Multi-to one-dimensional optimal transport. (English) Zbl 1380.49066
Summary: We consider the Monge-Kantorovich problem of transporting a probability density on \(\mathbb{R}^m\) to another on the line, so as to optimize a given cost function. We introduce a nestedness criterion relating the cost to the densities, under which it becomes possible to solve this problem uniquely by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level-set dynamics to develop and quantify a local regularity theory for this map and the Kantorovich potentials solving the dual linear program. We identify obstructions to global regularity through examples.
More specifically, fix probability densities \(f\) and \(g\) on open sets \(X \subset \mathbb{R}^m\) and \(Y \subset \mathbb{R}^n\) with \(m \geq n \geq 1\). Consider transporting \(f\) onto \(g\) so as to minimize the cost \(-s(x,y)\). We give a nondegeneracy condition on \(s \in C^{1,1}\) that ensures the set of \(x\) paired with [g-a.e.] \(y \in Y\) lie in a codimension-\(n\) submanifold of \(X\). Specializing to the case \(m >n = 1\), we discover a nestedness criterion relating \(s\) to \((f,g)\) that allows us to construct a unique optimal solution in the form of a map \(F:X \to \overline Y\). When \(s \in C^2 \cap W^{3,1}\) and \(g\) and \(f\) are bounded, the Kantorovich dual potentials (\(u,\upsilon\)) satisfy \(\upsilon \in C_{\mathrm{loc}}^{1,1}(Y)\), and the normal velocity \(V\) of \(F^{-1}(y)\) with respect to changes in \(y\) is given by \(V(x)=\frac{\upsilon''-s_{yy}}{|D_xs_y|}(x,F(x))\). Positivity of \(V\) locally implies a Lipschitz bound on \(f\); moreover, \(\upsilon \in C^2\) if \(F^{-1}(y)\) intersects \(\partial X \in C^1\) transversally. On subsets where this nondegeneracy, positivity, and transversality can be quantified, for each integer \(r \geq 1\) the norms of \(u,\upsilon \in C^{r+1,1}\) and \(F \in C^{r,1}\) are controlled by these bounds, \(\|g,\log f,\partial X\|_{C^{r-1,1}},\|\partial X\|_{W^{2,1}},\|s\|_{C^{r+1,1}}\), and the smallness of \(F^{-1}(y)\). We give examples showing regularity extends from \(X\) to part of \(\overline X\), but not from \(Y\) to \(\overline Y\). We also show that when \(s\) remains nested for all \((f,g)\), the problem in \(\mathbb{R}^m \times\mathbb{R}\) reduces to a supermodular problem in \(\mathbb{R} \times\mathbb{R}\).
More specifically, fix probability densities \(f\) and \(g\) on open sets \(X \subset \mathbb{R}^m\) and \(Y \subset \mathbb{R}^n\) with \(m \geq n \geq 1\). Consider transporting \(f\) onto \(g\) so as to minimize the cost \(-s(x,y)\). We give a nondegeneracy condition on \(s \in C^{1,1}\) that ensures the set of \(x\) paired with [g-a.e.] \(y \in Y\) lie in a codimension-\(n\) submanifold of \(X\). Specializing to the case \(m >n = 1\), we discover a nestedness criterion relating \(s\) to \((f,g)\) that allows us to construct a unique optimal solution in the form of a map \(F:X \to \overline Y\). When \(s \in C^2 \cap W^{3,1}\) and \(g\) and \(f\) are bounded, the Kantorovich dual potentials (\(u,\upsilon\)) satisfy \(\upsilon \in C_{\mathrm{loc}}^{1,1}(Y)\), and the normal velocity \(V\) of \(F^{-1}(y)\) with respect to changes in \(y\) is given by \(V(x)=\frac{\upsilon''-s_{yy}}{|D_xs_y|}(x,F(x))\). Positivity of \(V\) locally implies a Lipschitz bound on \(f\); moreover, \(\upsilon \in C^2\) if \(F^{-1}(y)\) intersects \(\partial X \in C^1\) transversally. On subsets where this nondegeneracy, positivity, and transversality can be quantified, for each integer \(r \geq 1\) the norms of \(u,\upsilon \in C^{r+1,1}\) and \(F \in C^{r,1}\) are controlled by these bounds, \(\|g,\log f,\partial X\|_{C^{r-1,1}},\|\partial X\|_{W^{2,1}},\|s\|_{C^{r+1,1}}\), and the smallness of \(F^{-1}(y)\). We give examples showing regularity extends from \(X\) to part of \(\overline X\), but not from \(Y\) to \(\overline Y\). We also show that when \(s\) remains nested for all \((f,g)\), the problem in \(\mathbb{R}^m \times\mathbb{R}\) reduces to a supermodular problem in \(\mathbb{R} \times\mathbb{R}\).
MSC:
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
| 49N60 | Regularity of solutions in optimal control |
| 93E20 | Optimal stochastic control |
