This paper extends the non-unique games framework introduced in Bandeira et al. (2020) to address symmetric structures. Precisely, we consider the problem \[ \begin{aligned} \underset{g_1,\ldots,g_n\in \mathcal{G}}{\operatorname{min}}\,\, \frac{1}{|S|}\sum_{i,j=1}^n \sum_{s\in S} f_{ij}(g_i s g_j^{-1}), \end{aligned} \] where $\mathcal{G}$ is a compact group, $S\subset \mathcal{G}$ is a finite subgroup of symmetry elements, and $f_{ij}:\mathcal{G}\rightarrow \mathbb{R}$ is a loss function. A notable example is common lines based orientation estimation of cryo-EM images for molecules with point group symmetry $S$, where $\mathcal{G}$ is the rotation group $\operatorname{SO}(3)$. With a semidefinite programming relaxation, we turn the problem into a convex program that is solved with the alternating direction method of multipliers. Unlike existing approaches which are tailored to specific symmetry groups, ours provides a unified computational framework for addressing all point group symmetries. We demonstrate the applicability of the proposed algorithm on the cryo-EM orientation estimation problem on both simulated and experimental datasets. In particular, our algorithm is effective in generating high-quality ab initio models from only a few class averages computed from experimental images.

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