Optimal transport, also known as Monge-Kantorovich transport or Wasserstein distance, is a mathematical framework that deals with the optimal way to transport mass from one configuration to another, minimizing the cost of the transportation. This field has gained significant attention across various disciplines due to its theoretical richness and a wide range of practical applications.

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$\textbf{c}$-transform. Concave functions

Introduction

Optimal transport is useful beyond classical problems of logistic and resource management. That method is somehow universal. I describe from my personal experience in NLP.

Optimal transport is useful tool for data scientist. I’ll bring case from my personal practice to prove.

Attacks

Adversarial attacks are a perfect subject of research as their allow to understand properties of models to depth, even providing solutions in closed form.

Business case. Matching scripts

Suppose we have call-center, where junior operators read scripts and more proficient colleagues speak freestyle. We want to match phrases from scripts to new original variations of freestylers.

Modern neural nets models BERT provides us with convenient of vector representation of sentences.

We actually have two distributions of sentences. First is for script sentences, second for

You can read more about embeddings here.

In normal practice we use cosine similarity

\[\text{similarity} = \cos(\text{emb}_1,\text{emb}_2)\]

All we need is to bring optimal connection.

About optimality

Optimality is actually one of the way of thinking and defining objects.

What’s more importantly you can relax

Entropy regularized optimal transport

\[\int_{x \in \Pi(\mu,\nu)}\]

Population dynamics

JKO flows:

JKO flows also were studied by Petr Mokrov et.al Large-Scale Wasserstein Gradient Flows.

I met Optimal Transport Modeling of Population Dynamics: Applications in Single-Cell by Charlotte Bunne and their excelent ariticle

Approach was elaborated in article https://arxiv.org/abs/2210.06662

Fluid approach

It will be instructive to consider another formulation of the optimal transport, originating in spirit from the fluid dynamics (Benamou & Brenier, 2000; Villani, 2003). Assume that at t = 0 we are given a set of ‘particles’ from the density ρ0 which move in such a way that at t = 1 their state is described by the density ρ1. Moreover, these particles move in such that they perform the least amount of work. Formally, they minimize the following action:

\[A= \int_{0}^{1} \left(\sum_{x} |\dot{x}(t)|^2\right)dt\]

In continuous limit obtaining:

\[\inf_{\rho,v} \left\{\int_{0}^1 \int \rho_t(x) |v_t(x)|^2 dxdt; \frac{\partial \rho_t}{\partial t} + \nabla \dot (\rho_t v_t) =0\right\}\]

where the infimum is taken over all time-dependent probability densities

Resources

Computational Optimal Transport by Marco Cuturi https://arxiv.org/pdf/1803.00567.pdf