Iterative Algorithm for Discrete Structure Recovery
It provides a unified solution for multiple discrete recovery problems, which is incremental as it generalizes existing methods like Lloyd's algorithm.
The paper tackles the problem of discrete structure recovery across various applications by proposing a general modeling and algorithmic framework with a simple iterative algorithm, achieving minimax rates in cases like clustering and compressed sensing.
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.