Overview

Minimize $$\mathcal{E}\colon X\to\mathbb{R}\cup\{+\infty\}$$

using a function $c(x,y)$ as a “movement limiter”

⏵ $X$ : possibly infinite-dimensional

⏵ $\mathcal{E},c$ : possibly nonsmooth 

Implicit and explicit methods with

a cost $c(x,y)$

1.

2.

Evolution variational inequalities (EVIs)

3.

NNCC spaces

Implicit method with cost $c(x,y)$

$(X,d)$ metric space

$X$ an arbitrary set

Proximal point method

Implicit method with cost $c(x,y)$

Remarks/

Motivations

• Tailored $c(x,y)$
• Regularizing operator
• Gradient flows   “$\dot x(t)=-\nabla \mathcal{E}(x(t))$”        ($\tau\to0$)

⏵ Define gradient flows in nonsmooth settings

⏵ $c(x,y)$ as a proxy for $d^2(x,y)$

(Ambrosio–Gigli–Savaré ’05)

(Rankin–Wong ’24)

Explicit method with cost $c(x,y)$

How to do an explicit method with cost $c(x,y)$ ?

$Y$ another set

c-concavity

Definition. $\mathcal{E}$ is c-concave if there exists $h\colon Y\to \mathbb{R}\cup\{+\infty\}$ s.t. $$\mathcal{E}(x)=\inf_{y\in Y}c(x,y)+h(y).$$

Smallest such $h$ is the c-transform $\mathcal{E}^c(y)=\sup_{x\in X} \mathcal{E}(x)-c(x,y).$

$\mathcal{E}$ is $c$-concave

$\mathcal{E}$ is not $c$-concave

$c(x,y)+\mathcal{E}^c(y)$

$\mathcal{E}$

$\mathcal{E}$

Explicit algorithm

Explicit algorithm

Suppose $\mathcal{E}$ is c-concave:

Nonsmooth settings

$\mathcal{E}$

(“Gradient descent with a general cost” L–Aubin-Frankowski ‘23)

Recap

Explicit: assume $\mathcal{E}$ is c-concave

Alternating Minimization (AM) of $\phi$

Implicit

Implicit and explicit methods

with a cost $c(x,y)$

1.

2.

Evolution variational inequalities (EVIs)

3.

NNCC spaces

Evolution Variational Inequalities (EVIs)

Definition. Let $\lambda\in[0,1)$. We say that $(x_n,y_n)_n$ satisfy the EVI if   $\forall n\geq 0$,

$$\forall x\in X,y\in Y,\quad(1-\lambda)\phi(x_n,y_n)+\phi(x,y_{n+1})\leq \phi(x,y)+(1-\lambda)\phi(x,y_n).$$

$X,Y$ two arbitrary sets,

$\phi\colon X\times Y\to\mathbb{R}\cup\{+\infty\}$ proper

⏵ Nonsmooth, intrinsic

⏵ Condition on $\phi$ and on the choice of iterates

T H E O R E M   (L–Aubin-Frankowski '23)

Background on EVIs

EVI $(\lambda=0)$

Euclidean

Consider implicit method

$\phi(x,y)$: extends the five-point property of Csiszár–Tusnády ’84

$(X,d)$ non-positively curved, Mayer/Jost

Ambrosio–Gigli–Savaré

$\mathcal{E}$ convex on geodesics

$\mathcal{E}$ convex

$\mathcal{E}$ convex on curves $x(t)$ such that $d^2(x,x_{n-1})$ is $1$-convex, i.e. $t\mapsto d^2(x(t),x_{n-1})-t^2 \,d^2(x(1),x(0))$ is convex

Implicit and explicit methods

with a cost $c(x,y)$

1.

2.

Evolution variational inequalities (EVIs)

3.

NNCC spaces

NNCC spaces

D E F I N I T I O N  (L–Todeschi–Vialard '24)

$(X\times Y,c)$ is an NNCC space if for each $(x_0,x_1,\bar y)\in X\times X\times Y$, there exists a path $x(\cdot)$ from $x_0$ to $x_1$ such that $\forall y\in Y$, $$c(x(t),\bar y)-c(x(t),y)\leq (1-t)[c(x_0,\bar y)-c(x_0,y)]+t[c(x_1,\bar y)-c(x_1,y)].$$

$(x(t),\bar y)$ is called a generalized c-segment.

$X, Y$ two arbitrary sets,   $c\colon X\times Y\to\mathbb{R}\cup\{+\infty\}$.

(Think:  $t\mapsto c(x(t),\bar y)-c(x(t),y)$  is convex)

NNCC spaces

History. Variant of the Ma–Trudinger–Wang (MTW) condition studied by Kim and McCann.

Original setting is smooth and finite-dimensional $c\in C^4(X\times Y)$.

Ma, Trudinger, Wang, Loeper, Kim, McCann, Villani, Figalli, Guillen, Kitagawa, Loeper

Basic finite-dim examples:

• $c(x,y)=\lVert x-y\rVert^2$
• $c(x,y)=$ Bregman divergence
• Any smooth reparametrization $c(x,y)=\lVert F(x)-G(y)\rVert^2$...
• Sphere

Theory. NNCC preserved by products, projections, pullbacks.

Stable under Gromov–Hausdorff.

EVIs in NNCC spaces

⏵ $(X\times X,c)$ NNCC space 

⏵ $\mathcal{E}(\cdot)-\mu\,c(\cdot,x_n)$ convex on generalized c-segments $(x(t),x_{n-1})$

Then EVI.

T H E O R E M   (L–Todeschi–Vialard '24)

(EVI)

Focus on implicit method $\phi(x,y)=\mathcal{E}(x)+c(x,y)$

⏵ Unique argmins

⏵ $c$ satisfies $\displaystyle\liminf_{t\to 0}\frac{c(x(t),x(0))}{t}=0.$

Examples of NNCC spaces

$X$, $Y$ Polish spaces, $c\in C(X\times Y)$.

If $(X\times Y,c)$ is an NNCC space then so is $(\mathcal{P}(X)\times \mathcal{P}(Y), \mathcal{T}_c)$.

Corollary: $(\mathcal{P}_2(X)\times \mathcal{P}_2(X), W_2^2)$ is an NNCC space when $X=$

$$\mathcal{T}_c(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int c(x,y)\,d\pi$$

T H E O R E M   (L–Todeschi–Vialard '24)

Generalized c-segments $(\mu(t),\nu)$:

  ⏵ $(T_0,S)$ optimal coupling of $(\mu_0,\nu)$ 

  ⏵ $(T_1,S)$ optimal coupling of $(\mu_1,\nu)$

  ⏵ $\forall \omega\in\Omega$, $t\mapsto (T_t(\omega),S(\omega))$ c-segment

  ⏵ $\mu(t)=(T_t)_\#\mathbb{P}$

$\nu$

$\mu$

Examples of NNCC spaces

Bures–Wasserstein

Gromov–Wasserstein        $\mathbf{X}=[X,f,\mu]$   and   $\mathbf{Y}=[Y,g,\nu]$

$$\operatorname{GW}^2(\mathbf{X},\mathbf{Y})=\inf_{\pi\in\Pi(\mu,\nu)}\int\lvert f(x,x')-g(y,y')\rvert^2\,d\pi(x,y)\,d\pi(x',y')\,.$$

Unbalanced OT

Hellinger, Fisher–Rao

Thank you!