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.

本文提出了一个有效的数值积分公式来计算Fisher-Bingham分布的归一化常数。该公式使用数值积分公式，并将其连续Euler变换转换为归一化常数的Fourier型积分表示。由于该方法快速且准确，因此可以应用于高维Fisher-Bingham分布的归一化常数的计算。更精确地，误差随着积分点的增加而呈指数衰减，并且计算成本随尺寸线性增加。此外，该公式对于计算归一化常数的梯度和Hessian矩阵很有用。因此，我们应用此公式来有效地计算高维数据的最大似然估计（MLE）。最后，我们将MLE应用于超球面变分自动编码器（S-VAE），这是一种基于深度学习的生成模型，可将潜在空间限制为单位超球面。我们使用训练有手写数字图像的S-VAE来估计每个标签的分布。该应用程序对于在模型中添加新标签很有用。