  We introduce a novel two-step approach for estimating a probability density
function (pdf) given its samples, with the second and important step coming
from a geometric formulation. The procedure involves obtaining an initial
estimate of the pdf and then transforming it via a warping function to reach
the final estimate. The initial estimate is intended to be computationally
fast, albeit suboptimal, but its warping creates a larger, flexible class of
density functions, resulting in substantially improved estimation. The search
for optimal warping is accomplished by mapping diffeomorphic functions to the
tangent space of a Hilbert sphere, a vector space whose elements can be
expressed using an orthogonal basis. Using a truncated basis expansion, we
estimate the optimal warping under a (penalized) likelihood criterion and,
thus, the optimal density estimate. This framework is introduced for
univariate, unconditional pdf estimation and then extended to conditional pdf
estimation. The approach avoids many of the computational pitfalls associated
with classical conditional-density estimation methods, without losing on
estimation performance. We derive asymptotic convergence rates of the density
estimator and demonstrate this approach using both synthetic datasets and real
data, the latter relating to the association of a toxic metabolite on preterm
birth.
