We study the problem of list-decodable mean estimation, where an adversary can corrupt a majority of the dataset. Specifically, we are given a set $T$ of $n$ points in $\mathbb{R}^d$ and a parameter $0< α 0$. All prior algorithms for this problem had additional polynomial factors in $\frac 1 α$. We leverage this result, together with additional techniques, to obtain the first almost-linear time algorithms for clustering mixtures of $k$ separated well-behaved distributions, nearly-matching the statistical guarantees of spectral methods. Prior clustering algorithms inherently relied on an application of $k$-PCA, thereby incurring runtimes of $Ω(n d k)$. This marks the first runtime improvement for this basic statistical problem in nearly two decades. The starting point of our approach is a novel and simpler near-linear time robust mean estimation algorithm in the $α\to 1$ regime, based on a one-shot matrix multiplicative weights-inspired potential decrease. We crucially leverage this new algorithmic framework in the context of the iterative multi-filtering technique of Diakonikolas et al. '18, '20, providing a method to simultaneously cluster and downsample points using one-dimensional projections -- thus, bypassing the $k$-PCA subroutines required by prior algorithms.

64 pages, 1 figure. v2 improves results on bounded-covariance clustering, polishes exposition