  Polynomial regression is a basic primitive in learning and statistics. In its
most basic form the goal is to fit a degree $d$ polynomial to a response
variable $y$ in terms of an $n$-dimensional input vector $x$. This is extremely
well-studied with many applications and has sample and runtime complexity
$\Theta(n^d)$. Can one achieve better runtime if the intrinsic dimension of the
data is much smaller than the ambient dimension $n$? Concretely, we are given
samples $(x,y)$ where $y$ is a degree at most $d$ polynomial in an unknown
$r$-dimensional projection (the relevant dimensions) of $x$. This can be seen
both as a generalization of phase retrieval and as a special case of learning
multi-index models where the link function is an unknown low-degree polynomial.
Note that without distributional assumptions, this is at least as hard as junta
learning.
  In this work we consider the important case where the covariates are
Gaussian. We give an algorithm that learns the polynomial within accuracy
$\epsilon$ with sample complexity that is roughly $N = O_{r,d}(n
\log^2(1/\epsilon) (\log n)^d)$ and runtime $O_{r,d}(N n^2)$. Prior to our
work, no such results were known even for the case of $r=1$. We introduce a new
filtered PCA approach to get a warm start for the true subspace and use
geodesic SGD to boost to arbitrary accuracy; our techniques may be of
independent interest, especially for problems dealing with subspace recovery or
analyzing SGD on manifolds.
