Minimizing movements with general costs,
and nonnegative cross-curvature in
infinite dimensions
Flavien Léger
joint works with Pierre-Cyril Aubin-Frankowski,
Gabriele Todeschi, François-Xavier Vialard
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
\begin{aligned}
x_n & \in\operatorname*{argmin}_{x\in X}g(x)+c(x,y_n)\\
y_{n+1}&\in\operatorname*{argmin}_{y\in Y}c(x_n,y)
\end{aligned}
I M P L I C I T S C H E M E
\inf_{y\in Y}c(x,y)=0
\inf_{y\in Y}c(x,y)=0
\inf_{y\in Y}c(x,y)=0
Goal: geometry in which convexity of \(g\) gives convergence rates
Explicit minimizing movements
\(f\colon X\to\mathbb{R}\cup\{+\infty\}\): c-concave energy
\begin{aligned}
y_{n+1}&\in\operatorname*{argmin}_{y\in Y}c(x_n,y)+f^c(y)\\
x_{n+1} & \in\operatorname*{argmin}_{x\in X}c(x,y_{n+1})
\end{aligned}
E X P L I C I T S C H E M E
f(x)=\inf_{y\in Y}c(x,y)+f^c(y)
(majorize)
(minimize)
Explicit minimizing movements
\(X,Y\) smooth manifolds, \(c\in C^1(X\times Y)\), \(f\in C^1(X)\) c-concave
\begin{aligned}
-\nabla_xc(x_n,y_{n+1})&=-\nabla f(x_n)\\
\nabla_xc(x_{n+1},y_{n+1})&=0
\end{aligned}
Under assumptions (twist, non-empty c-subdifferentials...), the explicit scheme can be written as
x_{n+1}-x_n=-\tau\nabla f(x_n)
c(x,y)=\frac{1}{2\tau}\lVert x-y\rVert^2
x_{n+1}=\exp_{x_n}(-\tau\nabla f(x_n))
c(x,y)=\frac{1}{2\tau}d_M^2(x,y)
\nabla u(x_{n+1})-\nabla u(x_n)=-\nabla f(x_n)
c(x,y)=u(x)-u(y)-\nabla u(y)(x-y)
x_{n+1}-x_n=-\nabla^2u(x_n)^{-1}\nabla f(x_n)
c(x,y)=u(y)-u(x)-\nabla u(x)(y-x)
EVI for Alternating Minimization
Alternating minimization (AM) of \(\phi\colon X\times Y\to\mathbb{R}\cup\{+\infty\}\)
\forall x\in X,\quad F(x_n)+D(x_n,y_n)+(1+\mu)D(x,y_{n+1})\leq F(x)+D(x,y_n)
(Csiszár–Tusnády ’84)
(L–Aubin-Frankowski ’23)
\begin{aligned}
x_n & \in\operatorname*{argmin}_{x\in X}\phi(x,y_n)\\
y_{n+1}&\in\operatorname*{argmin}_{y\in Y}\phi(x_n,y)
\end{aligned}
F(x)\coloneqq \inf_{y\in Y}\phi(x,y)
D(x,y)=\phi(x,y)-F(x)
\longrightarrow \quad\phi(x,y)=F(x)+D(x,y)
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
(\mu\geq 0)
Convergence rates
If \((x_n,y_n)\) satisfy the EVI then
\begin{aligned}
y_{n+1}&\in\operatorname*{argmin}_{y\in Y}\phi(x_n,y)\\
x_{n+1} & \in\operatorname*{argmin}_{x\in X}\phi(x,y_{n+1})
\end{aligned}
F(x_n)\leq F(x)+\frac{\phi(x,y_0)-\phi(x_0,y_0)}{n}
sublinear rates
when \(\mu=0\)
exponential rates
when \(\mu>0\)
(L–Aubin-Frankowski ’23)
F(x_n)\leq F(x)+\frac{\mu}{1+\mu}\frac{\phi(x,y_0)-\phi(x_0,y_0)}{(1+\mu)^n-1}
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)
\forall\xi\perp\eta,\quad\mathfrak{S}_c(x,y)(\xi,\eta)\geq 0
\forall(\xi,\eta),\quad\mathfrak{S}_c(x,y)(\xi,\eta)\geq 0
\(s\mapsto (x(s),\bar y)\) is a c-segment if \[\nabla_yc(x(s),\bar y)=(1-s)\nabla_yc(x(0),\bar y)+s\nabla_yc(x(1),\bar y).\]
D E F I N I T I O N
T H E O R E M (KIM–MCCANN '10)
Under assumptions (c-convex domains...) \(c\) has nonnegative cross-curvature iff for any c-segment \(s\mapsto (x(s),\bar y)\),
\forall y\in Y,\quad s\mapsto c(x(s),\bar y)-c(x(s),y)\text{ is convex}
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, \(c\colon X\times Y\to\mathbb{R}\cup\{\pm\infty\}\).
(1-s)[c(x(0),\bar y)-c(x(0),y)]+s[c(x(1),\bar y)-c(x(1),y)].
\forall y\in Y,\quad c(x(s),\bar y)-c(x(s),y)\leq
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
d^2(x(s),\bar y )\leq (1-s)d^2(x(0),\bar y)+s\,d^2(x(1),\bar y)-s(1-s)d^2(x(0),x(1))
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
\mathcal{T}_c(\mu,\nu)=\inf_{\pi\in\Pi(\mu,\nu)}\int c(x,y)\,d\pi
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 \(\mathbf{Y}=[Y,g,\nu]\in\mathbb{G}\)
\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')\,.
(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!
(Pisa 2024-12-03) NNCC
By Flavien Léger
