Optimisation Methods for Optimal Transport

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Optimal Transport is an elegant area of mathematics relating to the study of transporting the mass from one probability distribution to another in the most 'efficient' manner, with respect to a particular choice of cost function. It provides a principled mechanism for a cost function on the underlying space to induce a measure of distance between probability distributions on this space. Such problems occur frequently in machine learning systems, either as a loss function or to find an optimal mapping between distributions. While the mathematical theory of optimal transport has been well-developed over the past few decades, data-driven applications have become increasingly relevant due to recent computational advances that have allowed for tractable calculation of optimal mappings. Recent applications of optimal transport in machine learning systems have include methods for image generation, aligning single-cell data, and natural language processing, with many exciting applications yet to be explored. The solution to the optimal transport problem is highly dependent on the choice of underlying cost function. The majority of applications of computational optimal transport only consider using the standard quadratic cost, which is often an arbitrary choice and may not be a good cost function for the problem at hand. In this project, we aim to leverage both classical optimal transport theory and recent computational advances to design methods that can learn improved cost functions from observed data in a principled manner, which can thus lead to improved performance in subsequent downstream tasks. We will use elements of statistical learning theory to provide convergence guarantees, ensuring that our methods are both reliable and efficient.An application of optimal transport that could benefit from such methods is the analysis of single-cell omics data, which consists of measurements taken of individual cells in a population at a cost of destroying the cell in the process. The data is therefore recorded only as unlabelled snapshots that approximate the entire population. Optimal transport methods can be used to align the observed distributions, allowing the trajectories of individual cells in the population to be inferred. Given that the cell measurements are recorded according to a particular vector embedding, a good choice of cost function for this representation is not clear, so the ability to learn a cost function from existing data could enable improved performance. The ability to learn cost functions that are adapted for the problem at hand could be beneficial in many other applications of optimal transport and for a variety of different data structures, whenever the choice of cost function for the underlying space is unclear.As computational optimal transport methods play an important role in many machine learning systems, this proposal falls within the EPSRC's 'AI, Digitalisation and Data: Driving Value and Security' research priority.