For more than a century and a half it has been widely-believed that the physics of diffraction imposes certain fundamental limits on the resolution of an optical system. However our understanding of what exactly can and cannot be resolved has never risen above heuristic arguments which, even worse, appear contradictory.

In this work we remedy this gap by studying the diffraction limit as a statistical inverse problem and, based on connections to provable algorithms for learning mixture models, we rigorously prove upper and lower bounds on how many photons we need (and how precisely we need to record their locations) to resolve closely-spaced point sources. Moreover we show the emergence of a phase transition, which helps explain why the diffraction limit can be broken in some domains but not in others.

This is based on joint work with Sitan Chen.