  We consider the problem of identifying, from its first $m$ noisy moments, a
probability distribution on $[0,1]$ of support $k<\infty$. This is equivalent
to the problem of learning a distribution on $m$ observable binary random
variables $X_1,X_2,\dots,X_m$ that are iid conditional on a hidden random
variable $U$ taking values in $\{1,2,\dots,k\}$. Our focus is on accomplishing
this with $m=2k$, which is the minimum $m$ for which verifying that the source
is a $k$-mixture is possible (even with exact statistics). This problem, so
simply stated, is quite useful: e.g., by a known reduction, any algorithm for
it lifts to an algorithm for learning pure topic models.
  We give an algorithm for identifying a $k$-mixture using samples of $m=2k$
iid binary random variables using a sample of size $\left(1/w_{\min}\right)^2
\cdot\left(1/\zeta\right)^{O(k)}$ and post-sampling runtime of only
$O(k^{2+o(1)})$ arithmetic operations. Here $w_{\min}$ is the minimum
probability of an outcome of $U$, and $\zeta$ is the minimum separation between
the distinct success probabilities of the $X_i$s. Stated in terms of the moment
problem, it suffices to know the moments to additive accuracy
$w_{\min}\cdot\zeta^{O(k)}$. It is known that the sample complexity of any
solution to the identification problem must be at least exponential in $k$.
Previous results demonstrated either worse sample complexity and worse $O(k^c)$
runtime for some $c$ substantially larger than $2$, or similar sample
complexity and much worse $k^{O(k^2)}$ runtime.
