Abstract

Optimal Transport theory (OT) has gained much attention in the past decades due to its rich mathematical content and striking connections to other areas of math. It has also naturally found a place in data science as a way to compare and transform probability measures. Unfortunately, there remain significant computational burdens that restrict its effectiveness in large-scale data regimes. In this thesis, we consider Minibatch Optimal Transport (MiniOT), which breaks up the task of computing the optimal transport plan between datasets into several smaller tasks of computing the optimal transport plan between minibatches of the datasets. After obtaining the minibatch optimal transport plans, we can reconstruct a global transport plan by appropriately "stitching" the minibatch plans together. We prove that such an approach does indeed yield a transport plan, although not optimal. We also propose a way to obtain an approximate transport map from a transport plan. Using this construction, we can recast the problem of finding an approximate transport map between two distributions, given to us by finite samples, into a simple regression problem. By defining an appropriate loss function and parametrizing the transport map with a neural network, we can implement Stochastic Gradient Descent (SGD) to approximate the transport map. The code for this thesis can be accessed here: https://drive.google.com/drive/folders/1KRY8haBJwuE7I11f29MBCW_ZOfCTURNy?usp=share_link.

Degree

College and Department

Computational, Mathematical, and Physical Sciences; Mathematics

Rights

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