  We propose a general modeling and algorithmic framework for discrete
structure recovery that can be applied to a wide range of problems. Under this
framework, we are able to study the recovery of clustering labels, ranks of
players, signs of regression coefficients, cyclic shifts, and even group
elements from a unified perspective. A simple iterative algorithm is proposed
for discrete structure recovery, which generalizes methods including Lloyd's
algorithm and the power method. A linear convergence result for the proposed
algorithm is established in this paper under appropriate abstract conditions on
stochastic errors and initialization. We illustrate our general theory by
applying it on several representative problems: (1) clustering in Gaussian
mixture model, (2) approximate ranking, (3) sign recovery in compressed
sensing, (4) multireference alignment, and (5) group synchronization, and show
that minimax rate is achieved in each case.
