Hyperspherical Variational Autoencoders Using Efficient Spherical Cauchy Distribution

Abstract

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 M\"obius 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.