  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 $0< \alpha <\frac 1 2$ such that
an $\alpha$-fraction of the points in $T$ are i.i.d. samples from a
well-behaved distribution $\mathcal{D}$ and the remaining $(1-\alpha)$-fraction
are arbitrary. The goal is to output a small list of vectors, at least one of
which is close to the mean of $\mathcal{D}$. We develop new algorithms for
list-decodable mean estimation, achieving nearly-optimal statistical
guarantees, with running time $O(n^{1 + \epsilon_0} d)$, for any fixed
$\epsilon_0 > 0$. All prior algorithms for this problem had additional
polynomial factors in $\frac 1 \alpha$. 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 $\Omega(n d k)$. 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 $\alpha \to 1$ 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.
