  Given i.i.d samples from some unknown continuous density on hyper-rectangle
$[0, 1]^d$, we attempt to learn a piecewise constant function that approximates
this underlying density non-parametrically. Our density estimate is defined on
a binary split of $[0, 1]^d$ and built up sequentially according to discrepancy
criteria; the key ingredient is to control the discrepancy adaptively in each
sub-rectangle to achieve overall bound. We prove that the estimate, even though
simple as it appears, preserves most of the estimation power. By exploiting its
structure, it can be directly applied to some important pattern recognition
tasks such as mode seeking and density landscape exploration. We demonstrate
its applicability through simulations and examples.
