8.1 Sampling

Consider a Lie group $G$ and its associated Lie algebra vector space $\mathfrak{g}$, with $k$ degrees of freedom. We wish to represent Gaussian distributions over transformations in this group. Each such distribution has a mean transformation, $\mu\in G$, and a covariance matrix $\boldsymbol{\Sigma}\in\mathbb{R}^{k\times k}$. The algebra corresponds to tangent vectors around the identity element of the group. Thus, it is natural to express a sample $x$ from the desired distribution in terms of a sample $\boldsymbol{\delta}$ drawn from a zero-mean Gaussian and the mean transformation $\mu$:

$$\boldsymbol{\delta} \in \mathcal{N}\left(\mathbf{0};\boldsymbol{\Sigma}\right)$$ (215)

$$x = \exp\left(\boldsymbol{\delta}\right)\cdot\mu$$ (216)