Message

Let X = {X1, X2, ..., XN} be a random vector that follows multivariate Gaussian distribution, i.e., X ∼ N (μ,Σ). Directly sampling from the multivariate Gaussian distribution may be challenging. This challenge can be mitigated with the re-parameterization trick. Let Z be a random vector that follows the standard multivariate Gaussian distribution, i.e., Z ∼ N (0, I), where I is the identity covariance matrix. We can easily prove that x = Lz +μ, where L is the lower triangle matrix resulted from Cholesky decomposition of Σ, i.e., Σ = LL . As Z follows standard multivariate Gaussian distribution, we can sample each element of Z independently, yielding a sample zs, based on which we obtain the corresponding sample for X as xs = Lzs + μ.