Let C be a class of probability distributions over the discrete
domain [n]={1,...,n}. We show that if C satisfies a rather
general condition -- essentially, that each distribution in 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
C.
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.Comment: preliminary full version of soda'13 pape