Suppose that $H\in R^{m \times n}$ ($m\le n$) is rank-$r$, and the compact singular value decomposition of $H$ is $H=U\Sigma V^T$, where $U \in R^{m \times r}$,$V \in R^{n \times r}$, $\Sigma \in R^{r \times r}$.
Assume the $Y=H+N$, where the elements in noise $N$ are i.i.d. Gaussian with $ \mathcal{N}(0,\sigma^2)$.
The the $U_Y \in R^{m \times r}$ is the dominant-$r$ left singular vectors of $Y$. Then the semi-unitary matrix $U_Y$ relies on the noise $N$.
I want to calculate the expectation of $|| U_Y^T H||_F^2$, where the expectation is taken over the noise $N$.