  Given $iid$ observations from an unknown absolute continuous distribution
defined on some domain $\Omega$, we propose a nonparametric method to learn a
piecewise constant function to approximate the underlying probability density
function. Our density estimate is a piecewise constant function defined on a
binary partition of $\Omega$. The key ingredient of the algorithm is to use
discrepancy, a concept originates from Quasi Monte Carlo analysis, to control
the partition process. The resulting algorithm is simple, efficient, and has a
provable convergence rate. We empirically demonstrate its efficiency as a
density estimation method. We present its applications on a wide range of
tasks, including finding good initializations for k-means.
