  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. At the heart of our
algorithm is a new way to relax a system of polynomial equations which
corresponds to solving an improper learning problem where we are allowed to
output a Gaussian mixture model whose weights are low-degree polynomials.
