Before revision:  This is the statistically weakest possible notion of separation and allows, for e.g., clusteringof mixtures with components with the same mean with covariances differing in a single unknown direction or separated in Frobenius distance. The running time of our algorithm is d^{O(\log(\kappa)) \textrm{poly(k/\eta)} where \kappa is a measure of spread of the mixture in any direction.
After revision:  This is the statistically weakest possible notion of separation and (k/\eta)} where \kappa is a measure of spread of the mixture in any direction.