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For a neural net where the output is a Gaussian distribution, the output is usually parameterized as $(\mu=O_1, \sigma^2=e^{O_2})$. That is to say, the neural net will output the mean, and also output the log of the variance $O_2$. This is the natural choice to make all the outputs fit on the real line, and works well when training the neural net.

However, I want to output a bivariate Gaussian distribution (with 2 means $\mu_1$ and $\mu_2$, 2 variances $v_1$ and $v_2$, and a correlation $\rho$). The problem is, there are several natural ways to parameterize the variances and correlation (such that everything is a real number).

To name just 2:

  1. Each parameter gets its own output. \begin{align} v_1 &= e^{O_1} \\ v_2 &= e^{O_2} \\ \rho &= tanh(O_3) \end{align}

  2. Say that \begin{align} X_1 &= Z_1 + Z_c \\ X_2 &= Z_2 + Z_c \end{align} where $Z_i\sim\mathcal{N}(0, e^{O_i})$ are "private information" and $Z_c\sim\mathcal{N}(0, e^{O_3})$ is a "shared" gaussian information. Then we will have that \begin{align} v_1 &= e^{O_1} + e^{O_3} \\ v_2 &= e^{O_2} + e^{O_3} \\ \rho &= \frac{e^{O_3}}{\sqrt{v_1 \cdot v_2}} \end{align}

Which of these ways of parameterizing the output bivariate Gaussian distribution would be best? Or is there a better way? My main criteria would be what works best when training a neural network.

Personally, I think that the second way would be better, since the choice of correlation should be closely tied to the choice of variances, and the "private information"/"shared information" model for a bivariate Gaussian seems very natural.

Finally, I'm satisfied with just the bivariate case, but I may as well ask what one could do in the multivariate case?

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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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