A Multi-Task Decomposition-Based Evolutionary Algorithm for Tackling High-Dimensional Bi-Objective Feature Selection
Abstract
:1. Introduction
- First, a dynamic multi-task mechanism is designed and combined with the decomposition-based MOEA framework, which assigns multiple evolutionary search tasks for different subpopulations within the entire population and then conditionally merges them into a single task or an integrated population as the evolutionary process goes on, for tackling the large-scale bi-objective feature selection in a more effective way.
- Second, an adaptive decomposition-based MOEA framework is set up, which cooperates with the above multi-task mechanism via adaptively adjusting the ideal point for each subpopulation related to different tasks, so that each task has distinct search biases and focuses its computational resources on searching more productive areas in the objective space.
- Third, a series of comprehensive studies were conducted in experiments to analyze the optimization and classification performance of the proposed MTDEA algorithm against other state-of-the-art MOEAs, in terms of multiple indicators and using a variety of high-dimensional classification datasets.
2. Related Works
2.1. Bi-Objective Feature Selection Problem
2.2. Evolutionary Feature Selection Methods
2.3. Decomposition-Based MOEA Approaches
3. Proposed Algorithm
3.1. General Framework
3.2. Initialization Process
Algorithm 1 |
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Algorithm 2 |
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Algorithm 3 |
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3.3. Reproduction Process
Algorithm 4 |
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3.4. Environmental Selection Process
Algorithm 5 |
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3.5. Task Merging Process
Algorithm 6 |
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4. Experimental Setups
4.1. Classification Datasets
4.2. Comparison Algorithms
4.3. Performance Indicators
4.4. Parameter Settings
5. Experimental Studies
5.1. General Performance Studies
5.2. Nondominated Solution Distributions
5.3. Component Contribution Analyses
5.4. Computational Time Analyses
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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No. | Dataset | Feature | Sample | Class |
---|---|---|---|---|
1 | ORL | 1024 | 400 | 40 |
2 | Yale | 1024 | 165 | 15 |
3 | Colon | 2000 | 62 | 2 |
4 | SRBCT | 2308 | 83 | 4 |
5 | AR10P | 2400 | 130 | 10 |
6 | PIE10P | 2420 | 210 | 10 |
7 | Lymphoma | 4026 | 96 | 9 |
9 | DLBCL | 5469 | 77 | 2 |
8 | TOX171 | 5748 | 171 | 4 |
10 | Brain1 | 5920 | 90 | 5 |
11 | Leukemia | 7070 | 72 | 2 |
12 | CNS | 7129 | 60 | 2 |
13 | ALLAML | 7129 | 72 | 2 |
14 | Carcinom | 9182 | 174 | 11 |
15 | Nci9 | 9712 | 60 | 9 |
16 | Arcene | 10,000 | 200 | 2 |
17 | Pixraw10P | 10,000 | 100 | 10 |
18 | Orlraws10P | 10,304 | 100 | 10 |
19 | Brain2 | 10,367 | 50 | 4 |
20 | Prostate | 10,509 | 102 | 2 |
Metric | MTDEA | NSGA-II | MOEA/D | MOEA/AWA | MOEA/HD | SparseEA | DAEA | PRDH |
---|---|---|---|---|---|---|---|---|
HV | 1.012 | 5.215 | 3.917 | 5.290 | 6.275 | 7.530 | 2.192 | 4.567 |
MCE | 3.615 | 4.688 | 5.124 | 4.787 | 5.030 | 4.303 | 3.740 | 4.714 |
NSF | 1.000 | 5.270 | 3.112 | 5.640 | 6.461 | 7.949 | 2.110 | 4.457 |
Dataset | MTDEA | NSGA-II | MOEA/D | MOEA/AWA | MOEA/HD | SparseEA | DAEA | PRDH |
---|---|---|---|---|---|---|---|---|
ORL | 7.928e-01 | 6.103e-01 | 6.642e-01 | 6.576e-01 | 5.996e-01 | 5.859e-01 | 7.249e-01 | 6.350e-01 |
±1.80e-02 | ±1.08e-02 | ±1.53e-02 | ±1.79e-02 | ±1.25e-02 | ±9.94e-03 | ±1.75e-02 | ±1.18e-02 | |
Yale | 6.597e-01 | 4.878e-01 | 5.127e-01 | 5.199e-01 | 4.920e-01 | 4.811e-01 | 6.050e-01 | 5.071e-01 |
±3.18e-02 | ±1.34e-02 | ±2.62e-02 | ±2.40e-02 | ±3.39e-02 | ±1.64e-02 | ±2.48e-02 | ±2.15e-02 | |
Colon | 7.746e-01 | 5.500e-01 | 6.078e-01 | 5.466e-01 | 5.239e-01 | 4.987e-01 | 6.680e-01 | 5.554e-01 |
±3.84e-02 | ±2.65e-02 | ±4.78e-02 | ±3.55e-02 | ±3.30e-02 | ±2.01e-02 | ±3.95e-02 | ±3.10e-02 | |
SRBCT | 6.654e-01 | 2.841e-01 | 3.178e-01 | 2.983e-01 | 2.554e-01 | 2.479e-01 | 3.028e-01 | 2.768e-01 |
±1.70e-01 | ±2.05e-03 | ±2.03e-03 | ±2.74e-03 | ±1.81e-03 | ±1.69e-03 | ±1.96e-02 | ±3.17e-03 | |
AR10P | 4.972e-01 | 3.631e-01 | 3.709e-01 | 3.544e-01 | 3.449e-01 | 3.223e-01 | 4.314e-01 | 3.603e-01 |
±2.74e-02 | ±2.01e-02 | ±2.26e-02 | ±1.35e-02 | ±1.75e-02 | ±1.11e-02 | ±2.05e-02 | ±1.86e-02 | |
PIE10P | 8.127e-01 | 6.023e-01 | 6.458e-01 | 6.003e-01 | 5.883e-01 | 5.434e-01 | 6.982e-01 | 6.056e-01 |
±2.00e-02 | ±1.06e-02 | ±1.13e-02 | ±1.29e-02 | ±1.22e-02 | ±6.44e-03 | ±1.57e-02 | ±9.92e-03 | |
Lymphoma | 7.645e-01 | 5.603e-01 | 6.007e-01 | 5.598e-01 | 5.456e-01 | 5.067e-01 | 6.434e-01 | 5.626e-01 |
±2.18e-02 | ±9.91e-03 | ±9.76e-03 | ±1.59e-02 | ±8.85e-03 | ±1.07e-02 | ±1.57e-02 | ±6.99e-03 | |
TOX171 | 6.749e-01 | 4.829e-01 | 4.876e-01 | 4.697e-01 | 4.776e-01 | 4.575e-01 | 5.423e-01 | 4.907e-01 |
±2.21e-02 | ±8.58e-03 | ±1.97e-02 | ±1.13e-02 | ±1.65e-02 | ±1.19e-02 | ±1.92e-02 | ±1.23e-02 | |
DLBCL | 8.054e-01 | 5.852e-01 | 5.999e-01 | 5.722e-01 | 5.735e-01 | 5.469e-01 | 6.703e-01 | 5.982e-01 |
±3.06e-02 | ±2.02e-02 | ±1.82e-02 | ±2.17e-02 | ±1.67e-02 | ±1.83e-02 | ±2.12e-02 | ±1.77e-02 | |
Brain1 | 6.496e-01 | 4.718e-01 | 4.906e-01 | 4.657e-01 | 4.535e-01 | 4.312e-01 | 5.127e-01 | 4.744e-01 |
±1.38e-02 | ±3.11e-03 | ±1.09e-02 | ±1.06e-02 | ±1.00e-02 | ±1.78e-03 | ±8.53e-03 | ±2.57e-03 | |
Leukemia | 7.626e-01 | 5.360e-01 | 5.450e-01 | 5.295e-01 | 5.126e-01 | 4.972e-01 | 6.022e-01 | 5.454e-01 |
±2.70e-02 | ±8.95e-03 | ±1.94e-02 | ±1.89e-02 | ±1.79e-02 | ±9.84e-03 | ±1.67e-02 | ±2.04e-02 | |
CNS | 5.255e-01 | 3.781e-01 | 3.743e-01 | 3.690e-01 | 3.707e-01 | 3.669e-01 | 4.408e-01 | 3.844e-01 |
±5.88e-02 | ±3.28e-02 | ±3.32e-02 | ±3.09e-02 | ±3.25e-02 | ±1.57e-02 | ±3.03e-02 | ±2.94e-02 | |
ALLAML | 7.278e-01 | 5.205e-01 | 5.358e-01 | 5.084e-01 | 5.060e-01 | 4.853e-01 | 5.826e-01 | 5.280e-01 |
±2.93e-02 | ±1.52e-02 | ±1.34e-02 | ±1.60e-02 | ±1.44e-02 | ±1.52e-02 | ±1.83e-02 | ±1.32e-02 | |
Carcinom | 7.199e-01 | 5.180e-01 | 5.233e-01 | 5.098e-01 | 5.091e-01 | 4.871e-01 | 5.809e-01 | 5.193e-01 |
±1.40e-02 | ±1.09e-02 | ±1.55e-02 | ±1.00e-02 | ±1.10e-02 | ±8.18e-03 | ±1.18e-02 | ±7.20e-03 | |
Nci9 | 3.570e-01 | 2.406e-01 | 2.616e-01 | 2.544e-01 | 2.370e-01 | 2.254e-01 | 2.707e-01 | 2.451e-01 |
±2.33e-02 | ±2.54e-02 | ±2.94e-02 | ±2.80e-02 | ±2.21e-02 | ±2.00e-02 | ±2.75e-02 | ±2.71e-02 | |
Arcene | 5.132e-01 | 3.625e-01 | 3.724e-01 | 3.649e-01 | 3.445e-01 | 3.374e-01 | 3.859e-01 | 3.647e-01 |
±4.15e-03 | ±1.10e-03 | ±1.85e-03 | ±2.23e-03 | ±1.24e-03 | ±1.29e-03 | ±2.75e-03 | ±1.71e-03 | |
Pixraw10P | 8.104e-01 | 5.795e-01 | 5.911e-01 | 5.773e-01 | 5.640e-01 | 5.407e-01 | 6.327e-01 | 5.846e-01 |
±6.33e-03 | ±2.22e-03 | ±9.88e-03 | ±7.09e-03 | ±7.64e-03 | ±1.57e-03 | ±9.37e-03 | ±2.65e-03 | |
Orlraws10P | 7.491e-01 | 5.390e-01 | 5.447e-01 | 5.328e-01 | 5.297e-01 | 5.057e-01 | 5.951e-01 | 5.444e-01 |
±1.43e-02 | ±7.53e-03 | ±9.58e-03 | ±1.13e-02 | ±8.00e-03 | ±3.88e-03 | ±8.65e-03 | ±4.77e-03 | |
Brain2 | 5.603e-01 | 3.903e-01 | 3.824e-01 | 3.816e-01 | 3.782e-01 | 3.687e-01 | 4.335e-01 | 3.898e-01 |
±2.84e-02 | ±2.15e-02 | ±2.43e-02 | ±2.91e-02 | ±2.12e-02 | ±1.66e-02 | ±2.82e-02 | ±1.83e-02 | |
Prostate | 6.456e-01 | 4.629e-01 | 4.599e-01 | 4.574e-01 | 4.559e-01 | 4.419e-01 | 5.204e-01 | 4.693e-01 |
±2.38e-02 | ±1.29e-02 | ±1.51e-02 | ±1.58e-02 | ±1.16e-02 | ±8.71e-03 | ±1.53e-02 | ±1.52e-02 |
Dataset | MTDEA | NSGA-II | MOEA/D | MOEA/AWA | MOEA/HD | SparseEA | DAEA | PRDH |
---|---|---|---|---|---|---|---|---|
ORL | 1.375e-01 | ⋆ 1.442e-01 | ⋆ 1.437e-01 | 1.475e-01 | 1.517e-01 | 1.471e-01 | ⋆ 1.304e-01 | 1.487e-01 |
±1.49e-02 | ±2.01e-02 | ±1.35e-02 | ±1.30e-02 | ±2.03e-02 | ±1.33e-02 | ±1.49e-02 | ±1.39e-02 | |
Yale | 2.944e-01 | 3.478e-01 | 3.622e-01 | 3.367e-01 | 3.400e-01 | ⋆ 3.133e-01 | ⋆ 2.989e-01 | 3.500e-01 |
±3.60e-02 | ±2.53e-02 | ±3.82e-02 | ±3.90e-02 | ±5.20e-02 | ±2.78e-02 | ±3.01e-02 | ±3.13e-02 | |
Colon | 1.421e-01 | 2.132e-01 | 2.079e-01 | 2.079e-01 | 2.342e-01 | 1.974e-01 | 1.684e-01 | 2.132e-01 |
±4.86e-02 | ±4.35e-02 | ±6.50e-02 | ±5.53e-02 | ±5.26e-02 | ±3.77e-02 | ±5.01e-02 | ±4.97e-02 | |
SRBCT | 2.840e-01 | 6.400e-01 | 6.400e-01 | 6.400e-01 | 6.400e-01 | 6.400e-01 | 6.400e-01 | 6.400e-01 |
±2.15e-01 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | |
AR10P | 4.688e-01 | ⋆ 4.900e-01 | 5.200e-01 | 4.962e-01 | 5.150e-01 | 5.075e-01 | ⋆ 4.700e-01 | 5.063e-01 |
±3.52e-02 | ±3.38e-02 | ±3.77e-02 | ±2.47e-02 | ±3.08e-02 | ±2.00e-02 | ±3.10e-02 | ±3.02e-02 | |
PIE10P | 7.750e-02 | 9.667e-02 | 1.033e-01 | 9.583e-02 | 1.017e-01 | 1.017e-01 | ⋆ 8.583e-02 | 1.058e-01 |
±2.25e-02 | ±1.28e-02 | ±2.27e-02 | ±1.42e-02 | ±1.42e-02 | ±1.07e-02 | ±1.82e-02 | ±1.56e-02 | |
Lymphoma | 1.200e-01 | 1.350e-01 | ⋆ 1.300e-01 | ⋆ 1.333e-01 | ⋆ 1.300e-01 | ⋆ 1.300e-01 | ⋆ 1.250e-01 | 1.333e-01 |
±2.27e-02 | ±1.31e-02 | ±1.49e-02 | ±2.16e-02 | ±1.03e-02 | ±1.84e-02 | ±2.39e-02 | ±1.08e-02 | |
TOX171 | 2.038e-01 | 2.236e-01 | 2.406e-01 | 2.330e-01 | 2.274e-01 | ⋆ 2.132e-01 | ⋆ 2.179e-01 | 2.208e-01 |
±2.97e-02 | ±1.53e-02 | ±3.56e-02 | ±2.06e-02 | ±2.84e-02 | ±2.38e-02 | ±2.84e-02 | ±2.04e-02 | |
DLBCL | 7.174e-02 | ⋆ 5.870e-02 | ⋆ 7.174e-02 | ⋆ 6.957e-02 | ⋆ 6.304e-02 | 3.913e-02 | 4.130e-02 | 4.130e-02 |
±3.80e-02 | ±3.53e-02 | ±2.92e-02 | ±3.28e-02 | ±2.63e-02 | ±3.43e-02 | ±2.98e-02 | ±2.98e-02 | |
Brain1 | 2.593e-01 | ⋆ 2.593e-01 | ⋆ 2.593e-01 | ⋆ 2.593e-01 | ⋆ 2.593e-01 | ⋆ 2.593e-01 | ⋆ 2.593e-01 | ⋆ 2.593e-01 |
±1.20e-02 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | |
Leukemia | 1.091e-01 | 1.318e-01 | 1.455e-01 | 1.364e-01 | 1.455e-01 | ⋆ 1.273e-01 | ⋆ 1.205e-01 | ⋆ 1.227e-01 |
±3.73e-02 | ±1.40e-02 | ±2.80e-02 | ±2.95e-02 | ±2.80e-02 | ±1.87e-02 | ±2.67e-02 | ±3.64e-02 | |
CNS | 4.000e-01 | ⋆ 4.139e-01 | ⋆ 4.417e-01 | ⋆ 4.167e-01 | ⋆ 4.167e-01 | ⋆ 3.833e-01 | ⋆ 3.833e-01 | ⋆ 4.083e-01 |
±7.77e-02 | ±6.11e-02 | ±5.55e-02 | ±6.11e-02 | ±5.84e-02 | ±3.07e-02 | ±4.73e-02 | ±5.49e-02 | |
ALLAML | 1.455e-01 | ⋆ 1.568e-01 | ⋆ 1.591e-01 | 1.727e-01 | ⋆ 1.636e-01 | ⋆ 1.500e-01 | ⋆ 1.500e-01 | ⋆ 1.523e-01 |
±3.79e-02 | ±2.75e-02 | ±2.33e-02 | ±2.80e-02 | ±2.28e-02 | ±2.99e-02 | ±2.60e-02 | ±2.22e-02 | |
Carcinom | 1.356e-01 | ⋆ 1.423e-01 | 1.500e-01 | ⋆ 1.394e-01 | 1.500e-01 | ⋆ 1.385e-01 | ⋆ 1.327e-01 | 1.471e-01 |
±1.59e-02 | ±1.91e-02 | ±2.30e-02 | ±1.51e-02 | ±2.03e-02 | ±1.60e-02 | ±1.64e-02 | ±1.29e-02 | |
Nci9 | 6.289e-01 | 6.579e-01 | ⋆ 6.342e-01 | ⋆ 6.368e-01 | ⋆ 6.421e-01 | 6.553e-01 | ⋆ 6.316e-01 | ⋆ 6.526e-01 |
±3.18e-02 | ±4.68e-02 | ±5.26e-02 | ±5.09e-02 | ±4.39e-02 | ±4.00e-02 | ±4.83e-02 | ±4.95e-02 | |
Arcene | 4.333e-01 | ⋆ 4.333e-01 | ⋆ 4.333e-01 | ⋆ 4.333e-01 | ⋆ 4.333e-01 | ⋆ 4.333e-01 | ⋆ 4.333e-01 | ⋆ 4.333e-01 |
±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | ±1.14e-16 | |
Pixraw10P | 3.333e-02 | ⋆ 3.333e-02 | ⋆ 3.333e-02 | ⋆ 3.333e-02 | ⋆ 3.333e-02 | ⋆ 3.333e-02 | ⋆ 3.333e-02 | ⋆ 3.333e-02 |
±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | ±0.00e+00 | |
Orlraws10P | 1.033e-01 | ⋆ 1.050e-01 | ⋆ 1.117e-01 | ⋆ 1.067e-01 | ⋆ 1.050e-01 | ⋆ 1.017e-01 | ⋆ 1.033e-01 | ⋆ 1.017e-01 |
±1.84e-02 | ±1.22e-02 | ±1.63e-02 | ±1.37e-02 | ±1.22e-02 | ±7.45e-03 | ±1.03e-02 | ±7.45e-03 | |
Brain2 | 3.500e-01 | 3.767e-01 | 3.967e-01 | 3.833e-01 | 3.900e-01 | ⋆ 3.700e-01 | ⋆ 3.733e-01 | 3.833e-01 |
±3.67e-02 | ±3.91e-02 | ±4.03e-02 | ±5.24e-02 | ±3.91e-02 | ±3.40e-02 | ±4.54e-02 | ±2.96e-02 | |
Prostate | 2.355e-01 | ⋆ 2.419e-01 | 2.613e-01 | ⋆ 2.387e-01 | ⋆ 2.435e-01 | ⋆ 2.274e-01 | ⋆ 2.258e-01 | ⋆ 2.371e-01 |
±3.33e-02 | ±2.45e-02 | ±2.94e-02 | ±2.65e-02 | ±1.95e-02 | ±1.65e-02 | ±2.56e-02 | ±2.82e-02 |
Dataset | MTDEA | NSGA-II | MOEA/D | MOEA/AWA | MOEA/HD | SparseEA | DAEA | PRDH |
---|---|---|---|---|---|---|---|---|
ORL | 1.301e+02 | 3.498e+02 | 2.771e+02 | 3.257e+02 | 3.516e+02 | 4.001e+02 | 2.272e+02 | 3.054e+02 |
±2.37e+01 | ±2.55e+01 | ±1.95e+01 | ±8.52e+01 | ±1.56e+01 | ±3.34e+01 | ±3.23e+01 | ±1.35e+01 | |
Yale | 1.368e+02 | 3.340e+02 | 2.682e+02 | 3.149e+02 | 3.291e+02 | 3.848e+02 | 2.311e+02 | 2.966e+02 |
±2.19e+01 | ±2.99e+01 | ±1.46e+01 | ±5.93e+01 | ±1.68e+01 | ±2.03e+01 | ±4.03e+01 | ±1.94e+01 | |
Colon | 2.518e+02 | 6.990e+02 | 5.509e+02 | 7.224e+02 | 7.374e+02 | 8.752e+02 | 4.804e+02 | 6.866e+02 |
±2.74e+01 | ±1.50e+01 | ±5.59e+01 | ±4.67e+01 | ±2.43e+01 | ±3.37e+01 | ±3.17e+01 | ±1.98e+01 | |
SRBCT | 2.712e+02 | 8.142e+02 | 6.096e+02 | 7.275e+02 | 9.884e+02 | 1.034e+03 | 7.004e+02 | 8.581e+02 |
±5.61e+01 | ±1.24e+01 | ±1.23e+01 | ±1.66e+01 | ±1.10e+01 | ±1.03e+01 | ±1.19e+02 | ±1.93e+01 | |
AR10P | 3.889e+02 | 9.154e+02 | 7.826e+02 | 9.657e+02 | 9.311e+02 | 1.081e+03 | 6.920e+02 | 8.866e+02 |
±4.58e+01 | ±1.77e+01 | ±2.61e+01 | ±5.67e+01 | ±2.85e+01 | ±3.27e+01 | ±7.48e+01 | ±1.65e+01 | |
PIE10P | 3.518e+02 | 9.076e+02 | 7.648e+02 | 9.489e+02 | 9.400e+02 | 1.085e+03 | 6.721e+02 | 8.814e+02 |
±3.11e+01 | ±2.51e+01 | ±2.78e+01 | ±8.51e+01 | ±2.50e+01 | ±2.95e+01 | ±3.75e+01 | ±1.65e+01 | |
Lymphoma | 6.319e+02 | 1.600e+03 | 1.412e+03 | 1.607e+03 | 1.688e+03 | 1.890e+03 | 1.239e+03 | 1.598e+03 |
±4.38e+01 | ±2.11e+01 | ±2.43e+01 | ±7.12e+01 | ±4.38e+01 | ±1.56e+01 | ±7.37e+01 | ±2.49e+01 | |
TOX171 | 1.126e+03 | 2.510e+03 | 2.376e+03 | 2.594e+03 | 2.530e+03 | 2.768e+03 | 2.064e+03 | 2.465e+03 |
±7.98e+01 | ±6.23e+01 | ±3.48e+01 | ±8.09e+01 | ±3.03e+01 | ±3.54e+01 | ±6.63e+01 | ±4.24e+01 | |
DLBCL | 8.355e+02 | 2.300e+03 | 2.154e+03 | 2.342e+03 | 2.361e+03 | 2.626e+03 | 1.838e+03 | 2.287e+03 |
±5.62e+01 | ±2.54e+01 | ±9.42e+01 | ±7.18e+01 | ±4.26e+01 | ±2.31e+01 | ±7.16e+01 | ±2.78e+01 | |
Brain1 | 9.804e+02 | 2.492e+03 | 2.332e+03 | 2.544e+03 | 2.648e+03 | 2.838e+03 | 2.143e+03 | 2.470e+03 |
±9.27e+01 | ±2.65e+01 | ±9.29e+01 | ±9.00e+01 | ±8.53e+01 | ±1.52e+01 | ±7.27e+01 | ±2.19e+01 | |
Leukemia | 1.201e+03 | 3.041e+03 | 2.893e+03 | 3.081e+03 | 3.184e+03 | 3.423e+03 | 2.541e+03 | 3.010e+03 |
±5.41e+01 | ±3.44e+01 | ±1.06e+02 | ±9.67e+01 | ±5.71e+01 | ±3.21e+01 | ±8.82e+01 | ±4.22e+01 | |
CNS | 1.375e+03 | 3.102e+03 | 2.939e+03 | 3.193e+03 | 3.172e+03 | 3.449e+03 | 2.550e+03 | 3.064e+03 |
±8.44e+01 | ±7.67e+01 | ±6.44e+01 | ±8.03e+01 | ±4.51e+01 | ±4.15e+01 | ±5.91e+01 | ±8.10e+01 | |
ALLAML | 1.273e+03 | 3.085e+03 | 2.930e+03 | 3.112e+03 | 3.182e+03 | 3.450e+03 | 2.566e+03 | 3.043e+03 |
±7.06e+01 | ±2.61e+01 | ±4.67e+01 | ±7.95e+01 | ±4.63e+01 | ±3.33e+01 | ±5.94e+01 | ±4.01e+01 | |
Carcinom | 1.853e+03 | 4.097e+03 | 3.981e+03 | 4.240e+03 | 4.149e+03 | 4.493e+03 | 3.480e+03 | 4.079e+03 |
±9.64e+01 | ±3.82e+01 | ±6.53e+01 | ±7.39e+01 | ±3.06e+01 | ±3.26e+01 | ±1.01e+02 | ±8.56e+01 | |
Nci9 | 1.777e+03 | 4.288e+03 | 4.083e+03 | 4.230e+03 | 4.600e+03 | 4.727e+03 | 3.890e+03 | 4.245e+03 |
±1.07e+02 | ±3.25e+01 | ±3.29e+01 | ±2.61e+01 | ±2.86e+01 | ±1.76e+01 | ±4.30e+01 | ±1.69e+01 | |
Arcene | 1.686e+03 | 4.421e+03 | 4.240e+03 | 4.377e+03 | 4.748e+03 | 4.875e+03 | 3.996e+03 | 4.380e+03 |
±7.54e+01 | ±1.99e+01 | ±3.35e+01 | ±4.04e+01 | ±2.25e+01 | ±2.34e+01 | ±5.00e+01 | ±3.10e+01 | |
Pixraw10P | 1.807e+03 | 4.426e+03 | 4.295e+03 | 4.452e+03 | 4.603e+03 | 4.866e+03 | 3.823e+03 | 4.368e+03 |
±7.19e+01 | ±2.52e+01 | ±1.12e+02 | ±8.04e+01 | ±8.66e+01 | ±1.79e+01 | ±1.06e+02 | ±3.00e+01 | |
Orlraws10P | 1.971e+03 | 4.581e+03 | 4.463e+03 | 4.648e+03 | 4.697e+03 | 5.023e+03 | 3.890e+03 | 4.536e+03 |
±6.24e+01 | ±3.63e+01 | ±4.45e+01 | ±8.97e+01 | ±5.78e+01 | ±2.67e+01 | ±7.47e+01 | ±3.49e+01 | |
Brain2 | 2.038e+03 | 4.642e+03 | 4.584e+03 | 4.734e+03 | 4.729e+03 | 5.084e+03 | 3.935e+03 | 4.591e+03 |
±7.39e+01 | ±4.55e+01 | ±1.21e+02 | ±9.91e+01 | ±4.01e+01 | ±3.71e+01 | ±8.19e+01 | ±4.58e+01 | |
Prostate | 2.081e+03 | 4.713e+03 | 4.587e+03 | 4.820e+03 | 4.796e+03 | 5.154e+03 | 4.024e+03 | 4.670e+03 |
±9.25e+01 | ±5.72e+01 | ±5.63e+01 | ±8.07e+01 | ±4.31e+01 | ±4.61e+01 | ±1.16e+02 | ±6.92e+01 |
Dataset | HV Metric | MCE Metric | NSF Metric | |||
---|---|---|---|---|---|---|
MTDEA | Baseline | MTDEA | Baseline | MTDEA | Baseline | |
ORL | 7.9280e-01 | 6.2015e-01 | 1.3750e-01 | ⋆ 1.4250e-01 | 1.3005e+02 | 3.3955e+02 |
±1.797e-02 | ±2.020e-02 | ±1.493e-02 | ±1.402e-02 | ±2.373e+01 | ±2.964e+01 | |
Yale | 6.5971e-01 | 4.9766e-01 | 2.9444e-01 | 3.3556e-01 | 1.3675e+02 | 3.3045e+02 |
±3.181e-02 | ±2.479e-02 | ±3.596e-02 | ±3.808e-02 | ±2.190e+01 | ±1.961e+01 | |
Colon | 7.7459e-01 | 5.4779e-01 | 1.4211e-01 | 2.1316e-01 | 2.5180e+02 | 7.0630e+02 |
±3.839e-02 | ±2.682e-02 | ±4.860e-02 | ±4.345e-02 | ±2.739e+01 | ±1.718e+01 | |
SRBCT | 6.6537e-01 | 2.8661e-01 | 2.8400e-01 | 6.4000e-01 | 2.7120e+02 | 7.9880e+02 |
±1.696e-01 | ±1.828e-03 | ±2.152e-01 | ±1.139e-16 | ±5.614e+01 | ±1.110e+01 | |
AR10P | 4.9725e-01 | 3.5777e-01 | 4.6875e-01 | 4.9750e-01 | 3.8895e+02 | 9.2410e+02 |
±2.744e-02 | ±1.439e-02 | ±3.524e-02 | ±2.420e-02 | ±4.576e+01 | ±2.474e+01 | |
PIE10P | 8.1273e-01 | 5.9713e-01 | 7.7500e-02 | 1.0083e-01 | 3.5175e+02 | 9.1285e+02 |
±2.001e-02 | ±9.652e-03 | ±2.247e-02 | ±1.144e-02 | ±3.107e+01 | ±2.042e+01 | |
Lymphoma | 7.6449e-01 | 5.6584e-01 | 1.2000e-01 | ⋆ 1.3167e-01 | 6.3190e+02 | 1.5825e+03 |
±2.178e-02 | ±8.253e-03 | ±2.269e-02 | ±1.701e-02 | ±4.382e+01 | ±2.530e+01 | |
TOX171 | 6.7490e-01 | 4.8274e-01 | 2.0377e-01 | 2.2736e-01 | 1.1264e+03 | 2.5053e+03 |
±2.214e-02 | ±1.347e-02 | ±2.974e-02 | ±2.247e-02 | ±7.979e+01 | ±8.390e+01 | |
DLBCL | 8.0543e-01 | 5.7887e-01 | 7.1739e-02 | ⋆ 7.6087e-02 | 8.3550e+02 | 2.2779e+03 |
±3.059e-02 | ±2.118e-02 | ±3.805e-02 | ±3.419e-02 | ±5.622e+01 | ±3.618e+01 | |
Brain1 | 6.4956e-01 | 4.7324e-01 | 2.5926e-01 | ⋆ 2.5926e-01 | 9.8040e+02 | 2.4799e+03 |
±1.384e-02 | ±5.628e-03 | ±1.202e-02 | ±0.000e+00 | ±9.274e+01 | ±4.795e+01 | |
Leukemia | 7.6257e-01 | 5.4152e-01 | 1.0909e-01 | ⋆ 1.3182e-01 | 1.2007e+03 | 2.9922e+03 |
±2.699e-02 | ±1.669e-02 | ±3.731e-02 | ±2.912e-02 | ±5.407e+01 | ±3.736e+01 | |
CNS | 5.2552e-01 | 3.7420e-01 | 4.0000e-01 | ⋆ 4.2500e-01 | 1.3754e+03 | 3.0789e+03 |
±5.878e-02 | ±2.658e-02 | ±7.774e-02 | ±4.862e-02 | ±8.443e+01 | ±6.460e+01 | |
ALLAML | 7.2778e-01 | 5.2572e-01 | 1.4545e-01 | ⋆ 1.5227e-01 | 1.2732e+03 | 3.0622e+03 |
±2.933e-02 | ±1.071e-02 | ±3.789e-02 | ±2.224e-02 | ±7.057e+01 | ±4.950e+01 | |
Carcinom | 7.1986e-01 | 5.2429e-01 | 1.3558e-01 | ⋆ 1.3462e-01 | 1.8533e+03 | 4.0858e+03 |
±1.404e-02 | ±8.217e-03 | ±1.588e-02 | ±1.395e-02 | ±9.638e+01 | ±6.919e+01 | |
Nci9 | 3.5698e-01 | 2.4945e-01 | 6.2895e-01 | ⋆ 6.4737e-01 | 1.7768e+03 | 4.2062e+03 |
±2.326e-02 | ±2.105e-02 | ±3.183e-02 | ±3.856e-02 | ±1.071e+02 | ±4.303e+01 | |
Arcene | 5.1318e-01 | 3.6631e-01 | 4.3333e-01 | ⋆ 4.3333e-01 | 1.6858e+03 | 4.3514e+03 |
±4.154e-03 | ±1.934e-03 | ±1.139e-16 | ±1.139e-16 | ±7.540e+01 | ±3.511e+01 | |
Pixraw10P | 8.1039e-01 | 5.8583e-01 | 3.3333e-02 | ⋆ 3.3333e-02 | 1.8071e+03 | 4.3545e+03 |
±6.335e-03 | ±4.056e-03 | ±0.000e+00 | ±0.000e+00 | ±7.186e+01 | ±4.601e+01 | |
Orlraws10P | 7.4909e-01 | 5.4469e-01 | 1.0333e-01 | ⋆ 1.0333e-01 | 1.9710e+03 | 4.5222e+03 |
±1.430e-02 | ±6.752e-03 | ±1.842e-02 | ±1.026e-02 | ±6.243e+01 | ±3.812e+01 | |
Brain2 | 5.6026e-01 | 3.8877e-01 | 3.5000e-01 | 3.8333e-01 | 2.0377e+03 | 4.6294e+03 |
±2.838e-02 | ±1.692e-02 | ±3.667e-02 | ±2.962e-02 | ±7.393e+01 | ±8.246e+01 | |
Prostate | 6.4555e-01 | 4.6144e-01 | 2.3548e-01 | ⋆ 2.4839e-01 | 2.0812e+03 | 4.6716e+03 |
±2.378e-02 | ±1.361e-02 | ±3.326e-02 | ±2.585e-02 | ±9.247e+01 | ±4.519e+01 |
Dataset | MTDEA | NSGA-II | MOEA/D | MOEA/AWA | MOEA/HD | SparseEA | DAEA | PRDH |
---|---|---|---|---|---|---|---|---|
ORL | 7.289e+02 | 1.071e+03 | 9.947e+02 | 9.512e+02 | 1.084e+03 | 1.065e+03 | 8.724e+02 | 1.053e+03 |
±2.43e+01 | ±3.74e+01 | ±3.94e+01 | ±3.00e+01 | ±4.59e+01 | ±2.32e+01 | ±3.81e+01 | ±2.42e+01 | |
Yale | 1.434e+02 | 1.942e+02 | 1.860e+02 | 1.910e+02 | 2.032e+02 | 2.028e+02 | 1.790e+02 | 2.022e+02 |
±4.69e+00 | ±4.97e+00 | ±3.54e+00 | ±3.57e+00 | ±6.39e+00 | ±4.44e+00 | ±4.73e+00 | ±4.74e+00 | |
Colon | 6.170e+01 | 8.630e+01 | 7.946e+01 | 8.715e+01 | 8.672e+01 | 8.419e+01 | 8.861e+01 | 9.450e+01 |
±1.06e+00 | ±1.55e+00 | ±1.58e+00 | ±1.85e+00 | ±2.60e+00 | ±1.84e+00 | ±1.74e+00 | ±1.65e+00 | |
SRBCT | 1.653e+02 | 2.327e+02 | 2.078e+02 | 2.154e+02 | 2.618e+02 | 2.254e+02 | 2.078e+02 | 2.558e+02 |
±5.26e+00 | ±5.64e+00 | ±2.87e+00 | ±5.34e+00 | ±7.01e+00 | ±5.98e+00 | ±1.03e+01 | ±3.74e+00 | |
AR10P | 4.898e+02 | 6.246e+02 | 6.008e+02 | 6.226e+02 | 6.349e+02 | 5.974e+02 | 5.872e+02 | 6.314e+02 |
±2.02e+01 | ±9.67e+00 | ±1.52e+01 | ±1.60e+01 | ±1.30e+01 | ±1.74e+01 | ±1.11e+01 | ±1.02e+01 | |
PIE10P | 9.958e+02 | 1.462e+03 | 1.382e+03 | 1.469e+03 | 1.539e+03 | 1.411e+03 | 1.198e+03 | 1.415e+03 |
±3.44e+01 | ±9.16e+01 | ±6.56e+01 | ±1.03e+02 | ±9.56e+01 | ±6.73e+01 | ±5.89e+01 | ±5.14e+01 | |
Lymphoma | 6.800e+02 | 9.464e+02 | 8.585e+02 | 9.201e+02 | 9.785e+02 | 8.230e+02 | 8.358e+02 | 8.834e+02 |
±1.93e+01 | ±5.54e+01 | ±2.14e+01 | ±5.38e+01 | ±7.62e+01 | ±5.04e+01 | ±3.06e+01 | ±1.62e+01 | |
TOX171 | 2.417e+03 | 3.248e+03 | 3.065e+03 | 3.128e+03 | 3.300e+03 | 2.301e+03 | 2.826e+03 | 3.013e+03 |
±1.96e+02 | ±2.29e+02 | ±2.00e+02 | ±2.37e+02 | ±2.51e+02 | ±1.68e+02 | ±3.77e+01 | ±1.10e+02 | |
DLBCL | 8.218e+02 | 1.135e+03 | 1.083e+03 | 1.109e+03 | 1.159e+03 | 9.196e+02 | 1.032e+03 | 1.064e+03 |
±3.99e+01 | ±7.51e+01 | ±7.73e+01 | ±7.66e+01 | ±8.80e+01 | ±7.54e+01 | ±2.52e+01 | ±1.94e+01 | |
Brain1 | 1.224e+03 | 1.731e+03 | 1.715e+03 | 1.726e+03 | 1.789e+03 | 1.266e+03 | 1.498e+03 | 1.593e+03 |
±1.08e+02 | ±1.44e+02 | ±1.56e+02 | ±1.42e+02 | ±1.59e+02 | ±1.08e+02 | ±8.45e+01 | ±6.91e+01 | |
Leukemia | 1.150e+03 | 1.648e+03 | 1.584e+03 | 1.705e+03 | 1.708e+03 | 1.080e+03 | 1.379e+03 | 1.537e+03 |
±9.20e+01 | ±1.36e+02 | ±1.28e+02 | ±1.89e+02 | ±1.28e+02 | ±9.02e+01 | ±5.30e+01 | ±8.69e+01 | |
CNS | 9.885e+02 | 1.199e+03 | 1.145e+03 | 1.161e+03 | 1.217e+03 | 8.565e+02 | 1.077e+03 | 1.122e+03 |
±8.98e+01 | ±8.43e+01 | ±8.29e+01 | ±8.87e+01 | ±9.35e+01 | ±6.03e+01 | ±2.52e+01 | ±2.03e+01 | |
ALLAML | 1.175e+03 | 1.677e+03 | 1.648e+03 | 1.705e+03 | 1.731e+03 | 1.086e+03 | 1.407e+03 | 1.558e+03 |
±9.54e+01 | ±1.43e+02 | ±1.51e+02 | ±1.90e+02 | ±1.42e+02 | ±8.30e+01 | ±5.50e+01 | ±8.15e+01 | |
Carcinom | 4.527e+03 | 5.767e+03 | 5.719e+03 | 5.692e+03 | 5.860e+03 | 3.349e+03 | 4.934e+03 | 5.280e+03 |
±4.36e+02 | ±5.05e+02 | ±4.51e+02 | ±4.65e+02 | ±6.06e+02 | ±3.03e+01 | ±1.35e+02 | ±2.00e+02 | |
Nci9 | 1.348e+03 | 1.940e+03 | 1.891e+03 | 1.926e+03 | 2.276e+03 | 1.228e+03 | 1.818e+03 | 1.867e+03 |
±9.02e+01 | ±1.58e+02 | ±9.31e+01 | ±8.43e+01 | ±3.22e+02 | ±1.69e+02 | ±4.41e+01 | ±1.14e+02 | |
Arcene | 5.231e+03 | 7.230e+03 | 6.937e+03 | 7.047e+03 | 7.745e+03 | 4.329e+03 | 8.062e+03 | 6.646e+03 |
±1.97e+02 | ±5.42e+02 | ±4.00e+02 | ±3.64e+02 | ±9.04e+02 | ±4.43e+02 | ±1.38e+03 | ±3.93e+02 | |
Pixraw10P | 2.564e+03 | 3.430e+03 | 3.382e+03 | 3.429e+03 | 3.694e+03 | 1.989e+03 | 3.845e+03 | 3.276e+03 |
±1.83e+02 | ±1.94e+02 | ±2.31e+02 | ±2.45e+02 | ±3.92e+02 | ±2.61e+02 | ±5.98e+02 | ±2.56e+02 | |
Orlraws10P | 2.693e+03 | 3.628e+03 | 3.547e+03 | 3.598e+03 | 4.208e+03 | ⋆ 2.602e+03 | 3.944e+03 | 3.400e+03 |
±1.63e+02 | ±3.12e+02 | ±2.06e+02 | ±2.57e+02 | ±8.63e+02 | ±2.36e+02 | ±6.29e+02 | ±2.76e+02 | |
Brain2 | 1.230e+03 | 1.789e+03 | 1.708e+03 | 1.723e+03 | 1.978e+03 | 1.170e+03 | 1.775e+03 | 1.661e+03 |
±6.56e+01 | ±1.49e+02 | ±1.66e+02 | ±1.44e+02 | ±3.51e+02 | ±6.17e+01 | ±2.42e+02 | ±1.03e+02 | |
Prostate | 2.857e+03 | 3.906e+03 | 3.762e+03 | 3.775e+03 | 4.505e+03 | ⋆ 2.676e+03 | 3.546e+03 | 3.426e+03 |
±1.36e+02 | ±1.63e+02 | ±1.51e+02 | ±2.99e+02 | ±1.02e+03 | ±7.03e+02 | ±1.71e+02 | ±1.46e+02 |
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Xu, H.; Huang, C.; Lin, J.; Lin, M.; Zhang, H.; Xu, R. A Multi-Task Decomposition-Based Evolutionary Algorithm for Tackling High-Dimensional Bi-Objective Feature Selection. Mathematics 2024, 12, 1178. https://doi.org/10.3390/math12081178
Xu H, Huang C, Lin J, Lin M, Zhang H, Xu R. A Multi-Task Decomposition-Based Evolutionary Algorithm for Tackling High-Dimensional Bi-Objective Feature Selection. Mathematics. 2024; 12(8):1178. https://doi.org/10.3390/math12081178
Chicago/Turabian StyleXu, Hang, Chaohui Huang, Jianbing Lin, Min Lin, Huahui Zhang, and Rongbin Xu. 2024. "A Multi-Task Decomposition-Based Evolutionary Algorithm for Tackling High-Dimensional Bi-Objective Feature Selection" Mathematics 12, no. 8: 1178. https://doi.org/10.3390/math12081178