1 Introduction
2 Related work
2.1 Notations
Notations | Description |
---|---|
\(\textbf{X}^{(v)}\) | Data samples \(\in {\mathbb {R}}^{n\times d_v}, v=1,..., V\) |
\(\textbf{S}_v\) | The similarity matrix for each view |
\({\left\| \textbf{M}\right\| }_{2}\) | Frobenius or \(l_2\)-norm of a matrix \(\textbf{M}\) |
\(\textbf{H}\) | The consistent nonnegative embedding matrix |
\(\textbf{P}_v\) | The spectral embedding matrix for each view |
\(\textbf{L}_v\) | Laplacian matrix for each view |
\(\textbf{I}\) | Identity matrix |
\(\textbf{D}\) | Diagonal matrix |
V | Total number of views. |
K | Number of clusters |
n | Total number of samples |
\(w_v\) and \(\delta _v\) | The weight parameters |
\(\alpha \) and \(\lambda \) | Regularization parameters |
2.2 Related work
2.3 Review of the (NESE) method
3 Proposed approach
3.1 Optimization
Algorithm 1 | OCNE |
---|---|
Input: | Data samples \(\textbf{X}^{(v)} \in {\mathbb {R}}^{n\times d_v}, v=1,..., V\) |
The similarity matrix \(\textbf{S}_v\) for each view | |
Parameters \(\alpha \) and \(\lambda \) | |
Output: | The consistent nonnegative embedding matrix \(\textbf{H}\) |
The spectral embedding matrix \(\textbf{P}_v\) for each view | |
Initialization: | The weights \(w_v\) =\(\frac{1}{V}\) and \(\delta _v=1\) |
Repeat | |
Update \(\textbf{P}_v, \, v=1,..., V\) using (9) | |
Update \(\textbf{H}\) using (11) | |
Update \(w_v, \, v=1,..., V\) using (6) | |
Update \(\delta _v, \, v=1,..., V \) using (5) | |
Until convergence |
4 Performance evaluation
4.1 Experimental setup
View | COVIDx | COVIDx-2468 | COVIDx-6468 | |
---|---|---|---|---|
1 | (2048) ResNet50 | (2048) ResNet50 | (2048) ResNet50 | |
2 | (2048) ResNet101 | (2048) ResNet101 | (2048) ResNet101 | |
3 | (1664) DenseNet169 | (1664) DenseNet169 | (1664) DenseNet169 | |
\(\#\) of samples | 13,892 | 2468 | 6468 | |
\(\#\) of classes | 3 | 3 | 3 |
View | MSRCv1 | Caltech101-7 | MNIST-10000 | NUS |
---|---|---|---|---|
1 | (512) GIST | (512) GIST | (4096) VGG16 FC1 | (500) SIFT |
2 | (256) LBP | (928) LBP | (2048) Resnet50 | (73) Edge direction histogram |
3 | (24) Color moment | (48) Gabor | – | (128) Wavelet texture |
4 | (254) Centrist | (254) Centrist | – | (144) Color moment |
5 | (512) SIFT | (40) Wavelet moment | – | (64) Color histogram |
6 | – | (1984) HOG | – | (255) Block-wise color moment |
\(\#\) of samples | 210 | 1474 | 10,000 | 2400 |
\(\#\) of classes | 7 | 7 | 10 | 12 |
-
Auto-weighted Multi-View Clustering via Kernelized graph learning (MVCSK).
-
Spectral Clustering applied on the average of all views’ affinity matrices (SC Fused).
-
Multi-view spectral clustering via integrating Non-negative Embedding and Spectral Embedding approach (NESE).
-
Multi-View Spectral Clustering via Sparse graph learning (S-MVSC).
-
Consistency-aware and Inconsistency-aware Graph-based Multi-View Clustering approach (CI-GMVC).
-
Multi-view clustering via Adaptively Weighted Procrustes (AWP)
-
Multi-view Learning clustering with Adaptive Neighbors (MLAN) [25]
-
Self-weighted Multi-view Clustering with Multiple Graphs (SwMC) [18]
-
Parameter-free Auto-weighted Multiple Graph Learning (AMGL) [19]
-
Affinity Aggregation for Spectral Clustering (AASC) [36]
-
Graph Learning for Multi-View clustering (MVGL) [37]
-
Co-regulated Approach for Multi-View Spectral Clustering (CorSC) [4]
-
Co-training approach for multi-view Spectral Clustering (CotSC) [3].
4.2 Experimental results
4.3 Convergence study
Dataset | Method | ACC | NMI | Purity | ARI |
---|---|---|---|---|---|
COVIDx | SC Fused | 0.44 (± 0.03) | 0.08 (± 0.02) | 0.40 (± 0.02) | 0.07 (± 0.05) |
MVCSK | 0.43 (± 0.05) | 0.07 (± 0.03) | 0.55 (± 0.02) | 0.09 (± 0.03) | |
NESE | 0.62 (± 0.00) | 0.11 (± 0.00) | 0.71 (± 0.00) | 0.15 (± 0.00) | |
S-MVSC | 0.57 (± 0.01) | 0.11 (± 0.00) | 0.57 (± 0.02) | 0.15 (± 0.03) | |
CI-GMVC | 0.63 (± 0.00) | 0.10 (± 0.00) | 0.63 (± 0.00) | 0.08 (± 0.00) | |
OCNE | 0.65 (± 0.00) | 0.12 (± 0.00) | 0.72 (± 0.00) | 0.16 (± 0.00) |
Dataset | Method | ACC | NMI | Purity | ARI |
---|---|---|---|---|---|
NUS | SC-Best | 0.21 (± 0.01) | 0.09 (± 0.01) | 0.21 (± 0.01) | 0.07 (± 0.02) |
AWP | 0.28 (± 0.00) | 0.15 (± 0.00) | 0.29 (± 0.00) | 0.09 (± 0.00) | |
MLAN | 0.25 (± 0.00) | 0.15 (± 0.00) | 0.26 (± 0.00) | 0.04 (± 0.00) | |
SwMC | 0.15 (± 0.00) | 0.08 (± 0.00) | 0.17 (± 0.00) | 0.01 (± 0.00) | |
AMGL | 0.25 (± 0.01) | 0.13 (± 0.01) | 0.27 (± 0.01) | 0.07 (± 0.01) | |
AASC | 0.25 (± 0.00) | 0.13 (± 0.00) | 0.27 (± 0.00) | 0.06 (± 0.00) | |
MVGL | 0.15 (± 0.00) | 0.07 (± 0.00) | 0.16 (± 0.00) | 0.01 (± 0.00) | |
CorSC | 0.27 (± 0.01) | 0.14 (± 0.01) | 0.29 (± 0.01) | 0.09 (± 0.01) | |
CotSC | 0.29 (± 0.01) | 0.16 (± 0.01) | 0.30 (± 0.01) | 0.09 (± 0.01) | |
MVCSK | 0.26 (± 0.01) | 0.15 (± 0.00) | 0.28 (± 0.00) | 0.08 (± 0.00) | |
NESE | 0.30 (±0.00) | 0.17 (± 0.00) | 0.32 (± 0.00) | 0.10 (± 0.00) | |
OCNE | 0.30 (±0.00) | 0.17 (± 0.00) | 0.33(± 0.00) | 0.10(± 0.00) |
Dataset | Method | ACC | NMI | Purity | ARI |
---|---|---|---|---|---|
MSRCv1 | SC Fused | 0.77 (± 0.00) | 0.70 (± 0.00) | 0.79 (± 0.00) | 0.61 (± 0.00) |
MVCSK | 0.70 (± 0.02) | 0.59 (± 0.03) | 0.70 (± 0.02) | 0.50 (± 0.04) | |
NESE | 0.77 (± 0.00) | 0.72 (± 0.00) | 0.80 (± 0.03) | 0.64 (± 0.00) | |
S-MVSC | 0.60 (± 0.00) | 0.69 (± 0.02) | 0.74 (± 0.02) | 0.79 (± 0.01) | |
CI-GMVC | 0.74 (± 0.00) | 0.72 (± 0.00) | 0.77 (± 0.00) | 0.59 (± 0.00) | |
OCNE | 0.86 (±0.00) | 0.76 (±0.00) | 0.86 (± 0.00) | 0.72 (± 0.00) | |
Caltech101-7 | SC Fused | 0.53 (± 0.03) | 0.45 (± 0.03) | 0.60 (± 0.02) | 0.40 (± 0.03) |
MVCSK | 0.57 (± 0.02) | 0.51 (± 0.02) | 0.83 (± 0.01) | 0.45 (± 0.03) | |
NESE | 0.67 (± 0.00) | 0.55 (± 0.00) | 0.87 (± 0.00) | 0.52 (± 0.00) | |
S-MVSC | 0.64 (± 0.03) | 0.55 (± 0.02) | 0.72 (± 0.01) | 0.51 (± 0.03) | |
CI-GMVC | 0.74 (± 0.00) | 0.54 (± 0.00) | 0.85 (± 0.00) | 0.48 (± 0.00) | |
OCNE | 0.69 (± 0.00) | 0.58 (± 0.00) | 0.88 (± 0.00) | 0.56 (± 0.00) | |
MNIST-10000 | SC Fused | 0.20 (± 0.00) | 0.13 (± 0.00) | 0.20 (± 0.00) | 0.05 (± 0.00) |
MVCSK | 0.49 (± 0.00) | 0.41 (± 0.00) | 0.50 (± 0.00) | 0.29 (± 0.00) | |
NESE | 0.81 (± 0.00) | 0.83 (± 0.00) | 0.85 (± 0.00) | 0.76 (± 0.00) | |
S-MVSC | 0.77 (± 0.01) | 0.81 (± 0.01) | 0.81 (± 0.02) | 0.76 (± 0.07) | |
CI-GMVC | 0.66 (± 0.00) | 0.71 (± 0.00) | 0.71 (± 0.00) | 0.51 (± 0.00) | |
OCNE | 0.81 (± 0.00) | 0.83 (± 0.00) | 0.86 (± 0.00) | 0.78 (± 0.00) |
4.4 Parameter sensitivity
4.5 Performance on different subsets of COVIDx
Dataset | Method | Precision (Classes: 1, 2, 3) | Recall (Classes: 1, 2, 3) | Selectivity (Classes: 1, 2, 3) |
---|---|---|---|---|
COVIDx -2468 | SC1 (View 1) | 0.56/ 0.10/ 0.07 | 0.32/ 0.23/ 0.06 | 0.83/ 0.53/ 0.46 |
SC2 (View 2) | 0.58/ 0.22/ 0.62 | 0.54/ 0.47/ 0.32 | 0.74/ 0.50/ 0.86 | |
SC3 (View 3) | 0.53/ 0.15/ 0.54 | 0.63/ 0.34/ 0.13 | 0.62/ 0.56/ 0.93 | |
SC (All) | 0.35/ 0.11/ 0.20 | 0.22/ 0.18/ 0.21 | 0.73/ 0.65/ 0.41 | |
COVIDx -6468 | SC1 (View 1) | 0.83/ 0.03/ 0.47 | 0.57/ 0.21/ 0.25 | 0.89/ 0.56/ 0.75 |
SC2 (View 2) | 0.69/ 0.06/ 0.69 | 0.54/ 0.48/ 0.30 | 0.79/ 0.57/0.88 | |
SC3 (View 3) | 0.42/ 0.07/ 0.67 | 0.11/ 0.59/ 0.58 | 0.86/ 0.54/ 0.74 | |
SC (All) | 0.40/ 0.06/ 0.77 | 0.24/ 0.41/ 0.50 | 0.68/ 0.59/ 0.87 | |
COVIDx | SC1 (View 1) | 0.83/ 0.03/ 0.85 | 0.68/ 0.24/ 0.63 | 0.91/ 0.75/ 0.85 |
SC2 (View 2) | 0.60/ 0.02/ 0.70 | 0.61/ 0.09/ 0.50 | 0.73/ 0.81/ 0.71 | |
SC3 (View 3) | 0.22/ 0.05/ 0.65 | 0.23/ 0.63/ 0.13 | 0.45/ 0.54/ 0.91 | |
SC (All) | 0.66/ 0.02/ 0.64 | 0.65/ 0.17/ 0.30 | 0.78/ 0.65/ 0.77 |
Subset | Method | Precision (Classes: 1, 2, 3) | Recall (Classes: 1, 2, 3) | Selectivity (Classes: 1, 2, 3) |
---|---|---|---|---|
COVIDx -2468 | NESE \(v_1+v_2\) | 0.21/ 0.39/ 0.23 | 0.21/ 0.70/ 0.15 | 0.48/ 0.65/ 0.75 |
NESE \(v_1+v_3\) | 0.22/ 0.08/ 0.74 | 0.18/ 0.16/ 0.53 | 0.55/ 0.59/ 0.87 | |
NESE \(v_2+v_3\) | 0.31/ 0.24/ 0.60 | 0.07/ 0.63/ 0.59 | 0.89/ 0.52/ 0.73 | |
NESE (All) | 0.25/ 0.13/ 0.70 | 0.20/ 0.27/ 0.49 | 0.58/ 0.59/ 0.85 | |
OCNE \(v_1+v_2\) | 0.30/ 0.21/ 0.40 | 0.53/ 0.29/ 0.03 | 0.16/ 0.75/ 0.97 | |
OCNE \(v_1+v_3\) | 0.20/ 0.15/ 0.63 | 0.08/ 0.33/ 0.65 | 0.78/ 0.56/ 0.75 | |
OCNE \(v_2+v_3\) | 0.31/ 0.23/ 0.58 | 0.04/ 0.64/ 0.62 | 0.95/ 0.51/ 0.69 | |
OCNE (All) | 0.29/ 0.23/ 0.69 | 0.04/ 0.74/ 0.57 | 0.94/ 0.41/ 0.83 | |
COVIDx -6468 | NESE \(v_1+v_2\) | 0.22/ 0.06/ 0.63 | 0.17/ 0.49/ 0.22 | 0.48/ 0.52/ 0.88 |
NESE \(v_1+v_3\) | 0.27/ 0.04/ 0.77 | 0.12/ 0.38/ 0.51 | 0.70/ 0.53/ 0.86 | |
NESE \(v_2+v_3\) | 0.31/ 0.07/ 0.63 | 0.32/ 0.55/ 0.08 | 0.37/ 0.55/ 0.96 | |
NESE (All) | 0.24/ 0.06/ 0.63 | 0.19/ 0.57/ 0.16 | 0.44/ 0.51/ 0.91 | |
OCNE \(v_1+v_2\) | 0.33/ 0.06/ 0.75 | 0.14/ 0.48/ 0.53 | 0.75/ 0.53/ 0.84 | |
OCNE \(v_1+v_3\) | 0.33/ 0.06/ 0.75 | 0.14/ 0.48/ 0.53 | 0.75/ 0.53/ 0.84 | |
OCNE \(v_2+v_3\) | 0.31/ 0.07/ 0.63 | 0.32/ 0.55/ 0.08 | 0.37/ 0.55/ 0.96 | |
OCNE (All) | 0.26/ 0.06/ 0.63 | 0.23/ 0.59/ 0.10 | 0.41/ 0.50/ 0.95 | |
COVIDx | NESE \(v_1+v_2\) | 0.16/ 0.06/ 0.30 | 0.15/ 0.40/ 0.22 | 0.51/ 0.80/ 0.29 |
NESE \(v_1+v_3\) | 0.67/ 0.02/ 0.70 | 0.73/ 0.17/ 0.27 | 0.77/ 0.64/ 0.84 | |
NESE \(v_2+v_3\) | 0.24/ 0.04/ 0.71 | 0.30/ 0.55/ 0.08 | 0.38/ 0.56/ 0.96 | |
NESE (All) | 0.18/ 0.04/ 0.72 | 0.18/ 0.59/ 0.16 | 0.44/ 0.54/ 0.92 | |
OCNE \(v_1+v_2\) | 0.16/ 0.01/ 0.37 | 0.17/ 0.01/ 0.35 | 0.42/ 0.96/ 0.20 | |
OCNE \(v_1+v_3\) | 0.67/ 0.02/ 0.70 | 0.72/ 0.18/ 0.27 | 0.77/ 0.64/ 0.84 | |
OCNE \(v_2+v_3\) | 0.24/ 0.04/ 0.71 | 0.30/ 0.56/ 0.08 | 0.38/ 0.56/ 0.95 | |
OCNE (All) | 0.18/ 0.04/ 0.71 | 0.18/ 0.58/ 0.16 | 0.44/ 0.54/ 0.91 |