Wasserstein Distance: Bridging Measure Theory and Computational Limits via Kantorovich-Rubinstein Duality — E8 Intelligence Research

Abstract

FINDING: Optimal transport theory (Wasserstein distance) provides a geometric metric on probability spaces, with Kantorovich-Rubinstein duality linking it to Lipschitz functions and spectral gaps — a bridge between measure theory and computational limits. | MATH: Wasserstein-1 distance \( W_1(\mu,\nu) = \inf_{\gamma \in \Pi(\mu,\nu)} \int |x-y| \, d\gamma(x,y) \); Kantorovich-Rubinstein duality: \( W_1(\mu,\nu) = \sup_{\|f\|_{Lip} \le 1} \int f \, d\mu - \int f \, d\nu \); spectral gap \( \lambda_1 \) of a Markov generator bounds \( W_1 \) decay via \( e^{-\lambda_1 t} \). | CONNECTION: The Wasserstein space is a geodesic metric space — its curvature and low-dimensional embeddings (Murphy's talk) relate to intrinsic dimension reduction, echoing root system symmetries (A_n, D_n lattices) in optimal quantization; the duality constant 1 (Lipschitz norm) mirrors the golden ratio's self-duality in projective geometry, though no explicit 0.618/1.618 appears. | DEPTH: 7 — Profound for linking Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com