A Spectral Algorithm for List-Decodable Covariance Estimation in Relative Frobenius Norm

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 $α<1/2$ fraction of points in $T$ are i.i.d. samples from an unknown Gaussian $\mathcal{N}(μ, Σ)$, the goal is to output a list of $O(1/α)$ hypotheses at least one of which is close to $Σ$ in relative Frobenius norm. Our main result is a $\mathrm{poly}(d,1/α)$ sample and time algorithm for this task that guarantees relative Frobenius norm error of $\mathrm{poly}(1/α)$. 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.