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.

Summary: The main contribution of this paper is the derivation of an "accelerated" gradient descent scheme for computing the stationary point of a potential function on diffeomorphisms, inspired by the variational formulation of Nesterov's accelerated gradient methods [1]. The authors first derive the continuous time and space analogy of the Bregmann Lagrangian [1] for diffeomorphisms, then apply the discretization to solve image registration problems, empirically showing faster/better convergence than gradient descent. Pros: The paper is well-written. The proposed scheme of solving diffeomorphic registration by discretizing a variational solution similar to [1] is a novel contribution, to the best of my knowledge. The authors also show strong empirical support of the proposed method vs. gradient descent.

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.