2020/04/16 by Sitan Chen, Ankur Moitra, Chen, Sitan +1 · 2 citations
Biochemistry, Genetics and Molecular Biology · Engineering · #Advanced Fluorescence Microscopy Techniques #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #FOS: Mathematics #FOS: Physical sciences #Nonlinear Optical Materials Studies #Optics (physics.optics) #Spectroscopy Techniques in Biomedical and Chemical Research #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2004.07659
openalex publication_date 2020/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For more than a century and a half it has been widely-believed (but was never rigorously shown) 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 the statistical and algorithmic complexity needed to resolve closely spaced point sources. In particular we show that there is a phase transition where the sample complexity goes from polynomial to exponential. Surprisingly, we show that this does not occur at the Abbe limit, which has long been presumed to be the true diffraction limit.