It wants me to "unite", not "union bound" 😄

The results are based on a more general sum-of-squares dimension-reduction approach. Key idea: If you don't know how to identify the subspace you care about, try to identify its orthogonal complement! That is often much easier.

In both cases the previous best was d^k * f(w_min).

Second highlight: d^polylog(w_min) * f(w_min) algorithm for mixtures of arbitrary identical-covariance TVD-separated Gaussians. (A log in the exponent looks necessary in this case.) The algorithms are also robust to a dimension-independent fraction of corruptions.