  Multi-Agent Consensus Equilibrium (MACE) formulates an inverse imaging
problem as a balance among multiple update agents such as data-fitting terms
and denoisers. However, each such agent operates on a separate copy of the full
image, leading to redundant memory use and slow convergence when each agent
affects only a small subset of the full image. In this paper, we extend MACE to
Projected Multi-Agent Consensus Equilibrium (PMACE), in which each agent
updates only a projected component of the full image, thus greatly reducing
memory use for some applications.We describe PMACE in terms of an equilibrium
problem and an equivalent fixed point problem and show that in most cases the
PMACE equilibrium is not the solution of an optimization problem. To
demonstrate the value of PMACE, we apply it to the problem of ptychography, in
which a sample is reconstructed from the diffraction patterns resulting from
coherent X-ray illumination at multiple overlapping spots. In our PMACE
formulation, each spot corresponds to a separate data-fitting agent, with the
final solution found as an equilibrium among all the agents. Our results
demonstrate that the PMACE reconstruction algorithm generates more accurate
reconstructions at a lower computational cost than existing ptychography
algorithms when the spots are sparsely sampled.
