Group orbit recovery: Theory, algorithms, and applications

The group orbit recovery problem seeks to estimate the orbit of a signal from multiple noisy observations, each transformed by a random element of a group. These models can be viewed as Gaussian mixture models whose centers are structured by group actions. Motivated by applications in modern structural biology, such as single-particle cryo-electron microscopy, we focus on the notoriously challenging high-noise regime.

Our discussion unfolds along two complementary threads:

Fundamental Limits: We analyze the sample complexity and stability of orbit recovery, uncovering deep connections between information-theoretic bounds and algebraic structures.

Algorithms: We provide a thorough analysis of the standard expectation-maximization (EM) algorithm in the high-noise regime. We rigorously characterize its sample and iteration complexity and pinpoint the underlying reasons it stalls even near the optimum. Then, we introduce a new family of algorithms that attain the maximum likelihood estimator while being orders of magnitude faster than EM, delivering a scalable and practical solution for large-scale problems.

Tamir Bendory is an Associate Professor in the School of Electrical and Computer Engineering at Tel Aviv University. He earned his Ph.D. in Electrical Engineering from the Technion – Israel Institute of Technology in 2015. His research focuses on mathematical signal processing and data science, with an emphasis on developing theoretical and computational frameworks for applications in structural biology.