[Paper Review] On global Hölder estimates for optimal transportation

Alexander V. Kolesnikov|arXiv (Cornell University)|Oct 28, 2008

Point processes and geometric inequalities13 references4 citations

TL;DR

This paper establishes global Hölder continuity estimates for optimal transport maps between log-concave probability measures under mild convexity and growth conditions on the potential functions. It proves that under assumptions on the second-order difference quotients of the potential $ V $ and the modulus of convexity of $ W $, the transport map $ T $ is globally $ \frac{p+1}{q+1} $-Hölder continuous with a dimension-free constant, extending Caffarelli’s Lipschitz result to a broader Hölder regime.

ABSTRACT

We generalize a well-known result of L. Caffarelli on Lipschitz estimates for optimal transportation $T$ between uniformly log-concave probability measures. Let $T : \R^d o \R^d$ be an optimal transportation pushing forward $μ= e^{-V}dx$ to $ν= e^{-W}dx$. Assume that 1) the second differential quotient of $V$ can be estimated from above by a power function, 2) modulus of convexity of $W$ can be estimated from below by $A_q |x|^{1+q}$, $q \ge 1$. Under these assumptions we show that $T$ is globally Hölder with a dimension-free coefficient. In addition, we study optimal transportation $T$ between $μ$ and the uniform measure on a bounded convex set $K \subset \R^d$. We get estimates for the Lipschitz constant of $T$ in terms of $d$, ${diam(K)}$ and $D V, D^2 V$.

Motivation & Objective

• To generalize Caffarelli’s Lipschitz estimate for optimal transport maps to a global Hölder continuity framework.
• To establish dimension-free Hölder regularity for optimal transport maps between uniformly log-concave measures under growth and convexity conditions on the potentials.
• To derive concentration inequalities for target measures based on the Hölder regularity of the transport map.
• To analyze the Lipschitz constant of optimal transport from a log-concave measure to the uniform measure on a bounded convex set, in terms of dimension and diameter.

Proposed method

• Introduces a condition on the second-order difference quotient of $ V $: $ V(x+y) + V(x-y) - 2V(x) \leq A_p |y|^{p+1} $ for $ 0 \leq p \leq 1 $.
• Imposes a lower bound on the modulus of convexity of $ W $: $ W(x+y) + W(x-y) - 2W(x) \geq A_q |y|^{q+1} $ for $ q \geq 1 $.
• Uses the Monge-Ampère equation $ \rho_2(\nabla\varphi) \det D^2\varphi = \rho_1 $ to relate the transport map $ T = \nabla\varphi $ to the densities $ \mu = e^{-V}dx $, $ \nu = e^{-W}dx $.
• Applies a priori estimates from fully nonlinear PDE theory, particularly Hölder estimates for solutions of the Monge-Ampère equation.
• Introduces auxiliary functions $ b(t) $ and $ \delta(t) $ to quantify convexity and growth of $ W $, and uses convex conjugation $ b^* $ to derive functional inequalities.
• Employs Talagrand-type and modified log-Sobolev-type inequalities to derive concentration properties of the target measure $ \nu $.

Experimental results

Research questions

• RQ1 Under what conditions on the potentials $ V $ and $ W $ is the optimal transport map $ T $ globally Hölder continuous?
• RQ2 Can the dimension-free Hölder regularity of $ T $ be established when $ V $ satisfies a power-type upper bound on its second-order differences and $ W $ has a lower bound on its modulus of convexity?
• RQ3 What is the sharp dependence of the Hölder exponent on the growth parameters $ p $ and $ q $ of the potentials?
• RQ4 How do the Hölder estimates for $ T $ lead to concentration inequalities for the target measure $ \nu = e^{-W}dx $?
• RQ5 What is the Lipschitz constant of optimal transport from a log-concave measure to the uniform measure on a bounded convex set $ K $? How does it scale with $ d $ and $ \text{diam}(K) $?

Key findings

• Under the assumptions $ V(x+y)+V(x-y)-2V(x) \leq A_p |y|^{p+1} $ and $ W(x+y)+W(x-y)-2W(x) \geq A_q |y|^{q+1} $ with $ 0 \leq p \leq 1 \leq q $, the optimal transport map $ T $ is globally $ \frac{p+1}{q+1} $-Hölder continuous with a constant depending only on $ p, q, A_p, A_q $.
• For a measure $ \nu = e^{-W}dx $ satisfying $ W(x+y)+W(x-y)-2W(x) \geq C_\beta |y|^{2\beta} $ with $ \beta \geq 1 $, the concentration inequality $ \nu(B_r) \geq 1 - e^{-C|r|^{2\beta}} $ holds for all $ r > 0 $, where $ C $ depends only on $ \beta, C_\beta $.
• The optimal transport from the standard Gaussian measure to the uniform measure on a bounded convex set $ K \subset \mathbb{R}^d $ is $ C \sqrt{d} \cdot \text{diam}(K) $-Lipschitz for some universal constant $ C $.
• For a convex function $ W $, the optimal transport $ T = \nabla\varphi $ from the Gaussian measure to $ \nu = e^{-W}dx $ satisfies $ |\nabla\varphi(x) - \nabla\varphi(y)| \leq 8 \delta^{-1}(4|x-y|^2) $, where $ \delta(t) = \inf_{\|y\| \geq t} \{ W(x+y) + W(x-y) - 2W(x) \} $.
• The measure $ \nu $ satisfies the concentration inequality $ \nu(B_r) \geq 1 - \frac{1}{2} \exp\left( -\frac{1}{8} \delta(r/8) \right) $ for all $ r > 0 $, where $ B $ is any set with $ \nu(B) \geq 1/2 $.
• The concentration bound $ \nu(A_r) \geq 1 - 2e^{-2\tilde{b}(r/2)} $ holds for all $ r > 0 $, where $ \tilde{b} $ is the maximal convex minorant of $ b(t) = \inf_{\|y\| \geq t} \{ W(x+y) - W(x) - \langle \nabla W(x), y \rangle \} $.