  In this work we solve the problem of robustly learning a high-dimensional
Gaussian mixture model with $k$ components from $\epsilon$-corrupted samples up
to accuracy $\widetilde{O}(\epsilon)$ in total variation distance for any
constant $k$ and with mild assumptions on the mixture. This robustness
guarantee is optimal up to polylogarithmic factors. The main challenge is that
most earlier works rely on learning individual components in the mixture, but
this is impossible in our setting, at least for the types of strong robustness
guarantees we are aiming for. Instead we introduce a new framework which we
call {\em strong observability} that gives us a route to circumvent this
obstacle.
