  We give the first polynomial-time algorithm for robust regression in the
list-decodable setting where an adversary can corrupt a greater than $1/2$
fraction of examples.
  For any $\alpha < 1$, our algorithm takes as input a sample $\{(x_i,y_i)\}_{i
\leq n}$ of $n$ linear equations where $\alpha n$ of the equations satisfy $y_i
= \langle x_i,\ell^*\rangle +\zeta$ for some small noise $\zeta$ and
$(1-\alpha)n$ of the equations are {\em arbitrarily} chosen. It outputs a list
$L$ of size $O(1/\alpha)$ - a fixed constant - that contains an $\ell$ that is
close to $\ell^*$.
  Our algorithm succeeds whenever the inliers are chosen from a
\emph{certifiably} anti-concentrated distribution $D$. In particular, this
gives a $(d/\alpha)^{O(1/\alpha^8)}$ time algorithm to find a $O(1/\alpha)$
size list when the inlier distribution is standard Gaussian. For discrete
product distributions that are anti-concentrated only in \emph{regular}
directions, we give an algorithm that achieves similar guarantee under the
promise that $\ell^*$ has all coordinates of the same magnitude. To complement
our result, we prove that the anti-concentration assumption on the inliers is
information-theoretically necessary.
  Our algorithm is based on a new framework for list-decodable learning that
strengthens the `identifiability to algorithms' paradigm based on the
sum-of-squares method.
  In an independent and concurrent work, Raghavendra and Yau also used the
Sum-of-Squares method to give a similar result for list-decodable regression.
