# Gaussian approximation for penalized Wasserstein barycenters

## TL;DR

This paper establishes a Gaussian approximation for regularized Wasserstein barycenters in Fourier basis, providing explicit convergence rates depending on sample size and parameter dimension.

## Contribution

It introduces a Gaussian approximation for Fourier-based Wasserstein barycenters and derives finite-sample convergence rates with explicit dependence on key parameters.

## Key findings

- Random Fourier parameters of barycenters converge to a Gaussian distribution.
- Explicit convergence rates depend on the number of measures ($n$) and parameter dimension ($p$).
- Theoretical results facilitate understanding of statistical properties of Wasserstein barycenters.

## Abstract

In this work we consider regularized Wasserstein barycenters (average in Wasserstein distance) in Fourier basis. We prove that random Fourier parameters of the barycenter converge to some Gaussian random vector by distribution. The convergence rate has been derived in finite-sample case with explicit dependence on measures count ($n$) and the dimension of parameters ($p$).