List-Decodable Robust Mean Estimation and Learning Mixtures of Spherical Gaussians

We study the problem of list-decodable Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for these problems. {\bf List-Decodable Mean Estimation.} Fix any $d \in \mathbb{Z}_+$ and $0< \alpha <1/2$. We design an algorithm with runtime $O (\mathrm{poly}(n/\alpha)^{d})$ that outputs a list of $O(1/\alpha)$ many candidate vectors such that with high probability one of the candidates is within $\ell_2$-distance $O(\alpha^{-1/(2d)})$ from the true mean. The only previous algorithm for this problem achieved error $\tilde O(\alpha^{-1/2})$ under second moment conditions. For $d = O(1/\epsilon)$, our algorithm runs in polynomial time and achieves error $O(\alpha^{\epsilon})$. We also give a Statistical Query lower bound suggesting that the complexity of our algorithm is qualitatively close to best possible. {\bf Learning Mixtures of Spherical Gaussians.} We give a learning algorithm for mixtures of spherical Gaussians that succeeds under significantly weaker separation assumptions compared to prior work. For the prototypical case of a uniform mixture of $k$ identity covariance Gaussians we obtain: For any $\epsilon>0$, if the pairwise separation between the means is at least $\Omega(k^{\epsilon}+\sqrt{\log(1/\delta)})$, our algorithm learns the unknown parameters within accuracy $\delta$ with sample complexity and running time $\mathrm{poly} (n, 1/\delta, (k/\epsilon)^{1/\epsilon})$. The previously best known polynomial time algorithm required separation at least $k^{1/4} \mathrm{polylog}(k/\delta)$. Our main technical contribution is a new technique, using degree-$d$ multivariate polynomials, to remove outliers from high-dimensional datasets where the majority of the points are corrupted.