Learning mixtures of structured distributions over discrete domains

Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n]=\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}\mathrm{.}$ We show that if $\mathfrak{C}$ satisfies a rather general condition – essentially, that each distribution in $\mathfrak{C}$ can be well-approximated by a variable-width histogram with few bins – then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of $k$ unknown distributions from $\mathfrak{C}\mathrm{.}$ We analyze several natural types of distributions over $[n]$, including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems.