Coupling matrix manifolds assisted optimization for optimal transport problems

Contour Manifolds and Optimal Transport

Describing shapes by suitable measures in object segmentation, as proposed in [24], allows to combine the advantages of the representations as parametrized contours and indicator functions. The pseudo-Riemannian structure of optimal transport can be used to model shapes in ways similar as with contours, while the Kantorovich functional enables the application of convex optimization methods for ...

Optimization Algorithms on Matrix Manifolds

Statistical manifolds from optimal transport

Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures and they arise in various theoretical and applied problems. Using ideas in optimal transport, we introduce and study a parameterized family of $L^{(\pm \alpha)}$-divergences which includes the Bregman divergence corresponding to the Euclidean quadratic cost, a...

Haar Matrix Equations for Solving Time-Variant Linear-Quadratic Optimal Control Problems

‎In this paper‎, ‎Haar wavelets are performed for solving continuous time-variant linear-quadratic optimal control problems‎. ‎Firstly‎, ‎using necessary conditions for optimality‎, ‎the problem is changed into a two-boundary value problem (TBVP)‎. ‎Next‎, ‎Haar wavelets are applied for converting the TBVP‎, ‎as a system of differential equations‎, ‎in to a system of matrix algebraic equations‎...

Parabolic Optimal Transport Equations on Manifolds

We study a parabolic equation for finding solutions to the optimal transport problem on compact Riemannian manifolds with general cost functions. We show that if the cost satisfies the strong MTW condition and the stay-away singularity property, then the solution to the parabolic flow with any appropriate initial condition exists for all time and it converges exponentially to the solution to th...