标题： 超球面变分自编码器使用高效的球面柯西分布 显示英文标题

标题： Hyperspherical Variational Autoencoders Using Efficient Spherical Cauchy Distribution

摘要： 我们提出了一种新颖的变分自编码器（VAE）架构，该架构采用球面柯西（spCauchy）潜在分布。 与传统的高斯潜在空间或广泛使用的冯·米塞斯-费舍尔（vMF）分布不同，spCauchy提供了更自然的超球形潜在变量表示，在保持灵活性的同时更好地捕捉方向数据。 其重尾特性可防止过度正则化，确保潜在空间的有效利用，同时提供更具表现力的表示。 此外，spCauchy避免了vMF固有的数值不稳定性，这些不稳定性源于涉及贝塞尔函数的归一化常数计算。 相反，它通过莫比乌斯变换实现完全可微且高效的重参数化技巧，从而实现稳定和可扩展的训练。 KL散度可以通过一个快速收敛的幂级数进行计算，消除了与超几何函数比值评估相关的下溢或上溢问题。 这些特性使spCauchy成为VAE的一个有吸引力的替代方案，在高维生成建模中既具有理论优势又具有实际效率。 显示英文摘要

摘要： We propose a novel variational autoencoder (VAE) architecture that employs a spherical Cauchy (spCauchy) latent distribution. Unlike traditional Gaussian latent spaces or the widely used von Mises-Fisher (vMF) distribution, spCauchy provides a more natural hyperspherical representation of latent variables, better capturing directional data while maintaining flexibility. Its heavy-tailed nature prevents over-regularization, ensuring efficient latent space utilization while offering a more expressive representation. Additionally, spCauchy circumvents the numerical instabilities inherent to vMF, which arise from computing normalization constants involving Bessel functions. Instead, it enables a fully differentiable and efficient reparameterization trick via M\"obius transformations, allowing for stable and scalable training. The KL divergence can be computed through a rapidly converging power series, eliminating concerns of underflow or overflow associated with evaluation of ratios of hypergeometric functions. These properties make spCauchy a compelling alternative for VAEs, offering both theoretical advantages and practical efficiency in high-dimensional generative modeling.