On the Scalable Construction of Measure Transport Maps and Applications in Health Analytics
Characterizing and sampling from probability distributions is useful to reason about uncertainty in large, complex, and multi-modal datasets. One established and increasingly popular method to do so involves finding transformations or transport maps between one distribution to another. The computation of these transport maps is the subject of the field of Optimal Transportation, a rich area of mathematical theory that has led to many applications in machine learning, economics, and statistics. Finding these transport maps, however, usually comprises computational difficulties, particularly when datasets are large both in dimension and the number of samples to learn from.
Building upon previous work in our group, we introduce a formulation to find transport maps that is parallelizable and solvable with convex optimization methods. We show applications in the field of health analytics encompassing scalable Bayesian inference, density estimation, and generative models. We show how this formulation is scalable with the dimension of data and can be parallelized utilizing a sweep of architectures such as cloud computing services and Graphics Processing Units. Within the context of Bayesian inference, we present a distributed framework for finding the full posterior distribution associated with LASSO problems and show advantages compared to traditional methods of computing this posterior. Finally, we discuss novel methods to reduce the number of parameters necessary to approximate transport maps.