  We introduce Unbalanced Sobolev Descent (USD), a particle descent algorithm
for transporting a high dimensional source distribution to a target
distribution that does not necessarily have the same mass. We define the
Sobolev-Fisher discrepancy between distributions and show that it relates to
advection-reaction transport equations and the Wasserstein-Fisher-Rao metric
between distributions. USD transports particles along gradient flows of the
witness function of the Sobolev-Fisher discrepancy (advection step) and
reweighs the mass of particles with respect to this witness function (reaction
step). The reaction step can be thought of as a birth-death process of the
particles with rate of growth proportional to the witness function. When the
Sobolev-Fisher witness function is estimated in a Reproducing Kernel Hilbert
Space (RKHS), under mild assumptions we show that USD converges asymptotically
(in the limit of infinite particles) to the target distribution in the Maximum
Mean Discrepancy (MMD) sense. We then give two methods to estimate the
Sobolev-Fisher witness with neural networks, resulting in two Neural USD
algorithms. The first one implements the reaction step with mirror descent on
the weights, while the second implements it through a birth-death process of
particles. We show on synthetic examples that USD transports distributions with
or without conservation of mass faster than previous particle descent
algorithms, and finally demonstrate its use for molecular biology analyses
where our method is naturally suited to match developmental stages of
populations of differentiating cells based on their single-cell RNA sequencing
profile. Code is available at https://github.com/ibm/usd .
