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Consider the optimal transport map $T$ between $N(\mu_0,\Sigma_0)$ and $N(\mu_1,\Sigma_1)$. I believed that the optimal transport was given by:

$$ T(x) = \mu_1 + \Sigma_1^{1/2} \Sigma_0^{-1/2}(x-\mu_0) $$

However in Peyre's book "Computational optimal transport" and other resources they claim that the map is:

$$T(x) = \mu_1 + (\Sigma_0^{-1/2}(\Sigma_0^{1/2}\Sigma_1 \Sigma_0^{1/2})^{1/2} \Sigma_0^{-1/2})(x-\mu_0) $$

I was wondering what is wrong with the transport I wrote.

My attempt

To prove that the proposed map works I used the Monge-Ampere equation. It is a straight forward calculation and everything works nicely.

Kevin Ro
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  • See the "Quadratic normal case" here: https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)#Quadratic_normal_case – Quillo Feb 25 '23 at 17:05

1 Answers1

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Recall that for $T(x)$ to be optimal it has to be the gradient of a convex function $\phi(x).$ In your case if $T$ is the gradient of a function $\phi$ then $$ \Delta \phi = \Sigma_1^{1/2}\Sigma_0^{-1/2}, $$ but this matrix won't be in general positive definite or symmetric. Therefore you are considering cannot be optimal.

GBes
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