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 [ Proc. Natl. Acad. Sci. USA, 113 (2016), pp. E7351--E7358], 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 them 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 from 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.

Following the seminal work of Nesterov, accelerated optimization methods have been used to powerfully boost the performance of first-order, gradient based parameter estimation in scenarios where second-order optimization strategies are either inapplicable or impractical. Not only does accelerated gradient descent converge considerably faster than traditional gradient descent, but it also performs a more robust local search of the parameter space by initially overshooting and then oscillating back as it settles into a final configuration, thereby selecting only local minimizers with a basis of attraction large enough to contain the initial overshoot. This behavior has made accelerated and stochastic gradient search methods particularly popular within the machine learning community. In their recent PNAS 2016 paper, A Variational Perspective on Accelerated Methods in Optimization, Wibisono, Wilson, and Jordan demonstrate how a broad class of accelerated schemes can be cast in a variational framework formulated around the Bregman divergence, leading to continuum limit ODEs. We show how their formulation may be further extended to infinite dimensional manifolds (starting here with the geometric space of curves and surfaces) by substituting the Bregman divergence with inner products on the tangent space and explicitly introducing a distributed mass model which evolves in conjunction with the object of interest during the optimization process. The coevolving mass model, which is introduced purely for the sake of endowing the optimization with helpful dynamics, also links the resulting class of accelerated PDE based optimization schemes to fluid dynamical formulations of optimal mass transport.