Derviation of the equation 17 in supplemental document

[KIMGEONUNG]

Hi, thank you for awesome work.
I'm interested in the theoretical contents in the original paper, and taking hard time to understand
and derivate the equation 17.
Can you explain about the derivation of equation 17?

[KIMGEONUNG]

Specifically,

$$y = Hx + z, ~z\sim\mathcal{N}(0, \sigma^2_y I) $$

$$ \Rightarrow \bar{y} = V^T x + \Sigma^\dagger U^{-1}z$$

$$ \Rightarrow \bar{y} = \bar{x}_0 + \Sigma^\dagger U^{-1}z$$

at last expression if $U^{-1} = I$, the eq.17 is true, but how do I know that? or
Is there somthing wrong?