Suppose $v \in \mathbb{R}^n$, $\|v\|=1$, and $v^*=v+\epsilon$ be a noise corrupted version, with $\epsilon \in \mathbb{R}^n$ a random vector with entries Gaussian(0,1).
Can I derive an expression for the error obtained when projecting $v$ onto its noise-corrupted self, $\|v-\text{proj}_{v^*}(v)\|$?
My attempt:
$$\|\text{proj}_{v^*}(v)\| = |\frac{ (v, v+\epsilon)}{(v+\epsilon,v+\epsilon)}|\|v+\epsilon\|, $$
which simplifies to
$$\frac{|1+(v,\epsilon)|}{\|v+\epsilon\|}$$
I'd like to take the expected value, but I'm not sure how to simplify then. Any pointers?