Accelerated Optimization in the PDE Framework: Formulations for the Active Contour Case
This work provides a novel formulation for accelerating optimization in geometric settings, which could benefit researchers in computer vision and PDE-based modeling, though it appears incremental as it builds on existing variational frameworks.
The paper extends accelerated optimization methods to infinite-dimensional manifolds, specifically geometric spaces of curves and surfaces, by replacing the Bregman divergence with tangent space inner products and introducing a co-evolving distributed mass model, linking the approach to fluid dynamics and optimal mass transport.
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, 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 ODE's. We show how their formulation may be further extended to infinite dimension 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 co-evolving 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.