What I will present

Theory for minimizing movement schemes in infinite dimensions and in nonsmooth (nondifferentiable) settings, with a movement limiter given by a general cost function.

Main motivation: optimization on a space of measures $\mathcal{P}(M)$:

minimize   $E\colon \mathcal{P}(M)\to\mathbb{R}\cup\{+\infty\}$

Typical scheme:

where $D(\mu,\nu)=$

transport cost:   $W_2^2(\mu,\nu)$,  $\mathcal{T}_c(\mu,\nu)$,...

Bregman divergence:   $\operatorname{KL}(\mu,\nu)$,...

Csiszár divergence:   $\int_M (\sqrt{\mu}-\sqrt{\nu})^2$,...

...

What I will present

Theory for minimizing movement schemes in infinite dimensions and in nonsmooth (nondifferentiable) settings, with a movement limiter given by a general cost function.

1. Formulations for implicit and explicit schemes in a general setting

Setting

Minimize  $E\colon X\to\mathbb{R}\cup\{+\infty\}$,   where $X$ is a set (set of measures, metric space...).

Use   $D\colon X\times Y\to\mathbb{R}\cup\{+\infty\}$,   where $Y$ is another set (often $X=Y$).

Explicit minimizing movements

$\exists h\colon Y\to\mathbb{R}\cup\{+\infty\}$

Definition.

$E$ is c-concave if

c-concave

not c-concave

generalizes “$L$-smoothness”

Explicit minimizing movements

(majorize)

(minimize)

$\exists h\colon Y\to\mathbb{R}\cup\{+\infty\}$

Definition.

$E$ is c-concave if

Algorithm.

(Explicit scheme)

Assume $E$ c-concave.

(L–Aubin-Frankowski '23)

Explicit minimizing movements

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

Under certain assumptions, the explicit scheme can be written as

2. Convergence rates

EVI and convergence rates

Definition.

(Csiszár–Tusnády ’84)

(L–Aubin-Frankowski ’23)

Evolution Variational Inequality (or five-point property):

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

sublinear rates when $\mu=0$

exponential rates when $\mu>0$

Theorem.

(L–Aubin-Frankowski '23)

(Ambrosio–Gigli–Savaré ’05)

Variational c-segments and NNCC spaces

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

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

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

Definition.

(L–Todeschi–Vialard '24)

More: origins in regularity of optimal transport

(Ma–Trudinger–Wang ’05)

(Trudinger–Wang ’09)

(Kim–McCann ’10)

convexity of the set of c-concave functions

(Figalli–Kim–McCann '11)

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)

Variational c-segments $\approx$ generalized geodesics

Any Hilbert or Bregman cost is NNCC

Properties of NNCC spaces

Stable by products

Stable by quotients with “equidistant fibers”

Stable under Gromov–Hausdorff convergence

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

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

(connect to: Kim–McCann '12)

(connect to: Loeper ’09)

(L–Todeschi–Vialard '24)

Convergence rates

Suppose that for each $x\in X$ and $n\geq 0$,

Then sublinear ($\mu=0$) or linear ($\mu>0$) convergence rates. 

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

⏵ $s\mapsto E(x(s))-\mu \,D(x(s),y_{n+1})$ is convex

⏵ $\displaystyle\lim_{s\to 0^+}\frac{D(x(s),y_{n+1})}{s}=0$

Theorem.

(L–Aubin-Frankowski '23)

Thank you!