Maximum likelihood estimation of the Fisher-Bingham distribution via efficient calculation of its normalizing constant
This work addresses computational bottlenecks in statistical estimation for directional data, with incremental improvements in efficiency for specific domains like machine learning.
The paper tackled the problem of efficiently computing the normalizing constant for high-dimensional Fisher-Bingham distributions by proposing a fast and accurate numerical integration method, enabling maximum likelihood estimation and application to hyperspherical variational auto-encoders for tasks like adding new labels to models.
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.