Wasserstein Distance, Fourier Series and Applications [article]

Stefan Steinerberger
2020 arXiv   pre-print
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 𝔽_p and uniform distribution by ≲ p^-1/2 (the Polya-Vinogradov inequality implies ≲ p^-1/2logp). We also show for continuous f:𝕋→ℝ_ with mean value 0 ( f) ·( ∑_k=1^∞ |f̂(k)|^2/k^2)^1/2≳f^2_L^1(𝕋)/f_L^∞(𝕋). Moreover,
more » ... show that for a Laplacian eigenfunction -Δ_g ϕ_λ = λϕ_λ on a compact Riemannian manifold W_p(max{ϕ_λ, 0}dx, max{-ϕ_λ, 0} dx) ≲_p √(logλ/λ)ϕ_λ_L^1^1/p which is at most a factor √(logλ) away from sharp. Several other problems are discussed.
arXiv:1803.08011v3 fatcat:mw2lvnuqubfrdibmtskavks6de