Wasserstein Distance, Fourier Series and Applications

TL;DR

This paper explores the Wasserstein metric using Fourier analysis, providing bounds for distributions in finite fields, continuous functions, and eigenfunctions on manifolds, with applications in number theory and spectral geometry.

Contribution

It introduces novel bounds and inequalities for Wasserstein distances in various contexts, connecting Fourier analysis with probability distributions and eigenfunctions.

Findings

Bound on Earth Mover Distance for quadratic residues:  p^{-1/2}

Inequality relating roots of functions and Fourier coefficients

Wasserstein distance bound for Laplacian eigenfunctions:  \,rac{ ext{max}\{ ext{eigenfunction} ext{positive/negative parts} ight\,} \\lesssim_p \\sqrt{rac{ ext{log} \\lambda}{\\lambda}} \\| ext{eigenfunction}\\|_{L^1}^{

Abstract

We study the Wasserstein metric $W_p$, a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications. In particular, we bound the Earth Mover Distance $W_1$ between the distribution of quadratic residues in a finite field $\mathbb{F}_p$ and uniform distribution by $\lesssim p^{-1/2}$ (the Polya-Vinogradov inequality implies $\lesssim p^{-1/2} \log{p}$). We also show for continuous $f:\mathbb{T} \rightarrow \mathbb{R}_{}$ with mean value 0 $$ (\mbox{number of roots of}~f) \cdot \left( \sum_{k=1}^{\infty}{ \frac{ |\hat{f}(k)|^2}{k^2}}\right)^{\frac{1}{2}} \gtrsim \frac{\|f\|^{2}_{L^1(\mathbb{T})}}{\|f\|_{L^{\infty}(\mathbb{T})}}.$$ Moreover, we show that for a Laplacian eigenfunction $-\Delta_g \phi_{\lambda} = \lambda \phi_{\lambda}$ on a compact Riemannian manifold $W_p\left(\max\left\{\phi_{\lambda}, 0\right\}dx, \max\left\{-\phi_{\lambda}, 0\right\} dx\right) \lesssim_p \sqrt{\log{\lambda}/\lambda} \|\phi_{\lambda}\|_{L^1}^{1/p}$ which is at most a factor $\sqrt{\log{\lambda}}$ away from sharp. Several other problems are discussed.