  We consider the problem of identifying, from statistics, a distribution of
discrete random variables $X_1,\ldots,X_n$ that is a mixture of $k$ product
distributions. The best previous sample complexity for $n \in O(k)$ was
$(1/\zeta)^{O(k^2 \log k)}$ (under a mild separation assumption parameterized
by $\zeta$). The best known lower bound was $\exp(\Omega(k))$. It is known that
$n\geq 2k-1$ is necessary and sufficient for identification. We show, for any
$n\geq 2k-1$, how to achieve sample complexity and run-time complexity
$(1/\zeta)^{O(k)}$. We also extend the known lower bound of $e^{\Omega(k)}$ to
match our upper bound across a broad range of $\zeta$. Our results are obtained
by combining (a) a classic method for robust tensor decomposition, (b) a novel
way of bounding the condition number of key matrices called Hadamard
extensions, by studying their action only on flattened rank-1 tensors.
