  We study the problem of estimating the means of well-separated mixtures when
an adversary may add arbitrary outliers. While strong guarantees are available
when the outlier fraction is significantly smaller than the minimum mixing
weight, much less is known when outliers may crowd out low-weight clusters - a
setting we refer to as list-decodable mixture learning (LD-ML). In this case,
adversarial outliers can simulate additional spurious mixture components.
Hence, if all means of the mixture must be recovered up to a small error in the
output list, the list size needs to be larger than the number of (true)
components. We propose an algorithm that obtains order-optimal error guarantees
for each mixture mean with a minimal list-size overhead, significantly
improving upon list-decodable mean estimation, the only existing method that is
applicable for LD-ML. Although improvements are observed even when the mixture
is non-separated, our algorithm achieves particularly strong guarantees when
the mixture is separated: it can leverage the mixture structure to partially
cluster the samples before carefully iterating a base learner for
list-decodable mean estimation at different scales.
