Non-unique games over compact groups and orientation estimation in cryo-EM

Let G be a compact group and let fij ∈ L2(G). We define the Non-Unique Games (NUG) problem as finding g1,dots,gn ∈ G to minimize sumi,j=1n fij left( gi gj-1right). We devise a relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of fij over G, which can then be solved efficiently. The NUG framework can be seen as a generalization of the little Grothendieck problem over the orthogonal group and the Unique Games problem and includes many practically relevant problems, such as the maximum likelihood estimator to registering bandlimited functions over the unit sphere in d-dimensions and orientation estimation in cryo-Electron Microscopy.