  We study the problem of PAC learning a linear combination of $k$ ReLU
activations under the standard Gaussian distribution on $\mathbb{R}^d$ with
respect to the square loss. Our main result is an efficient algorithm for this
learning task with sample and computational complexity $(dk/\epsilon)^{O(k)}$,
where $\epsilon>0$ is the target accuracy. Prior work had given an algorithm
for this problem with complexity $(dk/\epsilon)^{h(k)}$, where the function
$h(k)$ scales super-polynomially in $k$. Interestingly, the complexity of our
algorithm is near-optimal within the class of Correlational Statistical Query
algorithms. At a high-level, our algorithm uses tensor decomposition to
identify a subspace such that all the $O(k)$-order moments are small in the
orthogonal directions. Its analysis makes essential use of the theory of Schur
polynomials to show that the higher-moment error tensors are small given that
the lower-order ones are.
