1. Minimizing movements with general cost

2. NNCC spaces

Implicit minimizing movements

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

$g\colon X\to\mathbb{R}\cup\{+\infty\}$: energy to minimize

I M P L I C I T   S C H E M E

Goal: geometry in which convexity of $g$ gives convergence rates

Explicit minimizing movements

$f\colon X\to\mathbb{R}\cup\{+\infty\}$: c-concave energy

E X P L I C I T   S C H E M E

(majorize)

(minimize)

Explicit minimizing movements

$X,Y$ smooth manifolds,   $c\in C^1(X\times Y)$,   $f\in C^1(X)$ c-concave

Under assumptions (twist, non-empty c-subdifferentials...), the explicit scheme can be written as

EVI for Alternating Minimization

Alternating minimization (AM) of $\phi\colon X\times Y\to\mathbb{R}\cup\{+\infty\}$

(Csiszár–Tusnády ’84)

(L–Aubin-Frankowski ’23)

EVI (five-point property) for AM:

Implicit:   $\phi(x,y)=g(x)+c(x,y)$

Explicit:   $\phi(x,y)=c(x,y)+f^c(y)$

D E F I N I T I O N

Convergence rates

If $(x_n,y_n)$ satisfy the EVI then

sublinear rates

when $\mu=0$

exponential rates

when $\mu>0$

(L–Aubin-Frankowski ’23)

EVI from convexity

Suppose that for every $y_0\in Y$ there exists $x_0\in \argmin_{x\in X}\phi(x,y_0)$, $y_1\in \argmin_{y\in Y}\phi(x_0,y)$, such that for each $x\in X$,

T H E O R E M   (L–Aubin-Frankowski '24)

Then EVI

⏵ there exists a variational c-segment $s\mapsto (x(s),y_0)$ on $(X\times Y,\phi)$ with $x(0)=x_0$ and $x(1)=x$

⏵ $s\mapsto F(x(s))-\mu \,D(x(s),y_1)$ is convex

⏵ $\displaystyle\lim_{s\to 0^+}\frac{D(x(s),y_1)}{s}=0$

1. Minimizing movements with general cost

2. NNCC spaces

Introduction

$X,Y$ $n$-dimensional manifolds,   $c\in C^4(X\times Y)$,   $\nabla^2_{xy}c(x,y)$ nonsingular

MTW condition

Nonnegative cross-curvature

(Ma–Trudinger–Wang ’05)

(Trudinger–Wang ’09)

(Kim–McCann ’10)

Variational c-segments and NNCC spaces

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

$s\mapsto (x(s),\bar y)$ is a variational c-segment if $c(x(s),\bar y)$ is finite and 

$(X\times Y,c)$ is a space with nonnegative cross-curvature (NNCC space) if variational c-segments always exist.

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

Properties of NNCC spaces

Stable by products

Stable by projections with “equidistant fibers”

Stable under Gromov–Hausdorff convergence

Metric cost $c(x,y)=d^2(x,y)$     NNCC$\implies$PC

NNCC $\iff$

the set of c-concave function with nonempty c-subdifferentials is convex

(connect to: Ambrosio–Gigli–Savaré ’05)

(connect to: Kim–McCann '12)

(connect to: Figalli–Kim–McCann '11)

(connect to: Loeper ’09)

Examples

Gromov–Wasserstein

Kullback–Leibler divergence, Hellinger, Fisher–Rao costs are NNCC

Transport costs

$(\mathbb{G}\times\mathbb{G},\operatorname{GW}^2)$ is NCCC

$(\mathcal{P}(X)\times\mathcal{P}(Y),\mathcal{T}_c)$  NNCC $\iff$ $(X\times Y,c)$   NNCC

(Polish spaces, lsc cost)

Ex: $W_2^2$ on $\mathbb{R}^n$, on $\mathbb{S}^n$...

$\mathbf{X}=[X,f,\mu]$   and   $\mathbf{Y}=[Y,g,\nu]\in\mathbb{G}$

(L–Todeschi–Vialard '24)

Rem: strengthens PC

Variational c-segments $\approx$ generalized geodesics

(Sturm)

Conclusion and perspectives

⏵ Variational c-segments: nonsmooth geometry for EVIs

⏵ Variational c-segments: applications to functional inequalities

⏵ NNCC spaces: applications to mechanism design

Thank you!