Probabilistic Models on Matrix Manifolds

Wrapped Gaussian distribution on the 1-sphere is its heat kernel, which "wraps" the Gaussian distribution around the unit circle: $f_{WN} (\theta; \mu, \sigma) = \sum_{k \in \mathbb{Z}} \exp[-(\theta-\mu+2\pi k)^2/(2\sigma^2)] / \sqrt{2 \pi \sigma^2}$. Heat kernels of complete Riemannian manifolds do not have closed forms in general, see Geometric Diffusion.

von Mises distribution on the 1-shpere is a closed-form approximation to the wrapped Gaussian distribution; von Mises–Fisher distribution [@Fisher1953] generalizes the von Mises distribution to n-spheres, and is the most commonly used distribution in directional/circular/spherical statistics: $f(x; a, b) \propto \exp(b \cos(x-a))$, where $x, a \in [0, 2\pi), b \ge 0$.