It was previously asked on here how to affine transform one multivariate Gaussian to another. In the question, it was assumed that dimensionalities of the two Gaussians matched each other. How would we do this if the dimensionalities differ? The answer to the original question no longer works, because there is now dimensional mismatch between the Cholesky factors of the two covariance matrices.
For simplicity, we'll assume centered Gaussian distributions $N(0, C_1)$ with positive definite $m \times m$ covariance matrix $C_1$ and $N(0, C_2)$ with positive definite $n \times n$ matrix covariance $C_2$. The linked answer computes Cholesky factors $L_1 L_1^\top = C_1$ and $L_2 L_2^\top = C_2$, then uses the affine transform $A = L_2 L_1^{-1}$. The transformed variables then have covariance $A C_1 A^\top = C_2$ which exactly matches the desired covariance.
One problem with this generalized setting is that the desired covariance is no longer achievable when $m < n$. In this case, I'm only looking for a solution that minimizes the KL divergence, or is optimal in some other sense. Would it make sense to compute the top-$m$ SVD approximation of $U_2 \Sigma_2 V^\top_2 = C_2$, and then use the resulting $m \times m$ matrix in place of $C_2$?