  Nonparametric estimation of mutual information is used in a wide range of
scientific problems to quantify dependence between variables. The k-nearest
neighbor (knn) methods are consistent, and therefore expected to work well for
large sample size. These methods use geometrically regular local volume
elements. This practice allows maximum localization of the volume elements, but
can also induce a bias due to a poor description of the local geometry of the
underlying probability measure. We introduce a new class of knn estimators that
we call geometric knn estimators (g-knn), which use more complex local volume
elements to better model the local geometry of the probability measures. As an
example of this class of estimators, we develop a g-knn estimator of entropy
and mutual information based on elliptical volume elements, capturing the local
stretching and compression common to a wide range of dynamical systems
attractors. A series of numerical examples in which the thickness of the
underlying distribution and the sample sizes are varied suggest that local
geometry is a source of problems for knn methods such as the
Kraskov-St\"{o}gbauer-Grassberger (KSG) estimator when local geometric effects
cannot be removed by global preprocessing of the data. The g-knn method
performs well despite the manipulation of the local geometry. In addition, the
examples suggest that the g-knn estimators can be of particular relevance to
applications in which the system is large, but data size is limited.
