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Maximum likelihood estimation of the Fisher–Bingham distribution via efficient calculation of its normalizing constant

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Abstract

This paper proposes an efficient numerical integration formula to compute the normalizing constant of Fisher–Bingham distributions. This formula uses a numerical integration formula with the continuous Euler transform to a Fourier-type integral representation of the normalizing constant. As this method is fast and accurate, it can be applied to the calculation of the normalizing constant of high-dimensional Fisher–Bingham distributions. More precisely, the error decays exponentially with an increase in the integration points, and the computation cost increases linearly with the dimensions. In addition, this formula is useful for calculating the gradient and Hessian matrix of the normalizing constant. Therefore, we apply this formula to efficiently calculate the maximum likelihood estimation (MLE) of high-dimensional data. Finally, we apply the MLE to the hyperspherical variational auto-encoder (S-VAE), a deep-learning-based generative model that restricts the latent space to a unit hypersphere. We use the S-VAE trained with images of handwritten numbers to estimate the distributions of each label. This application is useful for adding new labels to the models.

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Notes

  1. https://github.com/nicola-decao/s-vae-pytorch.

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Acknowledgements

We are grateful to Taichi Kiwaki for providing access to a GPU and giving advice about VAE. We thank Kazuki Matoya for general discussion about the application of MLE to VAE. We would like to thank Shun Sato for providing some advice about numerical computation. This work was supported by all members in the mathematical informatics 3rd laboratory of the University of Tokyo. Finally, we appreciate Tomonari Sei, who joined our discussion and gave a lot of advice. Ken’ichiro Tanaka is supported by the grant-in-aid of Japan Society of the Promotion of Science with KAKENHI Grant Number 17K14241.

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Correspondence to Yici Chen.

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Chen, Y., Tanaka, K. Maximum likelihood estimation of the Fisher–Bingham distribution via efficient calculation of its normalizing constant. Stat Comput 31, 40 (2021). https://doi.org/10.1007/s11222-021-10015-9

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