用球面柯西分布改进球形VAE，训练更稳定高效。
Hyperspherical Variational Autoencoders Using Efficient Spherical Cauchy Distribution
- 引入球面柯西分布实现可微重参数化，避免复杂计算。
- 在高浓度下逼近vMF分布，且计算速度更快。
- 适合需要稳定球形生成建模的图像与分子序列任务。
我们为超球形潜在空间中的变分自编码器提出球面柯西（spCauchy）潜在变量。spCauchy具有重尾全局特性，并通过莫比乌斯变换对球面上的均匀采样实现精确可微重参数化。在高浓度极限下，spCauchy在显式浓度参数映射下恢复了冯·米塞斯-费舍尔（vMF）分布的局部切空间几何，同时避免了vMF实现所需的高阶贝塞尔函数计算。训练时，到均匀球形先验的相对熵可通过快速收敛级数、稳定求积及高浓度渐近形式高效计算。我们进一步建立了浓度相关KL核心的单调性，并推导出解析边界与闭式代理，支持极端情形下的稳定近似。压力测试表明，该潜在层目标函数在CPU和GPU上均比vMF基线更稳定且评估更快。在图像和分子序列数据上的实验显示，spCauchy-VAE为超球形潜在表示的生成建模提供了鲁棒且可扩展的替代方案。
原文摘要 · Abstract (English)
We propose spherical Cauchy (spCauchy) latent variables for variational autoencoders on hyperspherical latent spaces. The spCauchy family has heavy-tailed global behavior and admits an exact differentiable reparameterization by applying a Möbius transformation to uniform samples on the sphere. We show that, in the high-concentration limit, spCauchy recovers the local tangent-space geometry of the von Mises-Fisher (vMF) distribution under an explicit concentration parameter mapping, while avoiding the high-order Bessel-function evaluations required by vMF implementations. For training, the Kullback-Leibler divergence to a uniform spherical prior admits rapidly convergent series, stable quadrature, and high-concentration asymptotic forms. We further establish monotonicity of the concentration-dependent KL core and derive analytic brackets with closed-form surrogates and error control, supporting stable approximation in extreme regimes. Stress-test benchmarks show that the resulting latent-layer objective remains stable and faster to evaluate than vMF baselines on CPU and GPU. Experiments on image and molecular sequence data demonstrate that spCauchy-VAEs provide a robust and scalable alternative for generative modeling with hyperspherical latent representations.
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