Variational PDEs for Acceleration on Manifolds and Application to Diffeomorphisms

We consider the optimization of cost functionals on manifolds and derive a variational approach to accelerated methods on manifolds. We demonstrate the methodology on the infinite-dimensional manifold of diffeomorphisms, motivated by registration problems in computer vision. We build on the variational approach to accelerated optimization by Wibisono, Wilson and Jordan, which applies 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 simple mechanical principles. Our approach can also be viewed as a generalization of the $L^2$ 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 the mass. We derive the theory, compute the PDEs for acceleration, and illustrate the behavior of this new accelerated optimization scheme.