  Learning from data in the presence of outliers is a fundamental problem in
statistics. In this work, we study robust statistics in the presence of
overwhelming outliers for the fundamental problem of subspace recovery. Given a
dataset where an $\alpha$ fraction (less than half) of the data is distributed
uniformly in an unknown $k$ dimensional subspace in $d$ dimensions, and with no
additional assumptions on the remaining data, the goal is to recover a succinct
list of $O(\frac{1}{\alpha})$ subspaces one of which is nontrivially correlated
with the planted subspace. We provide the first polynomial time algorithm for
the 'list decodable subspace recovery' problem, and subsume it under a more
general framework of list decoding over distributions that are "certifiably
resilient" capturing state of the art results for list decodable mean
estimation and regression.
