Abstract 

We consider the optimization of cost functionals on infinite dimensional manifolds and derive a variational approach to accelerated methods on manifolds. We demonstrate the methodology on the infinite-dimensional manifold of diffeomorphisms, motivated by optical flow problems in computer vision. We build on a variational approach to accelerated optimization in finite dimensions, and generalize that approach to infinite dimensional manifolds. We derive the continuum evolution equations, which are partial differential equations (PDE), and relate them to mechanical principles. A particular case of our approach can be viewed as a generalization of the optimal mass transport problem. Our approach evolves an infinite number of particles endowed with mass, represented as a mass density. The density evolves with the optimization variable, and endows the particles with dynamics. This is different than current accelerated methods where only a single particle moves and hence the dynamics does not depend on mass. We derive theory and the PDEs for acceleration, and illustrate the behavior of this new scheme.