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Abstract:
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We study the non-convex optimization landscape for maximum likelihood estimation in the Gaussian orbit recovery model. This model is motivated by applications in molecular microscopy, where each measurement of an unknown object is subject to an independent random rotation from a rotational group.
Fundamental properties of the likelihood landscape depend on the signal-to-noise ratio and the group structure. At low noise, the landscape is "benign" for any group, possessing no spurious local optima. At high noise, the landscape may develop spurious optima depending on the specific group, and we discuss positive and negative examples. The Fisher information transitions from resembling a single Gaussian in low noise to having a graded eigenvalue structure in high noise, determined by the algebra of invariant polynomials under the group action. In a neighborhood of the true object, the likelihood is strongly convex in a reparametrization by a transcendence basis of this algebra. We discuss implications for optimization, including slow convergence of EM and possible advantages of momentum-based acceleration and variable reparametrization for first- and second-order descent methods.
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