Partial Gaussian splitting in quadratic Wasserstein distance
Year: 2026
Type: preprint
Abstract: Let $\mu=e^{-V}\,dx$ be a smooth probability measure on $\mathbb{R}^n$ with $\nabla^2V\succeq I_n$. Suppose that $u_1,\ldots,u_k$ are centered, variance-one Sobolev functions whose gradients are orthogonal in $L^2(\mu)$, and let $D=\sum_{i=1}^k\left(\int|\nabla u_i|^2\,d\mu-1\right)$. We prove that there is a $k$-dimensional subspace $E$ for which the squared quadratic Wasserstein distance from $\mu$ to the product of a standard Gaussian on $E$, translated by the projected mean, and the actual marginal of $\mu$ on $E^\perp$ is at most $15D$. The estimate has no ambient-dimension factor and requires no smallness assumption on $D$. The proof combines the Caffarelli contraction and Gaussian near-affinity estimates with an aggregate selection of directions and an explicit coupling that preserves the complementary marginal. Preprint, version 1.0. This record contains the manuscript PDF and its LaTeX source with build instructions. The manuscript has not undergone journal peer review.
Author: Vaseghi, Sam
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221695007
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