On the Smoothed Minimum Error Entropy Criterion 
Recent studies suggest that the minimum error entropy (MEE) criterion can outperform the traditional mean square error criterion in supervised machine learning, especially in nonlinear and non-Gaussian situations. In practice, however, one has to estimate the error entropy from the samples since in...
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| Format: | Article |
| Language: | English |
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MDPI AG
2012-11-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/14/11/2311 |
| _version_ | 1798041034806001664 |
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| author | Badong Chen Jose C. Principe |
| author_facet | Badong Chen Jose C. Principe |
| author_sort | Badong Chen |
| collection | DOAJ |
| description | Recent studies suggest that the minimum error entropy (MEE) criterion can outperform the traditional mean square error criterion in supervised machine learning, especially in nonlinear and non-Gaussian situations. In practice, however, one has to estimate the error entropy from the samples since in general the analytical evaluation of error entropy is not possible. By the Parzen windowing approach, the estimated error entropy converges asymptotically to the entropy of the error plus an independent random variable whose probability density function (PDF) corresponds to the kernel function in the Parzen method. This quantity of entropy is called the smoothed error entropy, and the corresponding optimality criterion is named the smoothed MEE (SMEE) criterion. In this paper, we study theoretically the SMEE criterion in supervised machine learning where the learning machine is assumed to be nonparametric and universal. Some basic properties are presented. In particular, we show that when the smoothing factor is very small, the smoothed error entropy equals approximately the true error entropy plus a scaled version of the Fisher information of error. We also investigate how the smoothing factor affects the optimal solution. In some special situations, the optimal solution under the SMEE criterion does not change with increasing smoothing factor. In general cases, when the smoothing factor tends to infinity, minimizing the smoothed error entropy will be approximately equivalent to minimizing error variance, regardless of the conditional PDF and the kernel. |
| first_indexed | 2024-04-11T22:15:53Z |
| format | Article |
| id | doaj.art-e29ee42e112b4f128d4304ff4b3ce1aa |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T22:15:53Z |
| publishDate | 2012-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-e29ee42e112b4f128d4304ff4b3ce1aa2022-12-22T04:00:24ZengMDPI AGEntropy1099-43002012-11-0114112311232310.3390/e14112311On the Smoothed Minimum Error Entropy Criterion Badong ChenJose C. PrincipeRecent studies suggest that the minimum error entropy (MEE) criterion can outperform the traditional mean square error criterion in supervised machine learning, especially in nonlinear and non-Gaussian situations. In practice, however, one has to estimate the error entropy from the samples since in general the analytical evaluation of error entropy is not possible. By the Parzen windowing approach, the estimated error entropy converges asymptotically to the entropy of the error plus an independent random variable whose probability density function (PDF) corresponds to the kernel function in the Parzen method. This quantity of entropy is called the smoothed error entropy, and the corresponding optimality criterion is named the smoothed MEE (SMEE) criterion. In this paper, we study theoretically the SMEE criterion in supervised machine learning where the learning machine is assumed to be nonparametric and universal. Some basic properties are presented. In particular, we show that when the smoothing factor is very small, the smoothed error entropy equals approximately the true error entropy plus a scaled version of the Fisher information of error. We also investigate how the smoothing factor affects the optimal solution. In some special situations, the optimal solution under the SMEE criterion does not change with increasing smoothing factor. In general cases, when the smoothing factor tends to infinity, minimizing the smoothed error entropy will be approximately equivalent to minimizing error variance, regardless of the conditional PDF and the kernel.http://www.mdpi.com/1099-4300/14/11/2311entropysupervised machine learningminimum error entropy criterion |
| spellingShingle | Badong Chen Jose C. Principe On the Smoothed Minimum Error Entropy Criterion  Entropy entropy supervised machine learning minimum error entropy criterion |
| title | On the Smoothed Minimum Error Entropy Criterion  |
| title_full | On the Smoothed Minimum Error Entropy Criterion  |
| title_fullStr | On the Smoothed Minimum Error Entropy Criterion  |
| title_full_unstemmed | On the Smoothed Minimum Error Entropy Criterion  |
| title_short | On the Smoothed Minimum Error Entropy Criterion  |
| title_sort | on the smoothed minimum error entropy criterion amp 160 |
| topic | entropy supervised machine learning minimum error entropy criterion |
| url | http://www.mdpi.com/1099-4300/14/11/2311 |
| work_keys_str_mv | AT badongchen onthesmoothedminimumerrorentropycriterionamp160 AT josecprincipe onthesmoothedminimumerrorentropycriterionamp160 |