Before revision:  Concretely, our algorithm takes input an \epsilon-corrupted sample from a k-GMM and (k/\eta)} time, } outputs an approximate clustering that misclassifies at most O(k\epsilon)(\epsilon} +\eta  fraction of the points whenever every pair of components is separated by 1-\exp(- poly (k/\eta)  ) in total variation distance. 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.
After revision:  Concretely, our algorithm takes input an \epsilon-corrupted sample from a k-GMM and (k/\eta)} time, } outputs an approximate clustering that misclassifies at most O(k\epsilon)(\epsilon} +\eta fraction of the points whenever every pair of components is separated by 1-\exp(- poly (k/\eta) ) in total variation (TV) distance. Such a result was not previously known even for k=2. TV separation 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.