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Linearization of Monge-Ampère Equations and Statistical Applications

Alberto González-Sanz, Shunan Sheng

TL;DR

This work establishes that the optimal transport maps between a sufficiently regular reference measure and a smoothly evolving family of target measures form a C^1 curve in time. The time derivative of the transport map is characterized as the solution to a linearized Monge–Ampère equation with strictly oblique boundary conditions, derived via an implicit-function framework applied to the Monge–Ampère equation. The authors provide an extension mechanism to accommodate time-varying supports and derive both a transport-based quantile regression regularity result and a central limit theorem for smooth optimal transport maps. These results contribute to a deeper understanding of the differentiability and statistical behavior of transport maps under perturbations, enabling improved inference and stability analyses in applications requiring Fréchet differentiability of OT maps.

Abstract

Optimal transport has found numerous applications across data science, many of which require differentiating the optimal transport map with respect to the underlying probability densities in the Fréchet sense. In this work, we show that when the reference measure $Q$ is sufficiently regular in space and the curve of target measures $\{P_t\}_{t\in I}$ is both spatially regular and $\mathcal{C}^1$ in time, then the associated curve of optimal transport maps $\{\nabla φ_t\}_{t\in I}$ pushing $Q$ toward $P_t$ is itself a $\mathcal{C}^1$ curve. Moreover, we identify its time derivative as the solution to the \emph{linearized Monge--Ampère equation}, a second-order elliptic PDE with strictly oblique boundary conditions and a vanishing zero-order term. Our proof relies on applying the implicit function theorem to the Monge--Ampère equation with natural boundary conditions. As consequences, we establish regularity of the transport-based quantile regressor with respect to the covariates and derive a central limit theorem for smooth optimal transport maps.

Linearization of Monge-Ampère Equations and Statistical Applications

TL;DR

This work establishes that the optimal transport maps between a sufficiently regular reference measure and a smoothly evolving family of target measures form a C^1 curve in time. The time derivative of the transport map is characterized as the solution to a linearized Monge–Ampère equation with strictly oblique boundary conditions, derived via an implicit-function framework applied to the Monge–Ampère equation. The authors provide an extension mechanism to accommodate time-varying supports and derive both a transport-based quantile regression regularity result and a central limit theorem for smooth optimal transport maps. These results contribute to a deeper understanding of the differentiability and statistical behavior of transport maps under perturbations, enabling improved inference and stability analyses in applications requiring Fréchet differentiability of OT maps.

Abstract

Optimal transport has found numerous applications across data science, many of which require differentiating the optimal transport map with respect to the underlying probability densities in the Fréchet sense. In this work, we show that when the reference measure is sufficiently regular in space and the curve of target measures is both spatially regular and in time, then the associated curve of optimal transport maps pushing toward is itself a curve. Moreover, we identify its time derivative as the solution to the \emph{linearized Monge--Ampère equation}, a second-order elliptic PDE with strictly oblique boundary conditions and a vanishing zero-order term. Our proof relies on applying the implicit function theorem to the Monge--Ampère equation with natural boundary conditions. As consequences, we establish regularity of the transport-based quantile regressor with respect to the covariates and derive a central limit theorem for smooth optimal transport maps.
Paper Structure (10 sections, 19 theorems, 90 equations, 1 figure)

This paper contains 10 sections, 19 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation
  3. Main Result
  4. Applications
  5. Transport-based Quantile Regression
  6. Central Limit Theorem for Smooth Optimal Transport Map
  7. Proof of \ref{['Theorem:MainWithTime']}
  8. Fréchet Derivative of the Functional
  9. Linearized Monge--Ampère Equation
  10. The Implicit Function Theorem

Key Result

Theorem 3.2

Let $\Omega$ be a $\mathcal{C}^2$ uniformly convex domain and let $\Omega'\subset \mathbb{R}^d$ be a domain. If the curve $\{P_t\}_{t\in I}$ is $\mathcal{C}^1$ in $\mathcal{C}^{1, \gamma}(\overline{\Omega'})$ with $0<\gamma\leq 1$ (in the sense of Definition:smoothcourve) and let $\log(q)\in \mathca

Figures (1)

  • Figure 1: Multivariate quantile regression (two-dimensional variable of interest $\mathbf{Y}=(Y_1,Y_2)^\intercal$; univariate regressor $X$), showing the conditional medians (red) and the conditional quantile contours of order $\tau = 0.2$ (black), $\tau = 0.4$ (green), $\tau = 0.8$ (yellow). The number of samples generated is $n = 320,050$. The nonparametric estimator is that of delBarrioQuantile2024, trivially adapted to the change of reference measure.

Theorems & Definitions (33)

  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • Lemma 3.6
  • proof
  • Corollary 4.1
  • Lemma 4.2
  • ...and 23 more