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

Let G be a compact group and let f_ij ∈ C(G). We define the non-unique games (NUG) problem as finding g₁, ⋯, g_n ∈ G to minimize Pⁿ_ij=1 f_ij (gig^-1_j). We introduce a convex relaxation of the NUG problem to a semidefinite program (SDP) by taking the Fourier transform of f _ij over G. 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 of noisy cryo-electron microscopy (cryo-EM) projection images. We implement an SDP solver for the NUG cryo-EM problem using the alternating direction method of multipliers (ADMM). Numerical study with synthetic datasets indicate that while our ADMM solver is slower than existing methods, it can estimate the rotations more accurately, especially at low signal-to-noise ratio (SNR).