  Many machine learning problems can be seen as approximating a \textit{target}
distribution using a \textit{particle} distribution by minimizing their
statistical discrepancy. Wasserstein Gradient Flow can move particles along a
path that minimizes the $f$-divergence between the target and particle
distributions. To move particles, we need to calculate the corresponding
velocity fields derived from a density ratio function between these two
distributions. Previous works estimated such density ratio functions and then
differentiated the estimated ratios. These approaches may suffer from
overfitting, leading to a less accurate estimate of the velocity fields.
Inspired by non-parametric curve fitting, we directly estimate these velocity
fields using interpolation techniques. We prove that our estimators are
consistent under mild conditions. We validate their effectiveness using novel
applications on domain adaptation and missing data imputation.
