
- 100leaves [63]: It consists of 1600 samples from the UCI repository, each of which is one of a hundred species.
- Scene15 [64]: It consists of 4485 images of indoor and outdoor scenes with a total of three views.

#### 4.2. Evaluation Metrics

- (1).
- Accuracy: ACC refers to the ratio of the number of correctly clustered samples to the total number of instances $N$.

- (2).
- Adjusted Rand Index: The value range of the ARI is [−1, 1].$$\begin{array}{c}RI=\frac{TP+TN}{TP+FP+TN+FN}\\ ARI=\frac{RI-E\left[RI\right]}{\mathrm{max}(RI)-E\left[RI\right]}\end{array}$$
- (3).
- Normalized Mutual Information: $NMI$ measures the difference between cluster partitions through information theory. The value range is [0, 1]

- (4).
- $Precision$: It represents the probability of the true positive sample among all predicted positive samples.$$Precision=\frac{TP}{TP+FP}$$
- (5).
- F1-score: $F$ is the harmonic mean of precision and recall to comprehensively measure the clustering effect.$$\begin{array}{c}R=\frac{TP}{TP+FN}\\ F=\frac{2*Precision*R}{Precision+R}\end{array}$$
- (6).
- $Purity$: The general idea of cluster purity is to divide the number of correctly clustered instances by the total number of instances.

#### 4.3. Compared Methods

- SC [65] performs clustering on every single view and concatenates all views in the dataset into one view (Featconcat) for clustering.
- GBS [33] proposes a general graph-based multiview clustering system. The number of neighbors takes its default setting of 5.
- AMGL [66] is a parameter-free model for spectral embedding learning that automatically learns the weights for each view by solving a square root trace minimization problem.
- MVGL [67] uses it to explore the Laplacian rank-constrained graph after obtaining the similarity graph for each view, where the number of neighbors is set to a default value of 10.
- ASMV [32] adaptively jointly optimizes the data correlation between multiple features, and the number of neighbors is set to 15.
- CDMGC [36] is a graph clustering method of explicitly exploiting both multiview consistency and multiview diversity. The parameters in the experiment leverage the default values in the code provided by the author.
- CoMSC [68] is a multiview subspace clustering algorithm that groups objects and simultaneously removes data redundancy. In the experiment, the two parameters $\lambda $ and $c$ are respectively searched in $\{{2}^{-10},{2}^{-8},{2}^{-6},{2}^{-4},{2}^{-2},{2}^{0},$ ${2}^{2},{2}^{4},{2}^{6},{2}^{8},{2}^{10}\}$ and $\left\{k,2k,\dots ,20k\right\}$, where $k$ is the number of classes.

- The proposed MVCS-CP method performs better on six evaluation metrics on all datasets, basically being the best or second best. In the caltech101-20 dataset, it has the best performance on the four metrics of ARI, NMI, precision and purity with 8%, 2%, 4% and 6% improvement over the second-best results. As far as the caltech101-7 dataset is concerned, the three indicators of ACC, Precision and Purity are the best, and the remaining indicators ARI, NMI and F are the second best. In terms of the NUS dataset, except for the F indicator, the rest of the indicators perform the best. Compared with the better overall performance of CoMSC, the effect was increased by 5% (ACC), 2% (ARI), 2% (NMI), 2% (Precision) and 16% (Purity), respectively. Synthesizing the caltech101-20, caltech101-7 and NUS datasets, it can be concluded that the proposed MVCS-CP can achieve better results in processing more than five views. MVCS-CP performs optimally on all six metrics for the ORL dataset, with an average of 2% improvement for each metric over the second-best result. As for the 100leaves dataset, except for the NMI indicator, which is 0.5% lower than the second-best, all other indicators perform the best. The ORL and 100 leaves datasets have more clusters numbers (40 and 100 categories, respectively). Based on the above experimental results, it can be found that MVCS-CP can cope well with rich clusters number. On the 3sources dataset and the BBC dataset, the proposed algorithm demonstrates obvious improvement on all indicators. In addition, the BBC_Sport dataset has the best performance on the remaining five metrics except for Precision, which is the second-best. Combining the three datasets of 3sources, BBC and BBC_Sport, all of them have a higher dimension of the order of thousands. It can be seen that the proposed MVCS-CP can achieve satisfactory results when dealing with datasets with higher dimensions.
- Furthermore, the Scene dataset has the best performance on four metrics (ACC, ARI, Precision and Purity), especially on Purity, which is 7% better than the second-best result. And it is comparable to the best results on the NMI indicator. This illustrates that MVCS-CP can handle larger-scale datasets.
- Compared with the state-of-the-art multiview clustering algorithms, the MVCS-CP algorithm has better or comparable performance. This suggests that taking each view’s geometry and sparse representation into account yields better results.
- In terms of the single-view method, it is found that the multiview clustering algorithm is basically better than it, which shows that considering the multiple features of the dataset can be better clustered. However, on the BBC_Sport dataset, Featconcat performs the best in terms of Precision, which means that the multiview clustering method still needs further improvement.

#### 4.4. Running Time

#### 4.5. Comparison of the Number of Neighbors

#### 4.6. Parameter Analysis

#### 4.7. Result Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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Datasets | Objects | View | Clusters | d | d | d | d | d | d |
---|---|---|---|---|---|---|---|---|---|

Caltech101-20 | 2386 | 6 | 20 | 48 | 40 | 254 | 1984 | 512 | 928 |

Caltech101-7 | 1474 | 6 | 20 | 48 | 40 | 254 | 1984 | 512 | 928 |

NUS | 2400 | 6 | 12 | 64 | 44 | 73 | 128 | 155 | 500 |

ORL | 400 | 4 | 40 | 512 | 89 | 864 | 254 | - | - |

3sources | 169 | 3 | 6 | 3560 | 3631 | 3068 | - | - | - |

BBC | 685 | 4 | 5 | 4659 | 4633 | 4665 | 4684 | - | - |

BBC_Sport | 544 | 2 | 5 | 3183 | 3203 | - | - | - | - |

100leaves | 1600 | 3 | 100 | 64 | 64 | 64 | - | - | - |

Scene15 | 4485 | 3 | 15 | 20 | 59 | 40 | - | - | - |

Caltech101-20 | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 26.55 ± 1.46 | 11.73 ± 1.03 | 26.99 ± 0.4 | 36.13 ± 1.79 | 19.47 ± 1.01 | 52.46 ± 0.92 |

SC2 | 28.32 ± 1.38 | 16.27 ± 0.26 | 33.43 ± 0.32 | 46.73 ± 0.86 | 22.96 ± 0.32 | 59.62 ± 0.63 |

SC3 | 28.32 ± 1.38 | 16.27 ± 0.26 | 33.43 ± 0.32 | 46.73 ± 0.86 | 22.96 ± 0.32 | 59.62 ± 0.63 |

SC4 | 40.49 ± 1.13 | 30.05 ± 1.67 | 52.89 ± 0.99 | 71.02 ± 1.78 | 35.78 ± 1.67 | 75.43 ± 0.68 |

SC5 | 39.28 ± 1.72 | 27.47 ± 1.85 | 48.82 ± 0.98 | 67.2 ± 2.4 | 33.31 ± 1.81 | 73.19 ± 1.15 |

SC6 | 35.44 ± 2.75 | 24.18 ± 1.93 | 43.31 ± 1.37 | 60.39 ± 3.51 | 30.39 ± 1.8 | 68.64 ± 1.6 |

Featconcat | 49.97 ± 0.13 | 14.52 ± 0.35 | 20.2 ± 0.29 | 23.09 ± 0.19 | 36.51 ± 0.21 | 52.77 ± 0.12 |

AMGL | 52.73 ± 3.14 | 26.82 ± 2.82 | 52.19 ± 3.33 | 35.21 ± 3.1 | 40.67 ± 1.99 | 67.62 ± 1.88 |

MVGL | 60.69 ± 0 | 28.92 ± 0 | 50.73 ± 0 | 33.54 ± 0 | 44.15 ± 0 | 71.29 ± 0 |

ASMV | 41.17 ± 2.07 | 28.79 ± 2.06 | 54.23 ± 0.65 | 63.13 ± 2.75 | 35.25 ± 2 | 74.78 ± 0.7 |

GBS | 64 ± 0 | 34.08 ± 0 | 53.73 ± 0 | 37.07 ± 0 | 47.95 ± 0 | 73.34 ± 0 |

CoMSC | 53.98 ± 4.83 | 43.01 ± 6.31 | 59.47 ± 6.59 | 78.6 ± 4.91 | 78.21 ± 5.05 | 48.77 ± 2.32 |

CDMGC | 55.7 ± 9.49 | 22.72 ± 10.33 | 44.68 ± 8.29 | 29.28 ± 6.34 | 40.43 ± 6.76 | 65.08 ± 8.99 |

MVCS-CP | 60.6 ± 0.59 | 51.05 ± 2.2 | 61.36 ± 1.48 | 82.23 ± 1.56 | 56.56 ± 2.07 | 81.98 ± 1.01 |

Caltech101-7 | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 28.83 ± 2.1 | 7.94 ± 0.92 | 11.51 ± 0.5 | 48.77 ± 0.97 | 29.14 ± 1.45 | 65.88 ± 1.66 |

SC2 | 34.79 ± 2.24 | 19.67 ± 1.26 | 24.18 ± 0.59 | 66.23 ± 1.46 | 36.94 ± 1.13 | 73.09 ± 0.75 |

SC3 | 55.6 ± 0.22 | 2.81 ± 0.26 | 3.15 ± 0.37 | 39.44 ± 0.09 | 55.91 ± 0.03 | 56.61 ± 0.29 |

SC4 | 42.43 ± 2.57 | 29.55 ± 1.99 | 37.88 ± 1.62 | 78.48 ± 2.15 | 45.18 ± 1.9 | 81.25 ± 1.46 |

SC5 | 40.72 ± 0.39 | 28.11 ± 1.48 | 35.36 ± 0.7 | 77.99 ± 1.45 | 43.59 ± 1.37 | 81.41 ± 0.54 |

SC6 | 46.15 ± 3.24 | 30.32 ± 1.91 | 36.04 ± 1.17 | 78.42 ± 2.02 | 46.1 ± 1.64 | 80.62 ± 1.07 |

Featconcat | 54.04 ± 0.04 | 1.22 ± 0.08 | 1.47 ± 0.03 | 38.93 ± 0.03 | 55.69 ± 0.06 | 54.52 ± 0.06 |

AMGL | 64.46 ± 6.14 | 44.36 ± 5.82 | 54.6 ± 1.96 | 70.94 ± 6.65 | 63.71 ± 4.75 | 84.79 ± 0.77 |

MVGL | 57.06 ± 0 | 45.96 ± 0 | 53.17 ± 0 | 87.25 ± 0 | 60.37 ± 0 | 87.04 ± 0 |

ASMV | 40.77 ± 1.2 | 29.04 ± 1.22 | 41.55 ± 0.81 | 76.53 ± 0.75 | 45.2 ± 1.22 | 82.5 ± 0.53 |

GBS | 69.2 ± 0 | 59.43 ± 0 | 60.56 ± 0 | 88.58 ± 0 | 72.17 ± 0 | 88.47 ± 0 |

CoMSC | 63.28 ± 3.68 | 49.02 ± 3.96 | 53.62 ± 3.9 | 86.26 ± 4.95 | 63.49 ± 3.55 | 86.57 ± 1.32 |

CDMGC | 51.74 ± 11.66 | 5.97 ± 23.25 | 23.71 ± 16 | 42.53 ± 12.76 | 50.26 ± 10.59 | 61.8 ± 12.3 |

MVCS-CP | 69.95 ± 0.03 | 57.69 ± 0.07 | 56.13 ± 0.3 | 94.27 ± 1.29 | 69.99 ± 0.16 | 89.48 ± 0 |

NUS | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 21.25 ± 0.42 | 4.32 ± 0.35 | 8.74 ± 0.19 | 12.03 ± 0.37 | 12.71 ± 0.22 | 22.99 ± 0.58 |

SC2 | 20.76 ± 0.42 | 4.23 ± 0.18 | 8.75 ± 0.34 | 12.04 ± 0.2 | 12.42 ± 0.09 | 22.41 ± 0.34 |

SC3 | 18.7 ± 0.22 | 3.4 ± 0.17 | 7.18 ± 0.23 | 11.33 ± 0.17 | 11.62 ± 0.15 | 19.94 ± 0.2 |

SC4 | 23.43 ± 1.1 | 5.23 ± 0.36 | 10.02 ± 0.61 | 13.02 ± 0.31 | 13.21 ± 0.36 | 24.84 ± 0.84 |

SC5 | 21.03 ± 0.45 | 4.73 ± 0.22 | 9.64 ± 0.72 | 12.44 ± 0.2 | 12.98 ± 0.22 | 22.41 ± 0.65 |

SC6 | 11.43 ± 0.18 | 0.32 ± 0.01 | 4.61 ± 0.14 | 8.44 ± 0.01 | 15.31 ± 0.02 | 13.09 ± 0.2 |

Featconcat | 10.79 ± 0.23 | 0.32 ± 0.02 | 4.5 ± 0.16 | 8.44 ± 0.01 | 15.4 ± 0.02 | 12.75 ± 0.12 |

AMGL | 21.43 ± 0.96 | 4.15 ± 0.66 | 12.2 ± 0.96 | 10.68 ± 0.48 | 16.33 ± 0.2 | 23.37 ± 0.99 |

MVGL | 13 ± 0 | 0.36 ± 0 | 5.57 ± 0 | 8.46 ± 0 | 15.44 ± 0 | 13.83 ± 0 |

ASMV | 12.13 ± 1.2 | 0.71 ± 0.84 | 8.13 ± 2.21 | 9.14 ± 0.94 | 14.21 ± 0.15 | 22.46 ± 2.67 |

GBS | 16.5 ± 0 | 1.24 ± 0 | 7.88 ± 0 | 8.88 ± 0 | 15.92 ± 0 | 17.88 ± 0 |

CoMSC | 26.83 ± 2.65 | 8.32 ± 3.47 | 14.12 ± 3.47 | 15.84 ± 2.98 | 27.46 ± 2.76 | 16 ± 1.49 |

CDMGC | 11.96 ± 1.43 | 0.27 ± 0.25 | 4.14 ± 1.57 | 8.42 ± 0.12 | 15.42 ± 0.17 | 12.68 ± 1.54 |

MVCS-CP | 31.38 ± 0.83 | 10.49 ± 0.58 | 16.1 ± 0.29 | 17.52 ± 0.52 | 18.21 ± 0.52 | 32.42 ± 0.38 |

ORL | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 75.7 ± 1.95 | 68.01 ± 1.65 | 89.92 ± 0.68 | 58.42 ± 1.75 | 68.86 ± 1.6 | 80.6 ± 1.31 |

SC2 | 49.25 ± 2.02 | 35.32 ± 2.11 | 70.37 ± 1.16 | 34.84 ± 1.7 | 36.86 ± 2.07 | 53.3 ± 1.87 |

SC3 | 65.45 ± 1.16 | 58.83 ± 2.31 | 85.05 ± 0.81 | 51.09 ± 2.7 | 59.92 ± 2.23 | 71.8 ± 0.87 |

SC4 | 53.65 ± 2.06 | 37.44 ± 2.82 | 72.1 ± 1.5 | 36.77 ± 2.74 | 38.94 ± 2.76 | 57.15 ± 1.71 |

Featconcat | 74.4 ± 0.72 | 68.87 ± 1.18 | 89.37 ± 0.41 | 60.55 ± 1.72 | 69.67 ± 1.14 | 79.5 ± 0.71 |

AMGL | 72.91 ± 3.33 | 65.43 ± 6.51 | 89.69 ± 1.77 | 54.66 ± 7.71 | 66.39 ± 6.27 | 80.21 ± 2.54 |

MVGL | 73.75 ± 0 | 52.74 ± 0 | 87.15 ± 0 | 40.38 ± 0 | 54.17 ± 0 | 80.25 ± 0 |

ASMV | 67 ± 1.23 | 49.46 ± 0.67 | 81.08 ± 0.45 | 43.59 ± 1.37 | 50.79 ± 0.82 | 72.34 ± 0.71 |

GBS | 83.75 ± 0 | 76.32 ± 0 | 92.6 ± 0 | 68.75 ± 0 | 76.92 ± 0 | 86.75 ± 0 |

CoMSC | 86.5 ± 9.67 | 83.63 ± 13.03 | 94.42 ± 6.76 | 80.84 ± 11.97 | 84.01 ± 12.72 | 88.75 ± 9.78 |

CDMGC | 71.35 ± 1.9 | 47.16 ± 3.27 | 86.7 ± 0.85 | 33.95 ± 3.15 | 48.88 ± 3.12 | 79.2 ± 0.96 |

MVCS-CP | 89.5 ± 2.71 | 85.96 ± 0.46 | 94.87 ± 0.08 | 84.27 ± 0.82 | 86.28 ± 0.45 | 90.75 ± 1.41 |

3sources | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 30.3 ± 0.77 | −2.87 ± 0.42 | 6.34 ± 0.73 | 22.06 ± 0.19 | 34.37 ± 0.53 | 36.45 ± 0.9 |

SC2 | 37.4 ± 0.77 | 4.58 ± 0.44 | 10.37 ± 1.6 | 25.19 ± 0.2 | 38.27 ± 0.2 | 39.76 ± 1.28 |

SC3 | 31.95 ± 0 | −2 ± 0.19 | 7.07 ± 0.62 | 22.42 ± 0.08 | 35 ± 0.23 | 37.63 ± 0.79 |

Featconcat | 31.01 ± 1.36 | −0.37 ± 1.36 | 5.45 ± 2.12 | 23.09 ± 0.77 | 27.54 ± 1.84 | 37.28 ± 1.82 |

AMGL | 34.02 ± 2.69 | −1.66 ± 1.45 | 7.2 ± 2.95 | 22.58 ± 0.6 | 34.78 ± 0.56 | 39.25 ± 2.73 |

MVGL | 30.77 ± 0 | −3.38 ± 0 | 6.6 ± 0 | 21.86 ± 0 | 34.17 ± 0 | 37.87 ± 0 |

ASMV | 69.82 ± 4.7 | 60.01 ± 7.14 | 64.07 ± 4.56 | 65.99 ± 6.45 | 69.84 ± 5.23 | 77.51 ± 3.75 |

GBS | 69.23 ± 0 | 44.31 ± 0 | 54.8 ± 0 | 48.44 ± 0 | 60.47 ± 0 | 74.56 ± 0 |

CoMSC | 64.93 ± 4.39 | 53.44 ± 5.59 | 62.41 ± 3.63 | 68.11 ± 4.98 | 63.54 ± 4.4 | 78.27 ± 3.18 |

CDMGC | 34.91 ± 0 | −1.26 ± 0.05 | 6.31 ± 0.26 | 22.73 ± 0.02 | 35.77 ± 0.08 | 39.35 ± 0.31 |

MVCS-CP | 78.11 ± 0.74 | 65.86 ± 1.27 | 71.62 ± 1.41 | 80.31 ± 3.49 | 73.49 ± 0.69 | 85.21 ± 0.56 |

BBC | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 33.11 ± 2.04 | −1.4 ± 0.71 | 7.73 ± 2.46 | 22.88 ± 0.29 | 35.09 ± 0.39 | 36.15 ± 3.35 |

SC2 | 31.53 ± 0 | −0.66 ± 0 | 1.24 ± 0.13 | 23.2 ± 0 | 37.26 ± 0 | 33.02 ± 0.07 |

SC3 | 30.92 ± 1.38 | −0.71 ± 0.37 | 2.1 ± 0.16 | 23.17 ± 0.15 | 36.84 ± 0.6 | 33.28 ± 0.21 |

SC4 | 33.75 ± 0.28 | −0.29 ± 0.13 | 2.71 ± 0.32 | 23.34 ± 0.05 | 37.24 ± 0.13 | 35.07 ± 0.52 |

Featconcat | 33.26 ± 0.12 | −0.23 ± 0.03 | 1.19 ± 0.07 | 23.37 ± 0.01 | 37.59 ± 0.03 | 34.01 ± 0.18 |

AMGL | 35.66 ± 2.75 | 0.88 ± 1.22 | 2.23 ± 1.28 | 23.83 ± 0.51 | 37.22 ± 0.45 | 36.66 ± 2.93 |

MVGL | 35.04 ± 0 | 0.24 ± 0 | 3.82 ± 0 | 23.55 ± 0 | 37.49 ± 0 | 36.35 ± 0 |

ASMV | 63.94 ± 1.2 | 46.07 ± 3.02 | 46.82 ± 1.25 | 50.86 ± 0.86 | 0 ± 3.33 | 64.09 ± 1.21 |

GBS | 69.34 ± 0 | 47.89 ± 0 | 48.52 ± 0 | 50.12 ± 0 | 63.33 ± 0 | 69.34 ± 0 |

CoMSC | 70.18 ± 5.63 | 45.72 ± 8.07 | 51.49 ± 6.53 | 60.36 ± 6.92 | 57.99 ± 6.06 | 71.77 ± 3.89 |

CDMGC | 31.53 ± 1.24 | −0.69 ± 0.09 | 1.08 ± 1.03 | 23.19 ± 0.03 | 36.93 ± 0.13 | 32.99 ± 1.16 |

MVCS-CP | 74.89 ± 0.15 | 52.64 ± 0.19 | 51.76 ± 0.28 | 64.67 ± 0.11 | 63.56 ± 0.16 | 74.89 ± 0.15 |

BBC_Sport | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 35.59 ± 0.1 | −0.07 ± 0.06 | 1.33 ± 0.05 | 23.83 ± 0.02 | 38.25 ± 0.04 | 36.54 ± 0.08 |

SC2 | 36.76 ± 0 | 0.36 ± 0.02 | 1.78 ± 0.06 | 23.99 ± 0.01 | 38.41 ± 0 | 37.1 ± 0.08 |

Featconcat | 0.12 ± 0.12 | 1.4 ± 0.22 | 38.27 ± 0.08 | 96.04 ± 0.38 | 36.84 ± 0.28 | 23.9 ± 0.05 |

AMGL | 36.21 ± 0 | 0.15 ± 0 | 1.34 ± 0.3 | 23.91 ± 0 | 38.42 ± 0.04 | 36.58 ± 0 |

MVGL | 39.15 ± 0 | 1.89 ± 0 | 6.98 ± 0 | 24.59 ± 0 | 39.07 ± 0 | 39.52 ± 0 |

ASMV | 69.12 ± 6.7 | 40.78 ± 5.49 | 39.26 ± 5.09 | 48.07 ± 4.8 | 57.76 ± 3.04 | 69.3 ± 5.95 |

GBS | 80.7 ± 0 | 72.18 ± 0 | 72.26 ± 0 | 72.71 ± 0 | 79.43 ± 0 | 84.38 ± 0 |

CoMSC | 88.6 ± 0.81 | 72.37 ± 2.66 | 71.63 ± 1.84 | 80.28 ± 0.99 | 78.84 ± 2.15 | 88.6 ± 0.81 |

CDMGC | 36.03 ± 0.19 | 0.06 ± 0.14 | 1.43 ± 0.06 | 23.88 ± 0.05 | 38.33 ± 0.09 | 36.76 ± 0.19 |

MVCS-CP | 93.75 ± 0.41 | 84.18 ± 0.65 | 81.77 ± 0.44 | 88.2 ± 1.88 | 87.94 ± 0.78 | 93.75 ± 0.63 |

100leaves | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 41.78 ± 1.23 | 28.47 ± 1.17 | 67.71 ± 0.43 | 26.94 ± 1.16 | 29.2 ± 1.16 | 44.39 ± 1.29 |

SC2 | 33.3 ± 1.04 | 20.85 ± 0.84 | 62.44 ± 0.76 | 18.51 ± 0.76 | 21.72 ± 0.83 | 36.28 ± 1 |

SC3 | 45.96 ± 2.08 | 31.41 ± 1.85 | 70.13 ± 0.87 | 29.73 ± 1.91 | 32.1 ± 1.83 | 48.85 ± 1.86 |

Featconcat | 62.91 ± 2.45 | 52.85 ± 2.42 | 82.01 ± 1.03 | 49.96 ± 2.59 | 53.32 ± 2.39 | 66.23 ± 2.15 |

AMGL | 77.58 ± 2.5 | 47.47 ± 11.8 | 87.87 ± 2.17 | 34.87 ± 11.62 | 48.18 ± 11.58 | 81.25 ± 1.94 |

MVGL | 81.06 ± 0 | 51.55 ± 0 | 89.12 ± 0 | 37.95 ± 0 | 52.17 ± 0 | 83.31 ± 0 |

ASMV | 48.5 ± 0.41 | 23.8 ± 0.59 | 71.38 ± 0.51 | 16.36 ± 0.37 | 24.89 ± 0.19 | 54.06 ± 0.58 |

GBS | 82.44 ± 0 | 57.11 ± 0 | 91.15 ± 0 | 42.67 ± 0 | 57.65 ± 0 | 85.13 ± 0 |

CoMSC | 88.5 ± 6.83 | 86.56 ± 6.95 | 95.95 ± 4.84 | 82.92 ± 6.83 | 86.69 ± 6.2 | 90.88 ± 5.49 |

CDMGC | 88.61 ± 1.34 | 76.15 ± 9.08 | 94.54 ± 1.1 | 66.56 ± 12.45 | 76.42 ± 8.95 | 89.93 ± 1.04 |

MVCS-CP | 91.5 ± 0.74 | 86.82 ± 0.17 | 95.39 ± 0.13 | 84.1 ± 0.43 | 86.95 ± 0.16 | 92 ± 0.32 |

Scene15 | ACC | ARI | NMI | Precision | F | Purity |
---|---|---|---|---|---|---|

SC1 | 34.69 ± 0.7 | 19.64 ± 0.26 | 36.53 ± 0.19 | 24.87 ± 0.38 | 25.29 ± 0.23 | 40.08 ± 0.56 |

SC2 | 25.39 ± 0.43 | 10.05 ± 0.14 | 21.86 ± 0.29 | 14.06 ± 0.22 | 18.01 ± 0.1 | 27.87 ± 0.54 |

SC3 | 22.7 ± 0.99 | 8.82 ± 0.6 | 19.89 ± 0.18 | 14.77 ± 0.56 | 15.38 ± 0.57 | 28.54 ± 0.34 |

Featconcat | 14.46 ± 0.59 | 1.58 ± 0.33 | 10.49 ± 1.05 | 7.68 ± 0.17 | 13.78 ± 0.16 | 17.33 ± 0.66 |

AMGL | 32.78 ± 2.41 | 15.1 ± 1.73 | 30.79 ± 1.84 | 16.66 ± 1.6 | 23.38 ± 1.16 | 34.06 ± 2.09 |

MVGL | 23.21 ± 0 | 6.01 ± 0 | 20.44 ± 0 | 10 ± 0 | 17.16 ± 0 | 24.41 ± 0 |

ASMV | 34.09 ± 0.41 | 17.52 ± 0.48 | 33.74 ± 0.51 | 22.38 ± 0.54 | 23.51 ± 0.41 | 38.86 ± 0.69 |

GBS | 14 ± 0 | 0.42 ± 0 | 5.82 ± 0 | 7.11 ± 0 | 13.17 ± 0 | 14.65 ± 0 |

CoMSC | 43.15 ± 2.69 | 25.86 ± 1.97 | 41.24 ± 1.39 | 30.72 ± 2.02 | 47.29 ± 2.53 | 31.04 ± 1.79 |

CDMGC | 12.44 ± 0.73 | 0.19 ± 0.13 | 3.99 ± 0.84 | 7 ± 0.06 | 13.01 ± 0.09 | 12.97 ± 0.71 |

MVCS-CP | 45.84 ± 2.12 | 26.71 ± 1.01 | 41.18 ± 0.39 | 30.85 ± 1.01 | 31.95 ± 0.92 | 47.42 ± 0.81 |

Time(s) | AMGL | MVGL | ASMV | GBS | CoMSC | CDMGC | MVCS-CP |
---|---|---|---|---|---|---|---|

Caltech101-20 | 80.954 | 662.011 | 317.870 | 28.231 | 562.93 | 122.846 | 3.411 |

Caltech101-7 | 21.3763 | 150.587 | 288.339 | 8.102 | 282.196 | 51.511 | 1.467 |

NUS | 144.625 | 545.729 | 349.962 | 27.655 | 81.597 | 144.333 | 2.333 |

ORL | 0.809 | 5.115 | 8.740 | 0.459 | 4.441 | 2.612 | 0.062 |

3sources | 1.436 | 0.528 | 3.918 | 0.193 | 0.683 | 0.354 | 0.028 |

BBC | 4.837 | 8.776 | 22.629 | 29.819 | 9.583 | 5.269 | 6.742 |

BBC_Sport | 2.387 | 4.753 | 8.214 | 6.199 | 1.628 | 3.054 | 1.552 |

100leaves | 47.818 | 90.341 | 449.571 | 5.849 | 175.703 | 40.080 | 1.204 |

Scence15 | 616.978 | 3485.092 | 1100.982 | 97.190 | 432.381 | 641.264 | 8.988 |

Datasets | d_{1} | d_{2} | d_{3} | d_{4} | d_{5} | d_{6} |
---|---|---|---|---|---|---|

Caltech101-20 | 18 | 18 | 21 | 33 | 25 | 33 |

Caltech101-7 | 17 | 16 | 21 | 28 | 31 | 27 |

NUS | 13 | 2 | 4 | 5 | 5 | 16 |

ORL | 7 | 10 | 8 | 19 | - | - |

3sources | 8 | 8 | 8 | - | - | - |

BBC | 16 | 13 | 9 | 15 | - | - |

BBC_Sport | 14 | 16 | - | - | - | - |

100leaves | 17 | 26 | 14 | - | - | - |

Scene15 | 24 | 18 | 31 | - | - | - |

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**MDPI and ACS Style**

Zhang, X.; Liu, X.
Multiview Clustering of Adaptive Sparse Representation Based on Coupled P Systems. *Entropy* **2022**, *24*, 568.
https://doi.org/10.3390/e24040568

**AMA Style**

Zhang X, Liu X.
Multiview Clustering of Adaptive Sparse Representation Based on Coupled P Systems. *Entropy*. 2022; 24(4):568.
https://doi.org/10.3390/e24040568

**Chicago/Turabian Style**

Zhang, Xiaoling, and Xiyu Liu.
2022. "Multiview Clustering of Adaptive Sparse Representation Based on Coupled P Systems" *Entropy* 24, no. 4: 568.
https://doi.org/10.3390/e24040568