8.2 Transforming

This formulation allows convenient transformations of distributions through other group elements using the adjoint. Consider $x,y\in G$, and the distribution $\left(x,\boldsymbol{\Sigma}\right)$ with mean $x$. Consider $y\cdot\tilde{x},$ where $\tilde{x}$ is a sample drawn from $\left(x,\boldsymbol{\Sigma}\right)$:

$$\tilde{x} = \exp\left(\boldsymbol{\delta}\right)\cdot x$$ (217)

$$y\cdot\tilde{x} = y\cdot\exp\left(\boldsymbol{\delta}\right)\cdot x$$ (218)

$$= \exp\left(\mathrm{Adj}_{y}\boldsymbol{\delta}\right)\cdot y\cdot x$$ (219)

By the definition of covariance,

$$\boldsymbol{\Sigma}=\mathrm{E}\left[\boldsymbol{\delta}\cdot\boldsymbol{\delta}^{T}\right]$$ (220)

Given a linear transformation $L$, by linearity of expectation we have:

$$\mathrm{E}\left[\left(L\boldsymbol{\delta}\right)\cdot\left(L\boldsymbol{\delta}\right)^{T}\right] = \mathrm{E}\left[L\cdot\boldsymbol{\delta}\cdot\boldsymbol{\delta}^{T}\cdot L^{T}\right]$$ (221)

$$= L\cdot\mathrm{E}\left[\boldsymbol{\delta}\cdot\boldsymbol{\delta}^{T}\right]\cdot L^{T}$$ (222)

$$= L\cdot\boldsymbol{\Sigma}\cdot L^{T}$$ (223)

So we can express the parameters of the transformed distribution:

$$y\cdot\tilde{x}\in\mathcal{N}\left(y\cdot x;\:\mathrm{Adj}_{y}\cdot\boldsymbol{\Sigma}\cdot\mathbf{\mathrm{Adj}}_{y}^{T}\right)$$ (224)

Thus the mean of the transformed distribution is the transformed mean, and the covariance is mapped linearly by the adjoint.