Asymptotic Results on Low SNR Gaussian Mixtures and Interacting Particle Systems

In the first part of the talk I will discuss my work on poorly separated or “low SNR“ Gaussian mixture models (GMMs). This work is motivated by cryo-electron microscopy, in which one observes very noisy images of a molecule in different and unknown orientation. Even in well-separated GMMs, maximum likelihood estimation is difficult: the GMM log likelihood is non-convex, and the likelihood landscape is poorly understood. Through an asymptotic expansion of the log likelihood, we obtain valuable insight into this landscape. The expansion reveals that likelihood maximization in the low SNR regime reduces to a sequence of „moment matching“ stages. This insight is a stepping stone toward the analysis and improvement of Expectation Maximization and other algorithms for maximum likelihood estimation in a wide range of models.

The second part of the talk concerns an interacting particle model – a “stochastic lattice gas” – of crystal surface relaxation. The goal is to derive the partial differential equation governing the macroscopic dynamics, which emerges from the microscopic motion in a certain scaling limit. The PDE depends on the structure of the microscopic process’s local equilibrium state which, for the model I studied, is entirely novel. In the presence of this structure, the existence of a macroscopic limit is unclear. I will explain why the structure is nevertheless sufficient for the emergence of a macroscopic dynamics.

Data science, particularly sampling and optimization, is another research area in which interacting particle systems have emerged as a promising tool. At the end of this talk, I will briefly mention future work I intend to pursue in this area.

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