# Research

My research interests are focused mainly on different variations and their combinations of the famous Monge-Kantorovich Optimal Transport problem with applications of optimal transport in machine learning. Optimal Transport Problem (OTP) is the general problem of moving mass from a source distribution to a target distribution as efficiently as possible. There are several variations of this problem in literature, that stems from different interpretations of the physical mass transportation between many places, along different paths or with constraints with applications in image processing, data science, economics, chemical physics etc. Currently I am focusing on the following problems:

### Capacity Constraints to Barycenter Problem

I have introduced the notion of capacity constrained barycenters in Wasserstein space which is a generalization of the classical barycenter problem. As the name suggests, the capacity constrained barycenter problem introduces capacities to each of the two marginal problems associated. Under certain assumptions on the capacities, I have proven that the problem attains a minimizer. A dual formulation is also given and I have obtained the strong duality result. My current results are for the very natural cost function |x-y|^2 and can be extended to costs that satisfy triangle inequality. I research on other more general costs functions.

### Duality for Capacity Constrained Multi-Marginal OTP

The notion of the capacity constrained multi-marginal OTP already exists in the literature. Capacity constrained multi-marginal problem introduces capacities that limit the amount transported between the source and the targets. However, to the best of my knowledge, a dual problem is not discussed. I have introduced a dual formulation for the multi-marginal OTP and obtained the existence of dual maximizers and the strong duality result.

### Relations between Capacity Constrained Barycenter Problem and Capacity Constrained Multi-Marginal OTP

The relation between the classical barycenter problem and the multi-marginal OTP has been discussed by Agueh and Carlier. It has been explicitly shown how to find a minimizer of one problem, given a minimizer of the other. I wish to find a similar relation between these two problems under a capacity constrained setting.

### Entropy Regularized Barycenter Problem and Machine Learning

Entropic regularization provides us with an approximation of optimal transport, with lower computational complexity and easy implementation. It operates by adding an entropic regularization penalty to the original problem making it a strictly convex problem, hence guaranteeing a unique minimizer.

The Entropy Regularized barycenter problem and its duality has already been discussed in the literature. However, the existence of a minimizer for the problem has been proven using the strong duality result. I have proved the existence of a minimizer directly without using the duality result. The proof of the existence of the dual maximizers is still on-going. I also wish to find relations between the entropy regularized versions of the barycenter problem and the multi-marginal problem.

I am also interested in using machine learning techniques to obtain optimizers to different versions of entropy regularized OTP.