Approximation of splines in Wasserstein spaces (Q6621507)

For a curve \(\mu=(\mu_t)_t:[0,1]\to \mathcal{P}_2(\mathbb{R}^d)\) of probability measures with finite second moment, the classical energy functional is defined as\N\[\N\mathcal{E}((\mu_t)_t)=\inf_{v}\int_0^1\int_{\mathbb{R}^d}|v_t|^2d\mu_t\,dt,\N\]\Nwhile the continuous-time spline energy functional is\N\[\N\mathcal{F}((\mu_t)_t)=\inf_{v}\int_0^1\int_{\mathbb{R}^d}|\dot{v}_t+\frac{1}{2}\nabla|v_t|^2|^2d\mu_t\,dt.\N\]\NIn both the expressions the infimum is taken with respect to the vector fields \(v=(v_t)_t\) such that \((\mu,v)\) solves the continuity equation with prescribed starting and final measures \(\mu_0,\mu_1\), and notice that the term \(\dot{v}_t+\frac{1}{2}\nabla|v_t|^2\) naturally corresponds to the acceleration of the curve \((\mu_t)_t\). Finally, the regularized spline energy functional is the functional \(\mathcal{F}^\delta:=\mathcal{F}+\delta\mathcal{E}\).\N\NThe (regularized) spline interpolation problem in the Wasserstein space is to find a minimizer of the (regularized) spline energy functional subject to some prescribed interpolation constraints and suitable boundary conditions.\N\NUsing the notion of Wasserstein barycenter, the authors define a discretized version of the energy functionals and prove the existence of time-discrete regularized splines. Limited to Gaussian distributions, the spline interpolation problem is solved explicitly and the convergence of time-discrete spline energies to time-continuous ones in the sense of Mosco is proved. A numerical implementation of the minimization problem and experimental results are also provided, showing the robustness of the approach.