Relation between optimal transport cost and difference between topological invariants?

Question

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but I am not quite sure whether it has already been studied in some recent literature of Optimal Transport Theory (As far as I know, it hardly is)

The observation is as follows (which has been validated with some toy simulations).

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Problem (not rigorously stated)

Let $\mu, \nu$ probabilistic measures on regular manifolds $\mathcal{M}, \mathcal{N}$, $C^{\infty}(\mathcal{M}, \mathcal{N})$ the set of continuous mapping from $\mathcal{M}$ to $\mathcal{N}$, and $\Pi(\mu,\nu)$ the set of measures on $\mathcal{M}\times\mathcal{N}$ s.t. its marginal distributions are respectively $\mu, \nu$.

Consider the following optimization problem

$$ (*) = \min_{T\in{C^{\infty}(\mathcal{M}, \mathcal{N})}} \inf_{\gamma\in\Pi(\mu,\nu)} \int d^{2}(T(p), q) d\gamma(p,q) $$ where the cost function can be considered as the $L_2$ distance on $\mathcal{N}$'s total space as a real vector space.

My question is that whether $(*) \propto h(\chi(\mathcal{M}), \chi(\mathcal{N}))$ where $h$ is certain metric function and $\chi(\cdot)$ denotes Euler characteristic. Generally, would it be possible that the minimum cost of the Wasserstein game is deeply related with the difference between some topological invariants of underlying manifolds'?

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Look forward to any feedbacks and welcome discussions and potential references :D. I am willing to provide details of my toy experiments if one is interested in this problem.

Answer 1

Applications of OT to Algebraic Topology was the subject of my thesis available here https://github.com/jhmartel/Thesis2019

There remains many interesting questions to solve!

I found the topology of either source or target space was encoded in the topology of the singularity $Z$ of certain $c$-optimal transports from source $(X, \sigma)$ to target $(Y, \tau)$.

In case $Y=\partial X$ , we found the singularity $Z$ had the homotopy type of $Y$ if the cost was attractive (like quadratic cost $c=d^2/2$) and had the homotopy type of $X$ when the cost $c$ was repulsive (see above thesis). We described general technique for building strong deformation retracts in either case. (The argument depends on studying ``gradient flow to poles" defined by the dual Kantorovich $c$-concave potentials and not to zeros, which we found more useful and not requiring any Lowasiejicz type inequalities...).

Topology of $X,Y$ apparently controls the topology of singularities, i.e. nontrivial topology tends to force discontinuous optimal transports.

For example, one can prove the following: If $X=D^2$ unit disk, with boundary $Y=\partial X=S^1$, and if $X,Y$ are equipped with canonical volume measures $\sigma$, $\tau$ satisfying $\int_X \sigma\int_Y\tau$, and if $c: X\times Y\to \mathbb{R}$ is a repulsive cost (satisfying standard (Twist) property and nondegeneracy), then the singularity $Z$ of $c$-optimal transports has the homotopy type of the source $(X,\sigma)$. Therefore $Z$ is nonempty and has exactly one connected component. E.g., $Z$ cannot consist of two discrete points.

If rather we assume the cost $c$ is attractive (e.g. quadratic cost $c=d^2/2$), then the singularity $Z$ does not have the homotopy type of $X$, but is homotopic to an ``image" of the homotopy type of $Y$. Frequently the image is degenerate and $Z$ is empty, i.e. the optimal transport is regular and single-valued everywhere.

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Numerically I don't think there are any relations between Wasserstein distances (the positive real numbers found by minimal averaged squared distance of all couplings) and the topological integers or $\mathbb{R}$-valued invariants arising from (co)homology, e.g. volume, Euler characteristic, Betti numbers, etc..

However there definitely appears to be relation between the singularities of $c$-optimal transports for repulsive costs and their homological invariants. The idea is that the singularities of $c$-optimal transports are best defined by Kantorovich's contravariant singularity functor $Z=Z(c, \sigma, \tau):2^Y \to 2^X$, where $Z(Y_I):=\cap_{y\in Y_I} \partial ^c \psi(y)$, where $\partial ^c \psi$ is the $c$-subdifferential of the $c$-concave potential $\psi=\psi(y)$ arising from dual maximization program. The functor gives a $Y$-parameter cellulation of $X$, where the cells are the $c$-subdifferentials. When the cost $c$ is repulsive, then typically alot of the inclusions $Z(Y_J) \hookrightarrow Z(Y_I)$ are strong deformation retracts for subsets $Y_I \hookrightarrow Y_J$ of $Y$. This leads to a large codimension subset $Z_J$ for which $Z_J \hookrightarrow X$ is a strong deformation retract and $\dim(Z_J)=\dim(X)-j+1$. In this way we construct souls/spines of the source $X$ in the singularity of optimal transports.

There's much more to say, and it's very interesting question. I have several ongoing projects applying OT to Algebraic Topology. In fact, OT naturally leads to topology via the ideas of Dold-Thom Theorem, especially when we study the homotopy groups of ``electro neutral configurations", which is the additive abelian group of all finitely supported distributions $f=\sum n_x .x$, where $n_x\in \mathbb{Z}$ and $x\in X$ satisfying $\sum n_x=0$.

Answer 2

As for me, two trivial cases currently can be safely stated.

1. If $\mathcal{M}$ is $C^{\infty}$-diffeomorphic to $\mathcal{N}$, then $(*) = 0$.

2. In the toy case between $D^{1}$ and $S^{1}$, where $d\mu = p\delta_{0} + (1-p)d\text{Unif}_{D^{1}}$ and $d\nu = d\text{Unif}_{S^{1}}$, $ (*) = p\inf\int{d^{2}(y_{0}, y)}d\nu(y)$ exactly, since one can always construct a $C^{\infty}$-diffeomorphism from $D^{1}\backslash{\{(0,0)\}}$ to $S^{1}$.

Since $\beta_{1}(D^{1}) = 0$, $\beta_{1}(S^{1}) = 1$, $\beta_{1}(D^{1}\backslash{\{(0,0)\}}) = 0$, is there some possible relation between the different betti numbers and the non-vanishing $(*)$ in the toy case?