Path constrained unbalanced optimal transport
arxiv(2024)
摘要
Dynamical formulations of optimal transport (OT) frame the task of comparing
distributions as a variational problem which searches for a path between
distributions minimizing a kinetic energy functional. In applications, it is
frequently natural to require paths of distributions to satisfy additional
conditions. Inspired by this, we introduce a model for dynamical OT which
incorporates constraints on the space of admissible paths into the framework of
unbalanced OT, where the source and target measures are allowed to have a
different total mass. Our main results establish, for several general families
of constraints, the existence of solutions to the variational problem which
defines this path constrained unbalanced optimal transport framework. These
results are primarily concerned with distributions defined on a Euclidean
space, but we extend them to distributions defined over parallelizable
Riemannian manifolds as well. We also consider metric properties of our
framework, showing that, for certain types of constraints, our model defines a
metric on the relevant space of distributions. This metric is shown to arise as
a geodesic distance of a Riemannian metric, obtained through an analogue of
Otto's submersion in the classical OT setting.
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