# ($L_2$) distance for noisy data

I'm given a subspace $V$ and a set of $n$ corrupted observations $\tilde{x}_1 = x_1 +\epsilon_1,...,\tilde{x}_n = x_n + \epsilon_n \in \mathbb{R}^D$. Assume $D$ is large and that $\epsilon_i \sim N(0, \sigma^2)$.

Is it possible to estimate the true distances $d(x_i, U)$ of the samples from subspcae $V$? $d$ is either $L_2$ distance or some robust distance measure.