Before revision:  For k=2, our algorithms run in time and samples }poly (d) with no dependence on the spread \kappa. Such a results were not known prior to our work, even for k=2. More generally, our algorithm succeeds for mixtures of any distribution that satisfies two well-studied analytic assumptions - certifiable hypercontractivity and anti-concentration .
After revision:  For k=2, our algorithms run in time and samples }poly captures important special cases such as mixed linear regression and subspace clustering. Our main conceptual contribution is to distill two simple analytic properties - certifiable hypercontractivity and anti-concentration .