  We study the problem of robust mean estimation and introduce a novel Hamming
distance-based measure of distribution shift for coordinate-level corruptions.
We show that this measure yields adversary models that capture more realistic
corruptions than those used in prior works, and present an
information-theoretic analysis of robust mean estimation in these settings. We
show that for structured distributions, methods that leverage the structure
yield information theoretically more accurate mean estimation. We also focus on
practical algorithms for robust mean estimation and study when data
cleaning-inspired approaches that first fix corruptions in the input data and
then perform robust mean estimation can match the information theoretic bounds
of our analysis. We finally demonstrate experimentally that this two-step
approach outperforms structure-agnostic robust estimation and provides accurate
mean estimation even for high-magnitude corruption.
