Explanation of Variability through Optimal Transport

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1715753Tabak The investigator develops a methodology for the explanation of variability in data attributable to known and unknown external factors. This has applications in all data-intensive fields, including public health, economics, and environmental studies. It provides a natural tool for personalized medicine, as it can both filter from data the influence of each patient's characteristics, such as age and sex, and particularize the predictions to account for the specifics of each patient. It also permits the analysis of meteorological data taken at different locations and times and with different instruments, and of unstructured time series depending on known and unknown external factors. Graduate and postdoctoral students participate in the work of the project. The problems of amalgamating datasets and removing external factors of variability find a very natural formulation in an extension of the mathematical theory of optimal transport: both problems concern normalizing (i.e. transporting) the original data so that no signal is left that points to the dataset of provenance or the values of the confounding variables. On the other hand, the transport needs to be optimized, minimizing the removal of variability associated to other sources. This project extends the theory of optimal transport to handle these problems. The extensions include problems such as finding the barycenter of a continuum of probability densities of which only a finite amount of samples are known. Graduate and postdoctoral students participate in the work of the project.