Background on optimal transport

$$\mathrm{OT}_0(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\iint c(x,y)d\pi(x,y)$$

Primal formulation

Optimal transport

$\Pi(\mu,\nu)$: probability measures with marginals $\mu$ and $\nu$.

We assume that

$$\Sigma:=\mathrm{supp}\,\pi=\{(x,y(x)),x\in X\}$$

Optimal transport

$$X$$

$$Y$$

Optimal transport

Dual formulation

$$\mathrm{OT}_0(\mu,\nu) = \sup_{\phi,\psi}\int\phi\,d\nu - \int\psi\,d\mu$$

s.t.

$$u(x,y):=c(x,y)+\psi(x)-\phi(y)\ge 0$$

The $c$-divergence 

$$u(x,y)=c(x,y)+\psi(x)-\phi(y)\ge 0$$

$$\Sigma = \{(x,y) : u(x,y) = 0\}$$

Example: $c(x,y)=-x\cdot y$

$$u(x,y)=\psi(x)-\phi(y)-x\cdot y$$

$$=\psi(x|x(y))$$

Bregman divergence

Entropic transport

Primal formulation

$$X$$

$$Y$$

Dual formulation

Entropic transport

$$\longrightarrow \pi_\varepsilon(x,y) = e^{-\frac1\varepsilon(c(x,y)+\psi_\varepsilon(x)-\phi_\varepsilon(y))}\mu(x)\nu(y)$$

Our question

$\pi_0$ singular measure supported on $\Sigma$

$\pi_\varepsilon$ smooth measure supported on $X\times Y$

$$\mathrm{OT}_0(\mu,\nu)$$

$$\mathrm{OT}_\varepsilon(\mu,\nu)$$

What's known

• In general $$\phi_\varepsilon\to\phi_0\quad\text{as }\varepsilon\to 0$$  (Nutz & Wiesel ’21, Berman ’21, N Gigli & L Tamanini ’18)

• For general costs $$\mathrm{OT}_\varepsilon(\mu,\nu)\approx\mathrm{OT}_0(\mu,\nu)-\varepsilon\ln(2\pi\varepsilon)^{d/2}-\varepsilon H(\nu|m)$$ (S Pal ’19)
• For quadratic costs (Schrödinger problems )$$\mathrm{OT}_\varepsilon(\mu,\nu)\approx\mathrm{OT}_0(\mu,\nu)-\varepsilon\ln(2\pi\varepsilon)^{d/2}-\frac\varepsilon 2(H(\mu)+H(\nu))$$ $$+\frac{\varepsilon^2}{8} \int_0^1 \mathrm{FI}(\rho_t)\,dt$$ (G Conforti & L Tamanini ’21)

Background on the Kim–McCann geometry

(YH Kim & RJ McCann ’10)

$$c(x,y)+c(x+\xi,y+\eta) \le c(x+\xi,y)+c(x,y+\eta)$$

Quantifying a matching's stability

Kim and McCann's idea:

consider $$\hat g = -D^2_{xy}c$$ as a semi-metric over all $X\times Y$

Second fundamental form

$$h(U,V)=(\hat\nabla_UV)^\perp$$

Mean curvature    $H=\mathrm{tr}(h)$        (a normal vector field)

$$(T\Sigma\times T\Sigma\to T^\perp\Sigma)$$

Example: $c(x,y)=-x\cdot y$

$$\hat g=\begin{pmatrix}0&I_d\\I_d&0\end{pmatrix}$$

$$u(x,y)=\psi(x)-\phi(y)-x\cdot y$$

$$=\psi(x|x(y))$$

Bregman divergence

$$g=D^2\psi$$

Hessian metric

flat metric

In summary, we have 

On $X\times Y$

On $\Sigma$

Extrinsic curvatures

$\hat g$         semi-metric

$\hat m$        volume form

$\hat \nabla$        Levi-Civita connection

$\hat R$        scalar curvature

$g$         metric

$m$        volume form

$\nabla$        Levi-Civita connection

$R$        scalar curvature

$h$         second fundamental form

$H$        mean curvature

A new Laplace formula

$$\iint_{X\times Y}\frac{e^{-u(x,y)/\varepsilon}}{(2\pi\varepsilon)^{d/2}}f(x,y)\,d\hat m(x,y) = \int_\Sigma fdm\,+$$

$$\varepsilon\int_\Sigma \bigg[-\frac 18\hat\Delta f+ \frac 14 \hat\nabla_{\!H} f+ \frac{1}{16}\Big( |H|^2 - \frac{5}{3}|h|^2 -R + \frac{3}{4}\hat R\Big)f\bigg] \,dm  + \varepsilon^2\mathcal{R}(\varepsilon)$$

T H E O R E M

$u$ vanishes on $\Sigma$

$$\Sigma$$

$$e^{-u(x,y)/\varepsilon}$$

$$X$$

$$Y$$

Assumptions:

$$X=Y=\mathbb{R}^d$$

$$0<\lambda\le D^2u\le\Lambda$$

$$f\textrm{ and } D^2u \in W^{4,\infty}$$

Novelties:

1. Geometric expression

2. Quantitative remainder bound

$$\lvert\mathcal{R}(\varepsilon) \rvert \le C  \lVert D^2u\rVert_{W^{4,\infty}}^4 \iint_{X\times Y} \frac{e^{-\lambda\lvert y-y(x)\rvert^2/2\varepsilon}}{(2\pi\varepsilon/\lambda)^{d/2}} \lvert D_{\le 4}f\rvert (x,y)\,d\hat{m}(x,y)$$

Taylor expansion of the potentials

$$\mathrm{div}_\pi(\nabla V)=\mathrm{div}_\pi(H^\perp)$$

$H$ : mean curvature, $H^\perp$ tangent to $\Sigma$

Solve for $V$ on $\Sigma$

$\pi$ : optimal transport plan

(supported on $\Sigma$)

$$\int_\Sigma \mathrm{div}_\pi(\xi) f\,d\pi = -\int_\Sigma\xi\cdot\nabla f\,d\pi$$

D E F I N I T I O N

$$\mathrm{div}_\pi(\nabla V)=\mathrm{div}_\pi(H^\perp)$$

$$\psi_\varepsilon=\psi_0+\frac\varepsilon 2\ln\Big(\frac{m}{e^{-V}\mu}\Big) + o(\varepsilon)$$

T H E O R E M

Assumptions:

$$X=Y=\mathbb{R}^d$$

$$0<\lambda\le D^2u\le\Lambda$$

Log-concavity control over $\mu$ and $\nu$

$(\psi_\varepsilon)_\varepsilon$ uniformly bounded in $H^5$

Taylor expansion of the transport value

$$\mathrm{OT}_\varepsilon(\mu,\nu)=\mathrm{OT}_0(\mu,\nu) - \varepsilon\ln(2\pi\varepsilon)^{d/2}-\varepsilon H(\pi|m)$$

$$+\frac{\varepsilon^2}{8}  \int_\Sigma \Big[\lvert\nabla\ln(\pi/m)\rvert^2+\frac 14 \hat{R}+R+\frac{5}{3}|h|^2 -|\nabla V|^2\Big]d\pi + o(\varepsilon^2)$$

T H E O R E M

Example: Quadratic cost $c(x,y)=|x-y|^2$

$\varepsilon^2$ term was known (Conforti–Tamanini):

$$\frac{\varepsilon^2}{8} \int_0^1 \mathrm{FI}(\rho_t)\,dt$$

Strategy of proof

$\psi_\varepsilon$ : solution to dual problem $$\displaystyle\max_\psi J_\varepsilon(\psi)$$ $\widetilde\psi_\varepsilon$ : competitor

Proof strategy

Thanks!

The research leading to these results has received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement no. 866274)