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For the multivariate case, you could try outputting a vector $d$ of diagonal entries and an $n\times k$ matrix $R$, and let $\Sigma = \text{diag}(d) + RR^T$. This would be easy to sample, easy to backpropagate through, and also you can trade off computation cost with expressivity (as $k \rightarrow n$, you have a full covariance matrix).

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