  In astronomy and cosmology, significant effort is devoted to characterizing
and understanding spatial cross-correlations between points - e.g. galaxy
positions, high energy neutrino arrival directions, X-ray and AGN sources, and
continuous field - e.g. weak lensing and Cosmic Microwave Background (CMB)
maps. Recently, we introduced the $k$-nearest neighbor formalism to better
characterize the clustering of discrete (point) datasets. Here we extend it to
the point-field cross-correlation analysis. It combines $k$NN measurements of
the point data set with measurements of the field smoothed on many scales. The
resulting statistics are sensitive to all orders in the joint clustering of the
points and the field. We demonstrate that this approach, unlike the 2-pt
cross-correlation, can measure the statistical dependence of two datasets even
when there are no linear (Gaussian) correlations. We further demonstrate that
this framework is far more effective than the two-point function in detecting
cross-correlations when the continuous field is contaminated by high levels of
noise. For a particularly high level of noise, the cross-correlations between
halos and the underlying matter field in a cosmological simulation, between
$10h^{-1}{\rm Mpc}$ and $30h^{-1}{\rm Mpc}$, is detected at $>5\sigma$
significance using the technique presented here, when the two-point
cross-correlation significance is $\sim 1\sigma$. Finally, we show that the
$k$NN cross-correlations of halos and the matter field can be well-modeled on
quasilinear scales by the Hybrid Effective Field Theory (HEFT) framework, with
the same set of bias parameters as are used for the two-point
cross-correlations. The substantial improvement in the statistical power of
detecting cross-correlations with this method makes it a promising tool for
various cosmological applications.
