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One of the difficulties in 3D reconstruction of molecules from images insingle particle Cryo-Electron Microscopy (Cryo-EM), in addition to high levelsof noise and unknown image orientations, is heterogeneity in samples: in manycases, the samples contain a mixture of molecules, or multiple conformations ofone molecule. Many algorithms for the reconstruction of molecules from imagesin heterogeneous Cryo-EM experiments are based on iterative approximations ofthe molecules in a non-convex optimization that is prone to reaching suboptimallocal minima. Other algorithms require an alignment in order to performclassification, or vice versa. The recently introduced Non-Unique Gamesframework provides a representation theoretic approach to studying problems ofalignment over compact groups, and offers convex relaxations for alignmentproblems which are formulated as semidefinite programs (SDPs) with certificatesof global optimality under certain circumstances. In this manuscript, wepropose to extend Non-Unique Games to the problem of simultaneous alignment andclassification with the goal of simultaneously classifying Cryo-EM images andaligning them within their respective classes. Our proposed approach can alsobe extended to the case of continuous heterogeneity.

A Representation Theory Perspective on Simultaneous Alignment and Classification