Quantitative Stability of the Pushforward Operation by an Optimal Transport Map

TL;DR

The paper analyzes the quantitative stability of the pushforward operation induced by a fixed optimal transport map. It establishes a tight Hölder-type bound for how pushforwards depend on the base measures under minimal regularity assumptions, leveraging a novel bound on the size of singular sets of convex Lipschitz functions. The main result provides explicit exponents and constants for $p$-cost OT with $p\ge 2$, showing $W_q((T_\varphi)_\# \rho, (p_2)_\# \tilde{\gamma}) \lesssim W_r(\rho,\tilde{\rho})^{r/(q(r+1))}$ and related bounds, and it proves the optimality of these exponents via a 1D construction. The work also develops a quantitative singular-set size theorem and discusses implications for Linearized OT and ICNN-based generative models, highlighting both theoretical significance and practical impact for numerical OT, data analysis in LOT, and learning convex transport maps.

Abstract

We study the quantitative stability of the mapping that to a measure associates its pushforward measure by a fixed (non-smooth) optimal transport map. We exhibit a tight Hölder-behavior for this operation under minimal assumptions. Our proof essentially relies on a new bound that quantifies the size of the singular sets of a convex and Lipschitz continuous function on a bounded domain.

Figures (3)

• Figure 1: Linearized Optimal Transport barycenters, or generalized geodesic, between the discrete probability measures $\mu_0, \mu_1 \in \mathcal{P}([0,1]^2)$ (colored pixels indicate the position of support points). For $k \in \{0, 1\}$, the optimal transport map $\nabla \phi_k$ between the Lebesgue measure $\rho$ on $[0,1]^2$ and $\mu_k$ is computed using the Python package pysdot and Kitagawa2019ANA. Then for $N=70^2$, define $\frac{1}{N}\sum_{i=1}^{N} \delta_{x_i}$ to be a discrete approximation of $\rho$ on a uniform grid. For $t \in (0, 1)$, the interpolant is defined as $\mu_t = \frac{1}{N} \sum_{i=1}^{N}\delta_{y_i^t}$, where each $y_i^t$ is chosen in $\partial ( (1-t) \phi_0 + t \phi_1)(x_i)$.
• Figure 2: Illustration of Example \ref{['ex:holder-behavior']}.
• Figure 3: Graph of $\xi$ for $N=4$.

Theorems & Definitions (23)

• Proposition
• Example 1.2
• Example 1.3
• Example 1.4
• Theorem : Theorem \ref{['th:stability-pushforwards']}
• Theorem 2.1
• Corollary 2.2
• proof
• Remark 2.1: Singular sets of a convex Lipschitz function
• Remark 2.2: Tightness