2017/07/04 by Carla Tameling, Tameling, Carla, Max Sommerfeld +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · #60B12 #60F05 #62E20 (Primary) 90C08 #62G10 (Secondary) #90C31 #FOS: Mathematics #Machine Learning and Algorithms #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Single-cell and spatial transcriptomics #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1707.00973
openalex publication_date 2017/07/04 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We derive distributional limits for empirical transport distances between\nprobability measures supported on countable sets. Our approach is based on\nsensitivity analysis of optimal values of infinite dimensional mathematical\nprograms and a delta method for non-linear derivatives. A careful calibration\nof the norm on the space of probability measures is needed in order to combine\ndifferentiability and weak convergence of the underlying empirical process.\nBased on this we provide a sufficient and necessary condition for the\nunderlying distribution on the countable metric space for such a distributional\nlimit to hold. We give an explicit form of the limiting distribution for\nultra-metric spaces. Finally, we apply our findings to optimal transport based\ninference in large scale problems. An application to nanoscale microscopy is\ngiven.\n