Neural Optimal Transport with Lagrangian Costs
Keywords: neural optimal transport, Lagrangian costs, Riemannian geometry, amortized optimization
Abstract: Computational efforts in optimal transport traditionally revolve
around the squared-Euclidean cost. In this work, we choose to
investigate the optimal transport problem between probability measures
when the underlying metric space is non-Euclidean, or when the cost
function is understood to satisfy a *least action principle*,
also known as a *Lagrangian* cost.
These two generalizations are useful when connecting observations
from a physical system, where the transport dynamics are influenced
by the geometry of the system, such as obstacles, and allows
practitioners to incorporate *a priori* knowledge of the
underlying system. Examples include barriers for transport, or
enforcing a certain geometry, i.e., paths must be circular.
We demonstrate the effectiveness of this formulation on existing
synthetic examples in the literature, where we solve the optimal
transport problems in the absence of regularization, which is novel in
the literature.
Our contributions are of computational interest, where
we demonstrate the ability to efficiently compute geodesics and
amortize spline-based paths.
We demonstrate the effectiveness of this formulation on existing
synthetic examples in the literature, where we solve the optimal
transport problems in the absence of regularization.
Submission Number: 56
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