  We present two different approaches for parameter learning in several mixture
models in one dimension. Our first approach uses complex-analytic methods and
applies to Gaussian mixtures with shared variance, binomial mixtures with
shared success probability, and Poisson mixtures, among others. An example
result is that $\exp(O(N^{1/3}))$ samples suffice to exactly learn a mixture of
$k<N$ Poisson distributions, each with integral rate parameters bounded by $N$.
Our second approach uses algebraic and combinatorial tools and applies to
binomial mixtures with shared trial parameter $N$ and differing success
parameters, as well as to mixtures of geometric distributions. Again, as an
example, for binomial mixtures with $k$ components and success parameters
discretized to resolution $\epsilon$, $O(k^2(N/\epsilon)^{8/\sqrt{\epsilon}})$
samples suffice to exactly recover the parameters. For some of these
distributions, our results represent the first guarantees for parameter
estimation.
