  We study the problem of list-decodable Gaussian covariance estimation. Given
a multiset $T$ of $n$ points in $\mathbb R^d$ such that an unknown $\alpha<1/2$
fraction of points in $T$ are i.i.d. samples from an unknown Gaussian
$\mathcal{N}(\mu, \Sigma)$, the goal is to output a list of $O(1/\alpha)$
hypotheses at least one of which is close to $\Sigma$ in relative Frobenius
norm. Our main result is a $\mathrm{poly}(d,1/\alpha)$ sample and time
algorithm for this task that guarantees relative Frobenius norm error of
$\mathrm{poly}(1/\alpha)$. Importantly, our algorithm relies purely on spectral
techniques. As a corollary, we obtain an efficient spectral algorithm for
robust partial clustering of Gaussian mixture models (GMMs) -- a key ingredient
in the recent work of [BDJ+22] on robustly learning arbitrary GMMs. Combined
with the other components of [BDJ+22], our new method yields the first
Sum-of-Squares-free algorithm for robustly learning GMMs. At the technical
level, we develop a novel multi-filtering method for list-decodable covariance
estimation that may be useful in other settings.
