Before revision:  Our algorithm takes input n samples \alpha n (  \alpha \ll 1/2) are generated i.i.d. from Gaussian distribution \mathcal{N(0,\Sigma_*) on R^d with covariance \Sigma_* of rank r }  with a isotropic rank r covariance \Pi_* (the } and the rest are arbitrary, potentially adversarial outliers.
After revision:  Our algorithm takes input n samples \alpha n ( \alpha \ll 1/2) are generated i.i.d. from (0,\Sigma_*) on R^d with covariance \Sigma_* of rank r } with a isotropic rank r covariance \Pi_* (the } and the rest are arbitrary, potentially adversarial outliers.