Title: Hyperspherical Variational Autoencoders Leveraging an Efficient Spherical Cauchy Distribution

Abstract:

This paper introduces spherical Cauchy (spCauchy) latent variables designed for variational autoencoders operating within hyperspherical latent spaces. The spCauchy distribution is characterized by heavy-tailed global dynamics and allows for an exact, differentiable reparameterization technique. This is achieved by applying a M\"obius transformation to uniform samples drawn from the sphere. We demonstrate that as the concentration increases, the spCauchy distribution converges to the local tangent-space geometry of the von Mises-Fisher (vMF) distribution, facilitated by an explicit mapping of the concentration parameter. Crucially, this approach sidesteps the computationally expensive high-order Bessel-function calculations typically required by vMF implementations.

During the training phase, the Kullback-Leibler (KL) divergence relative to a uniform spherical prior can be computed using rapidly converging series, stable quadrature methods, and asymptotic forms for high-concentration scenarios. We also prove the monotonicity of the KL core with respect to concentration and provide analytic bounds using closed-form surrogates with rigorous error control, ensuring stable approximations even in extreme conditions. Performance benchmarks reveal that the latent-layer objective based on spCauchy is more stable and computationally faster than vMF baselines on both CPU and GPU architectures. Empirical results from experiments involving image and molecular sequence data confirm that spCauchy-VAEs offer a robust and scalable solution for generative modeling tasks utilizing hyperspherical latent representations.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC