  Let $V$ be any vector space of multivariate degree-$d$ homogeneous
polynomials with co-dimension at most $k$, and $S$ be the set of points where
all polynomials in $V$ {\em nearly} vanish. We establish a qualitatively
optimal upper bound on the size of $\epsilon$-covers for $S$, in the
$\ell_2$-norm. Roughly speaking, we show that there exists an $\epsilon$-cover
for $S$ of cardinality $M = (k/\epsilon)^{O_d(k^{1/d})}$. Our result is
constructive yielding an algorithm to compute such an $\epsilon$-cover that
runs in time $\mathrm{poly}(M)$.
  Building on our structural result, we obtain significantly improved learning
algorithms for several fundamental high-dimensional probabilistic models with
hidden variables. These include density and parameter estimation for
$k$-mixtures of spherical Gaussians (with known common covariance), PAC
learning one-hidden-layer ReLU networks with $k$ hidden units (under the
Gaussian distribution), density and parameter estimation for $k$-mixtures of
linear regressions (with Gaussian covariates), and parameter estimation for
$k$-mixtures of hyperplanes. Our algorithms run in time {\em quasi-polynomial}
in the parameter $k$. Previous algorithms for these problems had running times
exponential in $k^{\Omega(1)}$.
  At a high-level our algorithms for all these learning problems work as
follows: By computing the low-degree moments of the hidden parameters, we are
able to find a vector space of polynomials that nearly vanish on the unknown
parameters. Our structural result allows us to compute a quasi-polynomial sized
cover for the set of hidden parameters, which we exploit in our learning
algorithms.
