  We consider the problem of optimization of cost functionals on the
infinite-dimensional manifold of diffeomorphisms. We present a new class of
optimization methods, valid for any optimization problem setup on the space of
diffeomorphisms by generalizing Nesterov accelerated optimization to the
manifold of diffeomorphisms. While our framework is general for infinite
dimensional manifolds, we specifically treat the case of diffeomorphisms,
motivated by optical flow problems in computer vision. This is accomplished by
building on a recent variational approach to a general class of accelerated
optimization methods by Wibisono, Wilson and Jordan, which applies in finite
dimensions. We generalize that approach to infinite dimensional manifolds. We
derive the surprisingly simple continuum evolution equations, which are partial
differential equations, for accelerated gradient descent, and relate it to
simple mechanical principles from fluid mechanics. Our approach has natural
connections to the optimal mass transport problem. This is because one can
think of our approach as an evolution of an infinite number of particles
endowed with mass (represented with a mass density) that moves in an energy
landscape. The mass evolves with the optimization variable, and endows the
particles with dynamics. This is different than the finite dimensional case
where only a single particle moves and hence the dynamics does not depend on
the mass. We derive the theory, compute the PDEs for accelerated optimization,
and illustrate the behavior of these new accelerated optimization schemes.
