  Given a matrix $A \in \mathbb{R}^{m\times d}$ with singular values
$\sigma_1\geq \cdots \geq \sigma_d$, and a random matrix $G \in
\mathbb{R}^{m\times d}$ with iid $N(0,T)$ entries for some $T>0$, we derive new
bounds on the Frobenius distance between subspaces spanned by the top-$k$
(right) singular vectors of $A$ and $A+G$. This problem arises in numerous
applications in statistics where a data matrix may be corrupted by Gaussian
noise, and in the analysis of the Gaussian mechanism in differential privacy,
where Gaussian noise is added to data to preserve private information. We show
that, for matrices $A$ where the gaps in the top-$k$ singular values are
roughly $\Omega(\sigma_k-\sigma_{k+1})$ the expected Frobenius distance between
the subspaces is $\tilde{O}(\frac{\sqrt{d}}{\sigma_k-\sigma_{k+1}} \times
\sqrt{T})$, improving on previous bounds by a factor of
$\frac{\sqrt{m}}{\sqrt{d}} \sqrt{k}$. To obtain our bounds we view the
perturbation to the singular vectors as a diffusion process -- the Dyson-Bessel
process -- and use tools from stochastic calculus to track the evolution of the
subspace spanned by the top-$k$ singular vectors.
