Hypergraph-Regularized Lp Smooth Nonnegative Matrix Factorization for Data Representation
Abstract
:1. Introduction
1.1. Problem Statement
1.2. Research Contribution
2. Related Work
2.1. Nonnegative Matrix Factorization
2.2. Graph Regularization Smooth Nonnegative Matrix Factorization
2.3. Hypergraph Learning
3. Hypergraph-Regularized Smooth Nonnegative Matrix Factorization
3.1. The Objective Function
3.2. Optimization Method
3.3. Convergence Analysis
Algorithm 1 HGSNMF algorithm. |
Input: Data matrix . The number of neighbors k. The algorithm |
parameters r, p and regularization parameters , . The stopping criterion , and the maximum |
number of iterations maxiter. Let . |
Output: Factors and ; |
1: Initialize and ; |
2: Construct the weight matrix using (4), and calculate the |
matrix , ; |
3: for maxiter do |
4: Update and Update according to (8), (9), respectively. |
5: Compute the objective function value of (5) to denote . |
6: if |
Break and return . |
7: end if |
8: end for |
3.4. Computational Complexity Analysis
4. Numerical Experimentation
4.1. Data Sets
4.2. Evaluation Metrics
4.3. Performance Evaluations and Comparisons
4.4. Parameters Selection
4.5. The Converage Analysis
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
PCA | Principal component analysis |
LDA | Linear discriminant analysis |
SVD | Singular value decomposition |
NMF | Nonnegative matrix factorization |
HU | Hyperspectral unmixing |
ONMF | Orthogonal nonnegative matrix tri-factorizators |
GNMF | Graph regularized nonnegative matrix factorization |
DNMF | Graph dual regularization nonnegative matrix factorization |
HNMF | Hypergraph regularized nonnegative matrix factorization |
GSNMF | Graph regularized smooth nonnegative matrix factorization |
HGLNMF | Hypergraph regularized sparse nonnegative matrix factorization |
MHGNMF | Nonnegative matrix factorization with mixed hypergraph regularization |
DHPS-NMF | Dual hypergraph regularized partially shared nonnegative matrix factorization |
HGSNMF | Hypergraph regularized smooth nonnegative matrix factorization |
ACC | Accuracy |
NMI | Normalized mutual information |
MI | Mutual information |
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fladd | flmlt | fldiv | Overall | |
---|---|---|---|---|
NMF | 2+2 | |||
GNMF | 2+2 | 2+2 | ||
HNMF | 2+2 | 2+2 | ||
GSNMF | 2+2 | 2+2 | ||
HGLNMF | 2+2 | 2+2 | ||
HGSNMF | 2+2 | 2+2 |
Data Sets | Samples | Features | Classes |
---|---|---|---|
COIL20 | 1440 | 1024 | 20 |
YALE | 165 | 1024 | 15 |
ORL | 400 | 1024 | 40 |
Georgia | 750 | 1024 | 50 |
Mnist | 500 | 784 | 10 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
4 | 65.13 ± 16.60 | 69.63 ± 15.76 | 72.86 ± 14.86 | 79.53 ± 13.65 | 73.79 ± 13.51 | 79.52 ± 13.64 | |
6 | 67.70 ± 9.79 | 69.65 ± 11.27 | 72.79 ± 10.70 | 80.72 ± 10.58 | 68.63 ± 10.00 | 80.76 ± 10.54 | |
8 | 70.56 ± 5.89 | 69.44 ± 8.78 | 71.53 ± 8.60 | 80.94 ± 7.35 | 73.55 ± 7.28 | 81.08 ± 7.38 | |
10 | 76.02 ± 7.12 | 70.13 ± 6.95 | 76.01 ± 5.48 | 82.99 ± 5.84 | 76.36 ± 5.92 | 83.00 ± 5.75 | |
12 | 73.21 ± 4.82 | 70.91 ± 5.50 | 77.12 ± 5.68 | 82.16 ± 5.03 | 75.70 ± 5.33 | 82.31 ± 5.08 | |
14 | 74.10 ± 4.31 | 70.41 ± 4.66 | 77.06 ± 4.75 | 82.20 ± 4.24 | 76.88 ± 4.63 | 82.21 ± 4.19 | |
16 | 74.85 ± 3.65 | 72.70 ± 4.04 | 79.38 ± 4.52 | 84.05 ± 3.95 | 79.04 ± 4.27 | 83.99 ± 3.97 | |
18 | 73.28 ± 3.08 | 71.49 ± 3.00 | 78.03 ± 3.65 | 84.61 ± 3.15 | 79.36 ± 3.47 | 84.70 ± 3.16 | |
20 | 73.83 ± 2.52 | 71.95 ± 2.76 | 79.20 ± 3.05 | 84.08 ± 2.79 | 78.84 ± 2.76 | 84.05 ± 2.71 | |
Avg. | 72.0 | 71.70 | 76.00 | 82.36 | 75.80 | 82.40 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
4 | 71.45 ± 14.82 | 72.35 ± 16.50 | 73.41 ± 14.85 | 75.78 ± 17.88 | 75.33 ± 15.74 | 75.77 ± 17.88 | |
6 | 67.05 ± 10.72 | 68.87 ± 11.80 | 69.65 ± 12.61 | 74.42 ± 13.78 | 68.04 ± 10.78 | 74.44 ± 13.78 | |
8 | 64.56 ± 64.56 | 65.39 ± 9.95 | 64.35 ± 10.93 | 71.06 ± 11.77 | 67.36 ± 9.73 | 70.71 ± 11.75 | |
10 | 67.38 ± 9.76 | 63.27 ± 7.96 | 66.76 ± 7.47 | 70.73 ± 10.04 | 67.07 ± 8.44 | 70.61 ± 5.75 | |
12 | 63.73 ± 7.09 | 63.19 ± 7.02 | 66.81 ± 8.30 | 68.63 ± 8.43 | 65.81 ± 8.43 | 69.00 ± 5.08 | |
14 | 62.48 ± 6.42 | 60.01 ± 6.16 | 65.18 ± 7.57 | 68.04 ± 8.33 | 64.21 ± 7.77 | 68.03 ± 8.18 | |
16 | 61.78 ± 5.69 | 62.18 ± 6.43 | 65.47 ± 7.25 | 69.09 ± 7.62 | 66.03 ± 7.50 | 68.85 ± 7.66 | |
18 | 59.15 ± 6.18 | 59.68 ± 5.44 | 63.39 ± 6.86 | 69.29 ± 6.51 | 65.84 ± 6.70 | 69.65 ± 6.59 | |
20 | 58.18 ± 5.43 | 59.11 ± 4.60 | 63.86 ± 6.24 | 63.95 ± 6.00 | 68.56 ± 6.34 | 68.19 ± 6.83 | |
Avg. | 63.97 | 63.78 | 66.54 | 70.64 | 67.07 | 70.63 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
3 | 40.18 ± 23.03 | 28.96 ± 11.71 | 28.81 ± 12.06 | 36.08 ± 12.82 | 37.80 ± 16.40 | 36.12 ± 12.89 | |
5 | 35.72 ± 12.87 | 38.23 ± 10.25 | 38.48 ± 10.02 | 39.37 ± 8.83 | 40.01 ± 10.61 | 39.35 ± 8.85 | |
7 | 38.38 ± 6.51 | 38.17 ± 7.57 | 39.07 ± 6.84 | 39.33 ± 6.46 | 39.38 ± 5.23 | 42.32 ± 7.39 | |
9 | 41.80 ± 3.87 | 40.56 ± 5.00 | 38.93 ± 5.05 | 38.85 ± 4.88 | 39.18 ± 5.23 | 40.21 ± 4.35 | |
11 | 39.80 ± 4.40 | 41.82 ± 4.45 | 42.05 ± 4.25 | 43.88 ± 4.30 | 44.08 ± 4.62 | 44.04 ± 4.61 | |
13 | 44.13 ± 4.63 | 44.17 ± 3.03 | 44.59 ± 3.43 | 44.24 ± 3.12 | 44.53 ± 2.80 | 44.29 ± 3.07 | |
14 | 44.34 ± 3.84 | 43.27 ± 2.92 | 43.31 ± 3.10 | 44.21 ± 3.39 | 44.82 ± 3.02 | 44.21 ± 3.32 | |
15 | 44.48 ± 2.92 | 43.91 ± 2.90 | 44.36 ± 2.72 | 45.37 ± 2.38 | 45.32 ± 2.62 | 45.47 ± 2.35 | |
Avg. | 41.76 | 40.07 | 40.04 | 41.40 | 41.84 | 41.51 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
3 | 62.24 ± 14.91 | 59.30 ± 8.16 | 59.12 ± 8.11 | 61.49 ± 8.20 | 62.64 ± 11.14 | 61.91 ± 8.61 | |
5 | 50.24 ± 10.66 | 54.15 ± 8.92 | 53.78 ± 8.27 | 54.95 ± 7.83 | 54.82 ± 9.04 | 54.84 ± 7.82 | |
7 | 49.43 ± 6.26 | 47.40 ± 6.33 | 46.92 ± 6.90 | 47.21 ± 6.81 | 46.65 ± 6.14 | 47.44 ± 6.62 | |
9 | 44.40 ± 7.02 | 43.12 ± 4.87 | 42.48 ± 5.45 | 42.15 ± 4.74 | 42.81 ± 5.49 | 43.68 ± 4.98 | |
11 | 39.12 ± 4.80 | 41.37 ± 5.06 | 41.74 ± 4.65 | 43.43 ± 5.00 | 43.07 ± 5.29 | 43.50 ± 5.12 | |
13 | 40.41 ± 4.81 | 41.18 ± 3.73 | 41.11 ± 4.10 | 40.76 ± 3.83 | 41.05 ± 3.61 | 40.78 ± 3.91 | |
14 | 39.46 ± 4.26 | 39.05 ± 4.21 | 38.27 ± 3.72 | 39.92 ± 4.21 | 40.03 ± 3.75 | 39.94 ± 4.22 | |
15 | 38.52 ± 3.30 | 38.72 ± 3.51 | 38.67 ± 3.21 | 40.25 ± 3.39 | 39.52 ± 3.14 | 40.44 ± 3.24 | |
Avg. | 45.41 | 45.76 | 45.34 | 46.31 | 46.24 | 46.46 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
5 | 66.24 ± 12.00 | 67.18 ± 12.07 | 68.81 ± 11.16 | 68.58 ± 12.77 | 66.51 ± 13.18 | 68.97 ± 13.01 | |
10 | 70.22 ± 6.63 | 73.59 ± 5.90 | 72.11 ± 6.70 | 71.95 ± 5.92 | 72.39 ± 6.59 | 73.29 ± 6.55 | |
15 | 68.46 ± 4.21 | 75.23 ± 5.01 | 76.14 ± 5.18 | 75.26 ± 5.62 | 75.67 ± 5.27 | 75.26 ± 5.62 | |
20 | 69.87 ± 4.75 | 74.21 ± 4.34 | 74.44 ± 4.78 | 75.24 ± 4.25 | 75.49 ± 3.76 | 75.46 ± 4.19 | |
25 | 71.13 ± 3.48 | 75.51 ± 2.69 | 75.88 ± 3.13 | 76.03 ± 3.29 | 76.10 ± 3.17 | 76.06 ± 3.12 | |
30 | 71.03 ± 2.81 | 75.34 ± 3.12 | 75.55 ± 2.81 | 74.60 ± 2.67 | 74.69 ± 2.65 | 75.88 ± 2.80 | |
35 | 71.07 ± 1.82 | 75.07 ± 2.23 | 74.96 ± 2.06 | 74.46 ± 1.87 | 75.85 ± 2.18 | 74.52 ± 1.91 | |
40 | 71.45 ± 2.06 | 75.05 ± 1.90 | 75.26 ± 1.82 | 74.54 ± 1.87 | 75.40 ± 1.91 | 74.54 ± 1.91 | |
Avg. | 69.93 | 73.90 | 74.35 | 73.85 | 74.11 | 73.99 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
5 | 67.32 ± 14.91 | 68.12 ± 12.11 | 68.76 ± 12.31 | 68.70 ± 12.27 | 67.36 ± 12.72 | 68.97 ± 13.01 | |
10 | 62.72 ± 10.66 | 65.85 ± 9.86 | 64.05 ± 7.87 | 64.59 ± 7.22 | 64.46 ± 7.82 | 65.42 ± 7.50 | |
15 | 56.19 ± 5.80 | 63.55 ± 6.85 | 64.84 ± 6.96 | 63.99 ± 7.44 | 64.59 ± 7.32 | 63.99 ± 7.44 | |
20 | 55.29 ± 6.25 | 61.21 ± 5.83 | 61.56 ± 6.21 | 62.44 ± 5.87 | 62.67 ± 5.30 | 62.52 ± 5.57 | |
25 | 54.15 ± 4.50 | 60.58 ± 3.98 | 61.01 ± 4.84 | 61.31 ± 4.66 | 61.16 ± 4.96 | 61.32 ± 4.80 | |
30 | 52.52 ± 4.29 | 58.57 ± 4.67 | 58.88 ± 4.52 | 58.07 ± 4.42 | 58.38 ± 4.28 | 59.50 ± 4.20 | |
35 | 51.30 ± 3.20 | 57.83 ± 3.62 | 57.22 ± 3.46 | 56.95 ± 3.28 | 57.05 ± 3.21 | 58.35 ± 4.06 | |
40 | 50.68 ± 3.43 | 56.57 ± 3.39 | 56.73 ± 3.15 | 55.88 ± 3.32 | 57.20 ± 3.46 | 55.88 ± 3.37 | |
Avg. | 56.27 | 61.57 | 61.86 | 61.42 | 62.03 | 61.57 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
5 | 67.05 ± 11.33 | 59.40 ± 11.79 | 63.00 ± 12.27 | 60.93 ± 11.93 | 64.46 ± 10.83 | 60.99 ± 11.94 | |
10 | 60.25 ± 8.93 | 61.24 ± 8.51 | 57.48 ± 10.50 | 61.59 ± 10.12 | 57.63 ± 10.67 | 65.91 ± 9.15 | |
15 | 64.64 ± 5.32 | 60.57 ± 4.39 | 62.46 ± 4.85 | 61.89 ± 5.12 | 64.02 ± 5.72 | 61.99 ± 5.02 | |
20 | 67.12 ± 4.30 | 60.60 ± 3.71 | 62.58 ± 3.91 | 60.98 ± 3.81 | 64.44 ± 3.18 | 61.08 ± 3.82 | |
25 | 66.30 ± 3.31 | 59.31 ± 2.73 | 61.35 ± 2.62 | 61.33 ± 3.22 | 64.83 ± 2.98 | 61.32 ± 3.68 | |
30 | 66.01 ± 3.13 | 60.26 ± 2.56 | 63.20 ± 2.25 | 60.52 ± 3.04 | 64.61 ± 2.97 | 60.47 ± 2.99 | |
35 | 65.10 ± 2.13 | 59.93 ± 2.33 | 63.3 ± 1.80 | 59.21 ± 2.21 | 63.27 ± 2.37 | 59.20 ± 2.22 | |
40 | 66.06 ± 2.20 | 59.58 ± 2.34 | 62.84 ± 1.82 | 58.61 ± 2.38 | 63.57 ± 1.98 | 58.62 ± 2.48 | |
45 | 66.17 ± 1.35 | 59.99 ± 1.75 | 62.92 ± 1.55 | 58.22 ± 1.90 | 62.92 ± 1.66 | 58.25 ± 1.99 | |
50 | 66.36 ± 1.32 | 59.05 ± 1.56 | 62.11 ± 1.51 | 58.19 ± 1.33 | 63.18 ± 1.24 | 58.19 ± 1.25 | |
Avg. | 66.26 | 59.90 | 62.50 | 59.74 | 63.69 | 59.77 |
k | K-Means | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|---|
5 | 68.73 ± 11.50 | 66.68 ± 10.96 | 69.52 ± 11.15 | 68.00 ± 10.33 | 69.76 ± 10.50 | 67.68 ± 10.47 | |
10 | 61.71 ± 8.56 | 57.79 ± 8.48 | 59.25 ± 8.44 | 55.73 ± 9.09 | 59.07 ± 8.94 | 55.83 ± 9.23 | |
15 | 55.38 ± 6.27 | 53.33 ± 4.41 | 55.05 ± 5.38 | 55.07 ± 5.41 | 56.08 ± 6.15 | 55.21 ± 5.38 | |
20 | 55.14 ± 5.07 | 50.55 ± 4.58 | 52.30 ± 4.60 | 50.22 ± 4.45 | 54.93 ± 4.36 | 50.37 ± 4.40 | |
25 | 51.82 ± 4.40 | 46.8¡¤ ± 3.59 | 48.50 ± 3.42 | 48.25 ± 4.38 | 52.10 ± 4.09 | 48.311 ± 4.30 | |
30 | 49.67 ± 4.12 | 45.20 ± 3.49 | 48.31 ± 3.13 | 45.82 ± 3.48 | 50.17 ± 3.82 | 45.68 ± 3.33 | |
35 | 47.80 ± 3.28 | 43.78 ± 3.12 | 47.09 ± 2.92 | 42.57 ± 3.03 | 47.19 ± 3.31 | 42.53 ± 3.02 | |
40 | 47.88 ± 3.28 | 41.93 ± 2.95 | 45.35 ± 2.81 | 40.47 ± 3.26 | 46.27 ± 3.02 | 40.41 ± 3.45 | |
45 | 47.39 ± 2.26 | 41.07 ± 2.52 | 43.93 ± 2.44 | 38.53 ± 2.49 | 44.34 ± 2.50 | 38.58 ± 2.59 | |
50 | 46.18 ± 2.19 | 38.94 ± 2.22 | 42.14 ± 2.40 | 37.24 ± 1.92 | 43.78 ± 2.32 | 37.26 ± 1.90 | |
Avg. | 53.17 | 48.61 | 51.14 | 48.19 | 52.37 | 48.19 |
k | GNMF | HNMF | GSNMF | HGLNMF | GNTD | UGNTD | HGSNMF |
---|---|---|---|---|---|---|---|
2 | 58.89 ± 23.94 | 57.08 ± 24.38 | 42.94 ± 12.91 | 57.34 ± 24.27 | 67.11 ± 18.27 | 53.17 ± 36.19 | |
4 | 53.12 ± 12.67 | 55.64 ± 14.51 | 53.70 ± 12.26 | 55.61 ± 14.04 | 57.17 ± 11.60 | 56.32 ± 11.42 | |
6 | 45.17 ± 4.90 | 49.23 ± 5.85 | 48.53 ± 6.11 | 48.72 ± 5.99 | 49.20 ± 5.01 | 56.27 ± 7.69 | |
7 | 47.63 ± 6.82 | 45.51 ± 4.82 | 47.48 ± 4.48 | 46.88 ± 5.46 | 47.05 ± 5.14 | 55.69 ± 7.31 | |
8 | 48.55 ± 3.86 | 48.32 ± 4.38 | 49.76 ± 4.87 | 47.22 ± 3.83 | 47.38 ± 3.38 | 57.30 ± 6.01 | |
10 | 47.01 ± 4.39 | 45.06 ± 2.21 | 46.30 ± 3.65 | 44.46 ± 2.81 | 45.07 ± 4.08 | 56.11 ± 4.73 | |
Avg. | 50.06 | 50.14 | 48.12 | 50.04 | 52.16 | 55.81 |
k | GNMF | HNMF | GSNMF | HGLNMF | GNTD | UGNTD | HGSNMF |
---|---|---|---|---|---|---|---|
2 | 80.07 ± 28.05 | 87.96 ± 13.17 | 80.50 ± 17.55 | 88.13 ± 12.96 | 82.73 ± 17.28 | 91.57 ± 13.32 | |
4 | 70.89 ± 11.28 | 74.17 ± 12.67 | 71.51 ± 16.93 | 65.99 ± 24.92 | 67.74 ± 11.50 | 71.32 ± 26.19 | |
6 | 57.26 ± 5.48 | 60.41 ± 7.06 | 57.58 ± 7.37 | 59.36 ± 7.64 | 60.50 ± 6.74 | 44.47 ± 21.53 | |
7 | 57.52 ± 7.66 | 54.99 ± 5.84 | 54.82 ± 3.97 | 55.54 ± 5.69 | 55.68 ± 5.24 | 56.34 ± 7.57 | |
8 | 45.34 ± 2.08 | 55.63 ± 5.18 | 56.84 ± 5.27 | 49.11 ± 15.91 | 55.21 ± 4.34 | 56.78 ± 19.12 | |
10 | 50.72 ± 5.42 | 48.21 ± 3.68 | 48.28 ± 6.05 | 48.02 ± 4.00 | 47.93 ± 4.53 | 51.01 ± 6.48 | |
Avg. | 60.30 | 63.56 | 61.59 | 60.86 | 60.05 | 68 |
k | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|
3 | 1.15 | 2.51 | 3.14 | 4.08 | 3.15 | |
11 | 9.21 | 20.75 | 21.31 | 35.62 | 14.77 | |
13 | 17.63 | 41.12 | 42.18 | 67.89 | 28.70 | |
14 | 20.64 | 49.80 | 46.75 | 80.06 | 35.45 | |
15 | 22.52 | 55.35 | 54.18 | 96.94 | 43.62 |
k | NMF | GNMF | HNMF | GSNMF | HGLNMF | HGSNMF |
---|---|---|---|---|---|---|
10 | 25.39 | 47.79 | 35.77 | 69.7 | 45.07 | |
15 | 40.36 | 94.57 | 67.58 | 134.67 | 88.01 | |
20 | 56.96 | 123.30 | 89.73 | 183.88 | 120.07 | |
25 | 90.97 | 233.04 | 151.94 | 312.03 | 201.96 | |
30 | 114.43 | 343.62 | 196.97 | 429.94 | 214.84 |
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Xu, Y.; Lu, L.; Liu, Q.; Chen, Z. Hypergraph-Regularized Lp Smooth Nonnegative Matrix Factorization for Data Representation. Mathematics 2023, 11, 2821. https://doi.org/10.3390/math11132821
Xu Y, Lu L, Liu Q, Chen Z. Hypergraph-Regularized Lp Smooth Nonnegative Matrix Factorization for Data Representation. Mathematics. 2023; 11(13):2821. https://doi.org/10.3390/math11132821
Chicago/Turabian StyleXu, Yunxia, Linzhang Lu, Qilong Liu, and Zhen Chen. 2023. "Hypergraph-Regularized Lp Smooth Nonnegative Matrix Factorization for Data Representation" Mathematics 11, no. 13: 2821. https://doi.org/10.3390/math11132821
APA StyleXu, Y., Lu, L., Liu, Q., & Chen, Z. (2023). Hypergraph-Regularized Lp Smooth Nonnegative Matrix Factorization for Data Representation. Mathematics, 11(13), 2821. https://doi.org/10.3390/math11132821