  Let $\pi_{0}$ and $\pi_{1}$ be two distributions on the Borel space
$(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}))$. Any measurable function
$T:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ such that $Y=T(X)\sim\pi_{1}$ if
$X\sim\pi_{0}$ is called a transport map from $\pi_{0}$ to $\pi_{1}$. For any
$\pi_{0}$ and $\pi_{1}$, if one could obtain an analytical expression for a
transport map from $\pi_{0}$ to $\pi_{1}$, then this could be straightforwardly
applied to sample from any distribution. One would map draws from an
easy-to-sample distribution $\pi_{0}$ to the target distribution $\pi_{1}$
using this transport map. Although it is usually impossible to obtain an
explicit transport map for complex target distributions, we show here how to
build a tractable approximation of a novel transport map. This is achieved by
moving samples from $\pi_{0}$ using an ordinary differential equation with a
velocity field that depends on the full conditional distributions of the
target. Even when this ordinary differential equation is time-discretized and
the full conditional distributions are numerically approximated, the resulting
distribution of mapped samples can be efficiently evaluated and used as a
proposal within sequential Monte Carlo samplers. We demonstrate significant
gains over state-of-the-art sequential Monte Carlo samplers at a fixed
computational complexity on a variety of applications.
