Simple unbalanced optimal transport (Q6624340)

The problem of moving one mass (or density) to another by a diffeomorphism while minimizing a certain (quadratic) cost can be understood as construction of geodesics in an appropriate metric on the space of normalized densities (or on its completion), see [\textit{F. Otto}, Commun. Partial Differ. Equations 26, No. 1--2, 101--174 (2001; Zbl 0984.35089); \textit{C. Villani}, Topics in optimal transportation. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1106.90001)]. \textit{J.-D. Benamou} [M2AN, Math. Model. Numer. Anal. 37, No. 5, 851--868 (2003; Zbl 1037.65063)] considered the problem of constructing a natural extension of the action allowing to connect in the most economical way two densities of different total masses, which led to the domain of unbalanced optimal transport.\N\NUsually, the setting of unbalanced optimal transport involves a large extension\N\[\NG=\mathrm{Diff}\left( M\right) \ltimes\mathrm{C}_{+}^{\infty}\left( M\right)\N\]\Nof the group \(\mathrm{Diff}\left( M\right) \)\ of all diffeomorphisms of a manifold by means of a semi-direct product with the space of smooth positive functions, in which the large semi-direct product acts on densities by a change of coordinates and then by adjusting pointwise the obtained density by means of a function.\N\NThis paper instead introduces and investigates a much simpler small extension\N\[\N\mathsf{cone}\left( \mathrm{Diff}\left( M\right) \right) =\mathrm{Diff} \left( M\right) \times\mathbb{R}_{+}\N\]\Nof the same group \(\mathrm{Diff}\left( M\right) \), in which both the group of diffeomorphisms and the space of normalized densities have similar conical extensions\N\begin{align*}\N& \mathsf{cone}\left( \mathrm{Diff}\left( M\right) \right) \\\N\mathrm{Vol}\left( M\right) & =\mathsf{cone}\left( \mathrm{Dens}\left( M\right) \right)\N\end{align*}\Nby one extra parameter, the total mass \(m\)\ of the density.\N\NThe authors describe natural metrics and geodesics for those extensions. They also introduce special variables in which the convexity of the dynamical formulation for the simple conical extension as a generalization of the convexity of the standard optimal transport is demonstrated. The authors compare in more detail their small extension with two other larger extensions, namely, Wasserstein-Fisher-Rao [\textit{L. Chizat} et al., J. Funct. Anal. 274, No. 11, 3090--3123 (2018; Zbl 1387.49066); \textit{T. Gallouët} et al., ``Regularity theory and geometry of unbalanced optimal transport'', Preprint, \url{arXiv:2112.11056}] and a weighted sum of the Wasserstein and Fisher-Rao metrics, both of which can be viewed as extensions of the authors' model in, respectively, Lagrangian and Hamiltonian settings. The corresponding geodesics and candidates for their finite-dimensional counterparts are depicted.