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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2202.13059 |
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| _version_ | 1866912198333300736 |
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| author | Furuya, Takashi Kusumoto, Hiroyuki Taniguchi, Koichi Kanno, Naoya Suetake, Kazuma |
| author_facet | Furuya, Takashi Kusumoto, Hiroyuki Taniguchi, Koichi Kanno, Naoya Suetake, Kazuma |
| contents | Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated analytically. In this paper, we study the approximate entropy represented as the sum of the entropies of unimodal Gaussian distributions with mixing coefficients. This approximation is easy to calculate analytically regardless of dimension, but there is a lack of theoretical guarantees. We theoretically analyze the approximation error between the true and the approximate entropy to reveal when this approximation works effectively. This error is essentially controlled by how far apart each Gaussian component of the Gaussian mixture is. To measure such separation, we introduce the ratios of the distances between the means to the sum of the variances of each Gaussian component of the Gaussian mixture, and we reveal that the error converges to zero as the ratios tend to infinity. In addition, the probabilistic estimate indicates that this convergence situation is more likely to occur in higher-dimensional spaces. Therefore, our results provide a guarantee that this approximation works well for high-dimensional problems, such as neural networks that involve a large number of parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_13059 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Theoretical Error Analysis of Entropy Approximation for Gaussian Mixtures Furuya, Takashi Kusumoto, Hiroyuki Taniguchi, Koichi Kanno, Naoya Suetake, Kazuma Machine Learning Gaussian mixture distributions are commonly employed to represent general probability distributions. Despite the importance of using Gaussian mixtures for uncertainty estimation, the entropy of a Gaussian mixture cannot be calculated analytically. In this paper, we study the approximate entropy represented as the sum of the entropies of unimodal Gaussian distributions with mixing coefficients. This approximation is easy to calculate analytically regardless of dimension, but there is a lack of theoretical guarantees. We theoretically analyze the approximation error between the true and the approximate entropy to reveal when this approximation works effectively. This error is essentially controlled by how far apart each Gaussian component of the Gaussian mixture is. To measure such separation, we introduce the ratios of the distances between the means to the sum of the variances of each Gaussian component of the Gaussian mixture, and we reveal that the error converges to zero as the ratios tend to infinity. In addition, the probabilistic estimate indicates that this convergence situation is more likely to occur in higher-dimensional spaces. Therefore, our results provide a guarantee that this approximation works well for high-dimensional problems, such as neural networks that involve a large number of parameters. |
| title | Theoretical Error Analysis of Entropy Approximation for Gaussian Mixtures |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2202.13059 |