An inverse problem in optimal transport on closed Riemannian manifolds
Jian Zhai, Kelvin Shuangjian Zhang
TL;DR
The paper tackles the inverse optimal transport problem on a compact manifold with cost $c(x,y)=\tfrac{1}{2}d^2(x,y)$, showing that the Riemannian metric $g$ can be recovered uniquely up to a positive constant from the optimal transport maps. By linearizing the Monge–Ampère equation around the base state, the authors obtain the linear elliptic operator $P=\Delta_g+\langle\nabla\log h,\nabla\cdot\rangle_g$ and relate internal measurements to the linearized equation $P\varphi=\frac{f_1}{h}$; the homogeneous problem has one-dimensional null space, enabling metric recovery from data. Local reconstruction on convex subsets uses multiple Pu=0 solutions to determine $\tilde g=(\det g)^{1/n}g^{-1}$ and a gradient equation for $\log\beta$ fixes the scale, with global reconstruction achieved by patching via a partition of unity. The main result provides a rigorous identifiability statement for inverse OT on closed manifolds and connects to techniques from hybrid imaging for interior data-based PDE recovery.
Abstract
We consider the problem of recovering the Riemannian metric on a compact closed manifold from the optimal transport maps when the underlying cost function is the squared Riemann distance. We show that the metric can be uniquely determined up to a multiplicative constant.