# When Optimal Transport Meets
Information Geometry

Gabriel Khan  and  Jun Zhang<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Information geometry and optimal transport are two distinct geometric
frameworks for modeling families of probability measures. During the recent
years, there has been a surge of research endeavors that cut across these two
areas and explore their links and interactions. This paper is intended to
provide an (incomplete) survey of these works, including entropy-regularized
transport, divergence functions arising from $c$-duality, density manifolds
and transport information geometry, the para-Kähler and Kähler geometries
underlying optimal transport and the regularity theory for its solutions. Some
outstanding questions that would be of interest to audience of both these two
disciplines are posed. Our piece also serves as an introduction to the Special
Issue on Optimal Transport of the journal Information Geometry.

###### Contents

  1. 1 Optimal Transport: A Brief Overview
    1. 1.1 The Monge and Kantorovich formulations
    2. 1.2 Solving the Kantorovich dual problem
    3. 1.3 Existence theory to the Monge problem
    4. 1.4 The regularity of the Monge map & the MTW tensor
      1. 1.4.1 Local obstruction to regularity
    5. 1.5 The Wasserstein metric
    6. 1.6 Displacement interpolation
    7. 1.7 Otto calculus
  2. 2 Linking Optimal Transport to Information Geometry
    1. 2.1 Comparing Wasserstein geometry with information geometry
      1. 2.1.1 “Vertical” versus “horizontal” displacement
      2. 2.1.2 Topological considerations
    2. 2.2 Entropy-regularized transport
      1. 2.2.1 Computational optimal transport
    3. 2.3 Extending Legendre duality to $\lambda$-duality
    4. 2.4 Relating Otto calculus to information geometry
      1. 2.4.1 Upstairs and downstairs probability measures
      2. 2.4.2 Transport information geometry
    5. 2.5 Other papers included in this Special Issue
  3. 3 Kähler versus para-Kähler Geometries for Optimal Transport
    1. 3.1 The para-Kähler geometry of optimal transport
    2. 3.2 Kähler geometry of optimal transport
      1. 3.2.1 Some open problems related to this geometry
    3. 3.3 Kähler-Ricci flow and other smoothing flows
      1. 3.3.1 Smoothing flows in optimal transport
  4. 4 Conclusion

##  1\. Optimal Transport: A Brief Overview

###  1.1. The Monge and Kantorovich formulations

Optimal transport is a classic area of mathematics which combines ideas from
geometry, analysis, measure theory, and probability. Today, it is a thriving
area of research, both in the pure and applied settings. Furthermore, it has
many practical applications ranging from logistics, economics, computer
vision, imaging processing, and many other fields.

Optimal transport was first studied by Gaspard Monge in 1781 [Mon81]. In this
work, Monge was interested in finding the most efficient way to construct
military fortifications (or other architecture) from a distribution of raw
material. To study this topic from a mathematical standpoint, he posed the
following question, which is now known as the _Monge problem_ of optimal
transport.

######  Problem 1 (Monge’s problem).

A worker is tasked with moving a pile of rubble (deblais) into a prescribed
configuration (remblais). The worker must complete the job, but would like to
minimize the effort required, and so move the rubble in the most cost-
efficient way possible. How should this be done?

Although the Monge problem is a natural question and is relatively simple to
state, it turns out to be quite difficult to solve, and a satisfactory answer
for this problem took over 200 years to develop (see Section 1.3 for details).
As a result, much of the work in optimal transport instead uses a framework
developed by Kantorovich [Kan06] in the 1940’s. To contrast this with the
original problem of Monge, we will call this the “Kantorovich problem”
(although it is sometimes known as the Monge-Kantorovich problem.)

######  Problem 2 (Kantorovich problem of optimal transport).

Let $(X,\mu)$ and $(Y,\nu)$ be two Polish probability spaces and $c:X\times
Y\to\mathbb{R}\cup\\{\infty\\}$ be a lower semicontinuous cost function. The
Kantorovich problem seeks to find a coupling $\pi$ of $\mu$ and $\nu$ (i.e. a
probability measure on $X\times Y$ whose marginals are $\mu$ and $\nu$,
respectively) which achieves the smallest total cost

(1) $\inf_{\pi\in\Pi(\mu,\nu)}\int_{X\times Y}c(x,y)\,d\pi(x,y).$

Here, $\Pi(\mu,\nu)$ is the set of all couplings between $\mu$ and $\nu$.
Intuitively, $\mu$ is the shape of the original configuration and $\nu$ is the
target configuration, and transport plan at a point $x\in X$ is given by the
disintegration of the coupling $\pi$. The cost function $c$ gives the cost of
moving a unit of mass from a point $x\in X$ to a point $y\in Y$ and the
integral

$\int_{X\times Y}c(x,y)\,d\pi(x,y)$

is then the total cost of transportation according to the measure $\pi$. Which
cost function one uses depends on the context of the problem, but is fixed as
part of the problem statement. In Monge’s work, $X$ and $Y$ were taken to be
sets in Euclidean space and the cost function was simply the distance
$c(x,y)=|x-y|$. For many applications however, it is preferable to instead use
the squared-distance $c(x,y)=|x-y|^{2}$.

The Kantorovich framework is a flexible and powerful approach to study optimal
transport for several reasons. First, an optimal coupling exists under very
general circumstances.

######  Theorem 1.

Suppose that $\psi:X\to\mathbb{R}\cup-\infty$ and
$\phi:Y\to\mathbb{R}\cup-\infty$ are two upper-semicontinuous functions with
$\psi\in L^{1}(\mu)$ and $\phi\in L^{1}(\nu)$, such that
$c(x,y)\geq\psi(x)+\phi(y)$. Then there is a coupling $\pi\in\Pi(\mu,\nu)$
which minimizes the total cost.

In particular, if the cost function is non-negative, an optimal coupling
exists (in this case, we take $\psi$ and $\phi$ to be identically zero).

Second, the Kantorovich problem has a dual formulation, which plays an
essential role in proving both theoretic and numeric results in optimal
transport. Take $\tilde{c}(x,y)=-c(x,y)$, and we consider the problem of
maximizing

$\displaystyle\sup_{\pi\in\Pi(\mu,\nu)}\int_{X\times
Y}\tilde{c}(x,y)\,d\pi(x,y)=$
$\displaystyle\inf_{\widetilde{\phi},\widetilde{\psi}}\left\\{\left(\int_{X}\widetilde{\psi}(x)d\mu(x)+\int_{Y}\widetilde{\phi}(y)d\nu(y)\right)~{}\Big{|}~{}\begin{aligned}
\widetilde{\psi}\in L^{1}(\mu),~{}\widetilde{\phi}\in L^{1}(\nu),\\\
\widetilde{\psi}(x)+\widetilde{\phi}(y)\geq\tilde{c}(x,y)\end{aligned}\right\\}.$

An economic interpretation the Kantorovich dual problem is clearly explained
in Villani’s textbook ([Vil09], Chapter 5). Under relatively mild assumptions
(for instance, when $c$ is continuous, real-valued and satisfies some mild
integrability assumptions), the infimum on the right hand side is obtained.

Treating $\tilde{c}\equiv-c$ as the gain, the maximizing and minimizing
versions of the optimal transport problem are equivalent, with the optimal
couplings and solutions unchanged (apart from
$\widetilde{\psi}\equiv-\psi,\widetilde{\phi}\equiv-\phi$). This maximizing
notation is more congruent with information geometric convention of using
convex functions and treating the Legendre transform as a maximization
(instead of minimization) problem. For notational simplicity, we will drop the
$\,\widetilde{\mbox{}}\,$ on the functions $\psi,\phi$ in the rest of the
paper. See the Remark in the next subsection.

###  1.2. Solving the Kantorovich dual problem

Solutions to the Kantorovich dual problem are linked to a duality which
appears throughout optimal transport, dubbed $c$-duality. More precisely, the
functions $\psi$ and $\phi$ in the formulation of Kantorovich problem are
linked: $\phi=\psi^{c}$, where $\psi^{c}$ is the $c$-conjugate of $\psi$,
defined as

(2) $\psi^{c}(y)=\sup_{x\in X}\left(\tilde{c}(x,y)-\psi(x)\right)$

or equivalently,

(3) $\psi^{c}(y)=-\inf_{x\in X}\left(c(x,y)+\psi(x)\right).$

Furthermore, the optimal coupling $\pi$ is concentrated on the
$c$-subdifferential of $\psi$, which is defined as follows:

$\partial_{c}\psi=\left\\{(x,y)\in X\times
Y\,|\,\psi^{c}(y)+\psi(x)=\tilde{c}(x,y)\right\\}.$

Given an arbitrary $\psi$, the function $\psi^{c}$ defined by (2) or (3) is
said to be $c$-convex, and a $c$-convex function is exactly one that can be
realized as the $c$-conjugate of some other function.

Taking the $c$-conjugate again, we find that

$\psi^{cc}(x)=\sup_{y\in Y}\left(\tilde{c}(x,y)-\psi^{c}(y)\right)$

or

$\psi^{cc}(x)=-\inf_{y\in Y}\left(c(x,y)+\psi^{c}(y)\right).$

This pattern continues for further applications of $c$-conjugation. It is
readily shown (see the tutorial by Ambriosio and Gigli [AG13]) that

  1. (i).

$\psi^{ccc}=\psi^{c}$;

  2. (ii).

$\psi^{c}$ is always $c$-convex;

  3. (iii).

$\psi^{cc}=\psi$ when $\psi$ is $c$-convex.

The $c$-conjugate transform is a generalization of Legendre duality: for the
cost $c(x,y)=-x\cdot y$, that is, $\tilde{c}(x,y)=x\cdot y$, $c$-duality
becomes the regular Legendre duality: the $c$-subdifferential is the usual
subdifferential for a convex function, and the $c$-conjugate is precisely the
standard Legendre transformation. This link between optimal transport and the
analysis of $c$-convex functions lies at the heart of optimal transport.

#### Remark

The language of $c$-duality, as used in optimal transport, is often cast in
$c$-concave functions by standard references. In information geometry which
invokes the regular Legendre duality, convex functions are adopted for
convenience. Of course, a convex function can be made concave simply by
multiplying it by $(-1)$, i.e., changing its sign; however this is not the
case of $c$-convexity and $c$-concavity of functions. To facilitate comparison
in this article, we restate the results of $c$-duality in terms of
$c$-convexity instead of $c$-concavity.

###  1.3. Existence theory to the Monge problem

In Monge’s work, he made the additional assumption that $\pi$ is concentrated
on the graph of a measurable map $T:X\to Y$. In other words, each point in $X$
is sent to a unique point in $Y$. A solution to the Monge problem need not
exist in general. For instance, if $\mu$ is atomic but $\nu$ is continuous,
then there are no transport maps at all, let alone an optimal one.

One great success of the modern theory of optimal transport is to develop a
satisfactory existence theory for the Monge problem. In particular, Brenier
[Bre87], Gangbo-McCann [GM96], and Knott-Smith [KS84] established that for
fairly general cost functions and measures, the solution to the Kantorovich
problem is actually a solution to the Monge problem.

######  Theorem 2.

Let $X$ and $Y$ be two open domains of $\mathbb{R}^{n}$ and consider a cost
function $c:X\times Y\to\mathbb{R}$. Suppose that $d\mu$ is a smooth
probability density with respect to the Lebesgue measure supported on $X$ and
that $d\nu$ is a smooth probability density with respect to the Lebesgue
measure supported on $Y$. Suppose that the following conditions hold:

  1. i.

the cost function $c$ is of class $C^{4}$ with $\|c\|_{C^{4}(X\times
Y)}<\infty$;

  2. ii.

for any $x\in X$, the map $Y\ni y\to c_{x}(x,y)\in\mathbb{R}^{n}$ is
injective;

  3. iii.

for any $y\in Y$, the map $X\ni x\to c_{y}(x,y)\in\mathbb{R}^{n}$ is
injective111Items ii. and iii. are often known collectively as the _Twist_
condition.;

  4. iv.

$\det(c_{x,y})(x,y)\neq 0$ for all $(x,y)\in X\times Y$.

Here the subscript of $c$ denotes taking partial derivative, with respect to
$x$ if before the comma and with respect to $y$ if after the comma.

Then, there exists a $c$-convex function $\psi:X\to\mathbb{R}$ such that the
map $\mathbb{T}_{\psi}:X\to Y$ defined by
$\mathbb{T}_{\psi}(x):=c\textrm{-}\exp_{x}(\mathsf{D}\psi(x))$ is the unique
optimal transport map sending $\mu$ onto $\nu$. Furthermore,
$\mathbb{T}_{\psi}$ is injective $d\mu$-a.e. and is an Alexandrov solution to
the Jacobian equation

(4)
$|\det(\mathsf{D}\mathbb{T}_{\psi}(x))|=\frac{d\mu(x)}{d\nu(\mathbb{T}_{\psi}(x))}\hskip
14.45377ptd\mu-a.e.$

In order to express Equation (4) more concretely, we recall the notion of the
$c$-exponential map (denoted $c\textrm{-}\exp_{x}$).

######  Definition 1 ($c$-exponential map).

For any $x\in X,y\in Y,p\in\mathbb{R}^{n}$, the $c$-exponential map satisfies
the following identity.

$c\textrm{-}\exp_{x}(p)=y\iff p=-c_{x}(x,y)=\tilde{c}_{x}(x,y).$

The name $c$-exponential comes from the fact that on a Riemannian manifold
with the cost $c=d^{2}(x,y),$ this coincides with the standard exponential
map. For a more complete reference on the existence of solutions to the Monge
problem, see Chapters 10-12 of [Vil09].

###  1.4. The regularity of the Monge map & the MTW tensor

As shown in Theorem 2, for fairly general costs and measures, the solution to
the Kantorovich problem will also be a solution to the Monge problem, and it
is induced by the $c$-subdifferential of a potential function $\psi$.
Furthermore, this potential function will be a weak solution to a fully non-
linear degenerate-elliptic PDE known as the Jacobian equation. From here, the
natural follow-up question is to understand the behavior of the solution — is
it continuous? injective? smooth?

To answer these questions, one must find a priori estimates for the Jacobian
equation. A full discussion of this theory is outside the scope of this
survey, but let us provide a brief summary. A priori, the potential function
$\psi$ will be $c$-convex and locally Lipschitz, which implies that its
$c$-subdifferential (and thus the transport map) is defined Lebesgue almost-
everywhere. However, the transport map need not be continuous, even for smooth
measures. For example, for the squared-distance cost $c(x,y)=|x-y|^{2}$,
Caffarelli [Caf92] showed that for the transport between arbitrary smooth
measures to be continuous, it is necessary to assume that the target domain
$Y$ is convex. In other words, non-convexity of $Y$ is a global obstruction to
regularity.222There are easier ways to see that the optimal transport map
might fail to be continuous. For instance, if $X$ is connected whereas $Y$ is
disconnected (and the measures have full support), there are no continuous
transport maps at all, let alone an optimal one.

From an analytic perspective, the reason for this somewhat pathological
behavior is that Equation (4) is degenerate-elliptic. To explain its meaning,
let us specialize to the squared-distance cost in Euclidean space, in which
case the $c$-convexity of $\psi$ is convexity in the usual sense and the
Jacobian equation simplifies to the following:

(5)
$\det\left(\mathsf{D}^{2}\psi(x)\right)=\frac{d\mu(x)}{d\nu(\mathsf{D}\psi(x))}.$

For a convex function $\psi$, the linearized Monge-Ampére operator

$Lf=\sum_{i,j}\left(\left(\operatorname{det}\mathsf{D}^{2}\psi\right)\left(\mathsf{D}^{2}\psi\right)^{-1}\right)^{ij}\frac{\partial^{2}}{\partial
x^{i}\partial x^{j}}f$

is non-negative definite. Furthermore, the potential $\psi$ will satisfy
Equation (5) where we replace the left-hand side by its linearized Monge-
Ampére equation $L$. As such, to establish regularity for $\psi$, we need to
derive estimates for solutions to

$L\psi=\frac{d\mu(x)}{d\nu(\mathsf{D}\psi(x))},$

which is a quasi-linear equation.

However, without an a priori bound on $\mathsf{D}^{2}\psi$, $L$ can have
arbitrarily small eigenvalues, which prevents the use of standard elliptic
theory to prove regularity of $\psi$. On the other hand, if we were able to
establish an a priori $C^{2}$ estimate on $\psi$, then the linearized operator
would be uniformly elliptic, and we would be able to use Schauder estimates to
show that the transport is smooth.

This phenomena gives rise to a striking dichotomy. When the cost functions and
measures satisfy the hypotheses of Theorem 2 and the measures are smooth
(i.e., $C^{\infty}$), the optimal transport map is either $C^{\infty}$-smooth,
or else discontinuous.333This dichotomy only holds in the interior. It is
possible for the Monge map to be smooth in the interior, but become singular
near the boundary. In the latter case, work of Figalli and De Phillippis
[DPF15] shows that the discontinuity occurs on a set of measure zero and that
the transport is smooth elsewhere. Kim and Kitagawa [KK16] showed that for
costs which satisfy the MTW(0) condition (Definition 2), there cannot be
isolated singularities. However, there are many open questions about the
structure of the singular set.

######  Question 1 (De Phillippis-Figalli [DPF14]).

For an arbitrary smooth domain and cost function, does the singular set for
the transport have Hausdorff dimension at most $n-1$? If so, is this set
rectifiable?

####  1.4.1. Local obstruction to regularity

In the 1990s, Caffarelli, Delanoe, Urbas and several others developed a theory
for the regularity of the Monge problem when the cost function is the squared
distance in Euclidean space (see, e.g., [Caf92, Del91, Urb97]). Though
imposing the additional assumption that the target space $Y$ was convex, this
line of work was able to apply techniques from Monge-Ampére equations, due to
its simplified form (5), to derive the relevant a priori estimates. However,
for more general cost functions, the problem of regularity remained open.

In 2005, a breakthrough paper by Ma, Trudinger, and Wang [MTW05] established
smoothness for the transport for general costs under two additional
assumptions. The first was a global condition, that the supports of the
initial and target measures are relatively c-convex. The second was a local
condition, that a certain fourth-order quantity, known as the MTW tensor, is
positive. Later, Trudinger and Wang [TW09] extended their previous work to the
case when this tensor is non-negative (i.e., when $\mathfrak{S}(\xi,\eta)\geq
0$).

######  Definition 2 (MTW($\kappa$) condition).

A cost function $c$ is said to satisfy the MTW($\kappa$) condition if for all
vector-covector pairs $(\xi,\eta)$ with $\eta(\xi)=0$, the following
inequality holds.

(6)
$\mathfrak{S}(\xi,\eta):=\sum_{i,j,k,l,p,q,r,s}(c_{ij,p}c^{p,q}c_{q,rs}-c_{ij,rs})c^{r,k}c^{s,l}\xi^{i}\xi^{j}\eta^{k}\eta^{l}\geq\kappa\|\eta\|^{2}\|\xi\|^{2}$

Here, the notation $c_{I,J}$ denotes $\partial_{x^{I}}\partial_{y^{J}}c$ for
multi-indices $I$ and $J$. Furthermore, $c^{i,j}$ denotes the matrix inverse
of the mixed derivative $c_{i,j}$. Finally, all the derivatives and the MTW
tensor are understood to be taken at the point $(x,y)$.

It is worth remarking that in (6), all of the terms involve at least a third
derivative, so $\mathfrak{S}$ identically vanishes for the squared-distance
cost in Euclidean space, which explains why it did not appear in the earlier
work on the regularity problem.

Although the Ma-Trudinger-Wang theory was a significant breakthrough, it also
raised many questions. For instance, the geometric significance of
$\mathfrak{S}$ was not well understood and it was unclear whether it plays an
essential role in optimal transport or if it was merely a technical assumption
in Ma-Trudinger-Wang’s work.

These questions were studied by Loeper [Loe09], who showed that for costs
which are $C^{4}$, the MTW(0) condition is equivalent to requiring that the
$c$-subdifferential of an arbitrary $c$-convex function be connected. This
provides a geometric interpretation of the MTW tensor in terms of convex
analysis, and shows that the failure of MTW condition is a local obstruction
to regularity. More precisely, this work shows for cost functions which do not
satisfy the MTW condition, the space of smooth optimal transports fails to be
dense within the space of all optimal transports. In other words, given a cost
which does not satisfy the MTW(0) condition, it is possible to find smooth
measures $\mu$ and $\nu$ for which the Monge optimal transport is
discontinuous.

There are many open questions remaining in the regularity theory of optimal
transport, and we will mention several in Section 3. However, let us mention
one prominent conjecture due to Villani.

######  Conjecture 3 (Villani’s Conjecture).

If $(M,g)$ is a Riemannian manifold such that the cost $c(x,y)=d(x,y)^{2}$
satisfies the MTW condition, then all of the injectivity domains of $M$ are
convex.

###  1.5. The Wasserstein metric

Though optimal transport is well defined for a very general class of cost
functions, many important applications consider the case when the cost is some
power of the distance between points on a metric space $(X,d)$. This gives
rise to the notion of the Wasserstein metric, which provides a notion of
distance between probability measures. The name “Wasserstein metric” is also
referred as the Kantorovich-Rubinstein distance, the Monge-Kantorovich
distance (both of which might be more historically accurate) or the Earth
Mover Distance (which is much more descriptive) [San15].

To define the Wasserstein metric, we consider the space of probability
measures with finite $p$-th moment:

(7) $\mathbb{P}_{p}(X)=\left\\{\textrm{probability
measures}\,\mu\,\Big{|}\,\int_{X}d(x,x_{0})^{p}d\mu(x)<\infty\textrm{ for any
}x_{0}\in X\right\\}$

For two probability measures $\mu,\nu\in\mathbb{P}_{p}(X)$, we can define
their $p$-Wasserstein distance as

$W_{p}(\mu,\nu)=\left(\inf_{\pi\in\Pi(\mu,\nu)}\int_{M\times
M}d(x,y)^{p}d\pi\right)^{1/p}.$

This is a genuine metric in that it is positive-definite, finite, symmetric,
and satisfies the triangle inequality. Note that here “metric” is used in the
sense of a distance function, and is different from its usage of Riemannian
metric (an inner product) or a Finsler metric (a smooth, homogeneous and
subadditive “norm” function) at each point of the manifold.

###  1.6. Displacement interpolation

Displacement interpolation is a continuous-time version of optimal transport
which interpolates between the initial and final measure [McC97]. For this,
one must assume that the source space $X$ and the target space $Y$ are the
same. However, the setting for displacement interpolation is quite general
otherwise, and can be defined when $X$ is a Polish metric space and the cost
functions is of the form

(8)
$c(x,y)=\inf\left\\{\mathcal{L}(\gamma)\,|\,\gamma_{0}=x,\,\,\gamma_{1}=y\right\\}$

where $\gamma$ is a path connecting $x$ to $y$ and $\mathcal{L}$ is a
Lagrangian action

$\mathcal{L}(\gamma)=\int_{0}^{1}L\left(\gamma_{t},\dot{\gamma}_{t},t\right)dt.$

In this section, we will focus on the case where the cost is the distance to
some power, so will specialize to the associated action

$\mathcal{L}_{p}(\gamma)=\int_{0}^{1}\frac{1}{p}\,\left|\dot{\gamma}_{t}\right|^{p}dt.$

Using the action $\mathcal{L}_{p}$, it is possible to induce on
$\mathbb{P}_{p}(X)$ (where $(X,d)$ is a priori merely a metric space) the
structure of a _length space_ , where the distance $d(x,y)$ can be realized as
the infimum as the lengths of paths connecting $x$ and $y$ as end-points:

$d(x,y)=\inf_{\gamma}\left\\{\ell(\gamma)~{}\big{|}\,\gamma_{0}=x,\,\gamma_{1}=y\right\\}$

where $\ell(\gamma)$ is defined to be the intrinsic length of the path

$\ell(\gamma)=\sup_{P}\left\\{\sum_{P}d\left(\gamma_{p_{i-1}},\gamma_{p_{i}}\right)~{}|~{}P\textrm{
a partition of $[0,1]$ with }p_{i}\in P\right\\}.$

######  Theorem 4 ([Vil09] Corollary 7.22).

Let $(X,d)$ be a complete separable, locally compact length space. Then, given
any two measures $\mu_{0},\mu_{1}\in\mathbb{P}_{p}(X)$, and a continuous curve
$\left(\mu_{t}\right)_{0\leq t\leq 1}$ in $\mathbb{P}_{p}(X)$, the following
properties are equivalent:

  1. (1)

$\mu_{t}$ is the pushforward by the flow of minimizing, constant-speed
geodesics from $x$ to $\mathbb{T}(x)$, where $\mathbb{T}:X\to X$ is the
solution to the optimal transport problem.444If the solution to this transport
problem is not Monge, this should be interpreted in the sense of couplings
where mass may split at the initial time along various geodesics.

  2. (2)

$\left(\mu_{t}\right)_{0\leq t\leq 1}$ is a geodesic curve in the space
$\mathbb{P}_{p}(X)$.

Moreover, if $\mu_{0}$ and $\mu_{1}$ are given, there exists at least one such
curve.

In other words, displacement interpolation provides geodesics in the space of
probability measure, and this interpolation is induced by geodesics in the
underlying space $X$. Furthermore, one can understand the geometry of $X$ by
studying the behavior of the measures along displacement interpolation. For
instance, the convexity of certain entropy functionals along geodesics in
$\mathbb{P}_{p}(X)$ is equivalent to Ricci lower bounds555More precisely, most
of these bounds involve both the curvature and the dimension, so are known as
curvature-dimension bounds. on the original space $X$ [vRS05, OT11].

###  1.7. Otto calculus

While the length space structure of $\mathbb{P}_{p}(X)$ holds for arbitrary
values of $p$, the case $p=2$ is distinguished. When the space $X$ is a
Riemannian manifold, Otto [Ott01] showed that the space $\mathbb{P}_{2}(X)$
(metrized by the $2$-Wasserstein distance) also admits the structure of a
formal Riemannian metric. For ease of exposition, let us look at the special
case of $X\subseteq\mathbb{R}^{n}$ a part of Euclidean space, and consider the
subset of $\mathbb{P}_{2}(\mathbb{R}^{n})$ which can be represented as
probability density functions over $\mathbb{R}^{n}$ with respect to the
Lebesgue measure $dx\equiv d\mbox{vol}_{x}$:

$\mathcal{M}=\mathbb{P}_{2}(X)=\left\\{\rho\,|\,\rho(x)>0,\,\forall x\in
X\subset\mathbb{R}^{n},\,\int_{X}\rho\,dx=1\right\\}.$

More precisely, the function $\rho$ is the Radon-Nikodym derivative of a
measure which is Lebesgue absolutely continuous. One can then consider the
tangent space at a point $\rho\in\mathbb{P}_{2}(X)$ as

$T_{\rho}\mathcal{M}=\left\\{s\,|\,s\text{ is a function on }X,\,\int
sdx=0\right\\}.$

To see why this can be understood as the tangent space, consider the “nearby”
probability measure $\rho_{\epsilon}=\rho+\epsilon s$. The integral of
$\rho_{\epsilon}$ will be 1: $\int\rho_{\epsilon}\,dx=1$. Furthermore, by
restricting $s$ and $\rho$ to reside in a suitable function space (e.g.,
$C_{c}^{\infty}(\mathbb{R}^{n})$) and to satisfy
$\textrm{supp}(s)\subset\textrm{supp}(\rho)$, we can ensure that for
$\epsilon$ sufficiently small, $\rho+\epsilon s$ is non-negative. So

$s(x)=\lim_{\epsilon\rightarrow
0}\frac{\rho_{\epsilon}(x)-\rho(x)}{\epsilon}.$

One can then use the fluid-dynamic elliptic equation

(9) $-\mathsf{div}\cdot(\rho\,\mathsf{grad}\,\phi)=s$

to obtain a second set of “coordinates” for the tangent fibers, which is
denoted $\phi$:

$T_{\rho}\mathcal{M}\cong\left\\{\text{ functions }\phi\text{ on
}\mathbb{R}^{n}\right\\}/\\{\phi_{1}-\phi_{2}\equiv const\\}.$

In other words, functions which differ by constants are deemed identical and
are then identified with the original coordinates $s$ via the solution to an
elliptic equation. Here, since this computation is merely formal, one omits
any discussion of which Sobolev spaces each of these families of functions
reside in, or what the boundary conditions should be.

In this setting, Otto was able to prescribe a Riemannian metric $g$ on
$\mathbb{P}_{2}(X)$, defined as the quantity

$g_{\rho}\left(s_{1},s_{2}\right)=\int_{X}(\mathsf{grad}\,\phi_{1}\cdot\mathsf{grad}\,\phi_{2})\,\rho\,dx\,\,.$

Here, the dot-product is the standard inner-product operation on
$X=\mathbb{R}^{n}$. The Riemannian metric $g$ induces the 2-Wasserstein
distance on $\mathbb{P}_{2}(\mathbb{R}^{n})$ as its distance function, so that
$\mathbb{P}_{2}(\mathbb{R}^{n})$ admits a formal Riemannian structure. If $X$
is a Riemannian manifold instead of a subset of Euclidean space
$\mathbb{R}^{n}$, then the dot-product $\cdot$ can be replaced by a Riemannian
metric $\langle\cdot,\cdot\rangle$ on $X$, and Equation (9) is defined in
terms of this metric (on $X$) as well. Before moving on, it is worth
mentioning contemporaneous work of Benamou and Brenier [BB00], which studied
this structure from a fluid mechanical perspective.

One ought to clearly distinguish the two spaces involved in Otto’s framework:

  * •

$X$, which we call the “downstairs” manifold, that is a Riemannian manifold
itself endowed with a Riemannian metric $\langle\cdot,\cdot\rangle$, a special
case being a Euclidean space; each point $x\in X$ is the instantiation of a
random variable;

  * •

$\mathbb{P}_{2}(X)$, which we call the “upstairs” space and which admits a
formal Riemannian metric $g_{\rho}(\cdot,\cdot)$; each point
$\rho\in\mathcal{M}=\mathbb{P}_{2}(X)$ is a probability measure on $X$.

See Section 2.4.1 for examples. So

  * •

$\mathsf{grad}$ and $\mathsf{div}$ are operators on the downstairs manifold
$X$;

  * •

$s$ and $\phi$ are functions on $X$, linked through Equation (9) with respect
to any given $\rho$ (i.e., treating $\rho$ as fixed);

  * •

$\rho$ is an element of $\mathcal{M}=\mathbb{P}_{2}(X)$ while $s$ is an
element of $T\mathcal{M}$.

Otto calculus plays a central role in modern optimal transport — one of its
hallmarks is to show that Ricci curvature lower bounds of $X$ gives rise to
geodesic convexity of certain entropy functionals on $\mathbb{P}_{2}(X)$.
Using ideas from Otto calculus, Lott, Sturm, Villani and others have developed
notions of Ricci curvature bounds on non-smooth metric-measure spaces. For an
overview of this theory, we refer the reader to the following survey [Vil16].
We also refer to the following paper of Lott [Lot08] for curvature
calculations of the Wasserstein metric. It is worth emphasizing that the
Riemannian structure is merely formal, as $\mathbb{P}_{2}(X)$ is generally not
a Banach manifold. As such, many of the rigorous proofs in this line of work
do not directly use the Otto calculus, but instead follow a motto of Villani
to “[t]hink Eulerian, prove Lagrangian” ([Vil09], page 444). In other words,
one uses the Otto calculus to derive results, but establishes them rigorously
using displacement interpolation.

In the case where the “upstairs” space of probability measures
$\mathbb{P}_{2}(X)$ admits a Kullback-Leibler divergence, then the gradient
flow (on the upstairs space, hence subscripted by $W_{2}$) of this KL
divergence can be calculated:

$\frac{\partial\rho_{t}}{\partial
t}=-\mathsf{grad}_{W_{2}}K(\rho_{t}||\rho_{0}),$

which gives rise to the Fokker-Planck equation

$\frac{\partial\rho_{t}}{\partial
t}=\nabla\cdot(\rho_{t}\nabla\phi)+\Delta\rho_{t}.$

where $\rho_{0}\simeq e^{-\phi}$ is the Gibbs measure (see Chapter 24 of
[Vil09], and see [JKO98] for a proof of convergence for this equation). There
are many results relating well-known PDEs to the gradient flows of certain
functionals in $\mathbb{P}_{2}(X)$ (or more generally $\mathbb{P}_{p}(X)$
[Ott01]), and this is a thriving area of functional analysis.

##  2\. Linking Optimal Transport to Information Geometry

In this section, we survey some areas of active research at the interface of
information geometry and optimal transport. Due to the anticipated familiarity
with information geometry by the audience of this Journal, we do not provide
any review of the framework of information geometry (see references like
[Ama12, AN00, AJVLS17]). Rather, we start by addressing some key differences
between Wasserstein geometry and information geometry.

###  2.1. Comparing Wasserstein geometry with information geometry

Both Wasserstein geometry and information geometry provide ways to quantify
the proximity of probability measures, the former through solutions to optimal
transport and the latter using divergence (or contrast) functions, such as
Kullback-Leibler (KL) divergence (or more generally, Csiszar’s $f$-divergence
[Csi64]). A divergence function is not a distance, in that it is not
necessarily symmetric or satisfy the triangular inequality. However, it is
still a non-negative function of two probability density functions $p$ and $q$
(or probability measures $\mu$ and $\nu$), and equals zero if and only if
$p=q$ almost surely.

There are several important distinctions between Wasserstein distance and
divergence functions. Firstly, they are defined on different classes of
probability measures. The KL divergence is only defined for probability
measures $\mu,\nu$ are which are absolutely continuous with respect to some
reference measure $\rho_{0}$ (or at least one with respect to another). More
precisely, on a sample space $X$, we can express the KL divergence as the
following

(10)
$K[p||q]=\int_{X}\frac{d\mu}{d\rho_{0}}\log\left(\frac{d\mu}{d\rho_{0}}/\frac{d\nu}{d\rho_{0}}\right)\,d\rho_{0}=\int_{X}\log\frac{d\mu}{d\nu}\,d\mu.$

As such, it is necessary to restrict our attention to probability measures
whose Radon-Nikodym derivatives (with respect to $\rho_{0}$) exist and live in
an appropriate Orlicz space (so that the integral (10) converges).

On the other hand, the $p$-Wasserstein distance is defined for any two
probability measures with finite $p$-th moment ($1\leq p<\infty)$, without any
assumptions about absolute continuity. In fact, the source and target
distributions may well have different supports on the sample space $X$. In
other words, the Wasserstein distance can be finite even when the support of
$\mu$ and $\nu$ is disjoint, which is not the case for the KL divergence.

####  2.1.1. “Vertical” versus “horizontal” displacement

Transport-based Wasserstein distances and information geometric divergence
functions both measure the “distance” between two probability
measures/densities, but in very different ways.

Figure 1. Vertical versus horizontal displacement

To explain this, let us first reframe the notion of a divergence in a slightly
different way. We suppose that $\mu$ and $\nu$ are absolutely continuous with
respect to a reference measure $\rho_{0}$ with Radon-Nikodym derivatives
$p(x)=\frac{d\mu}{d\rho_{0}}$ and $q(x)=\frac{d\nu}{d\rho_{0}}$. In this
setting, we can express the divergence $D$ between $\mu$ and $\nu$ in terms of
the following integral:

$D[p||q]=\int k(p(x),q(x))\,d\rho_{0}(x)$

where $k(a,b)$ is some non-negative smooth function on $\mathbb{R}_{\geq
0}^{2}$ which is equal to zero if and only if $a=b$. For instance, to
construct the KL-divergence, we use the function

$k(p(x),q(x))=q(x)-p(x)-p(x)\log\frac{q(x)}{p(x)},$

which satisfies $k(p(x),q(x))\geq 0,\,\forall x\in X.$ Divergence functions of
this form are sometimes called extended KL divergences [ZR95], and their
importance for generating the dualistic information geometry was recognized
and made use of extensively by the second named author [Zha04]. Furthermore,
the notion of gauge freedom [NZ18] can be introduced, allowing an arbitrary
monotone function to be used in defining the divergence function.

From this integral representation, we see that divergences $D$ compare the
“heights” $p(x)$ and $q(x)$ of the probability density functions at each point
$x$, and use the non-negative function $k(p(x),q(x))$ to measure the
difference in these heights. Then, we integrate these differences over the
random variable $x$ with respect to the reference measure $\rho_{0}$. In this
way, divergences $D$ measure displacement accumulated “vertically.” From this,
it immediately follows that these divergences remain invariant under measure-
preserving mappings of the underlying sample space (for example, when the
random variable $x$ in the sample space is relabeled, i.e., $X$ undergoes a
permutation).

On the other hand, the Wasserstein distance is defined by the cost of
optimally transporting one of the measures to the other. As such, this is
“horizontal” measure of displacement, in which mass from the source measure
$\mu$ is moved horizontally through space to the target measure $\nu$. Since
conjugation by a measure-preserving mapping generally does not preserve the
cost function or the optimal map, measure-preserving maps will not preserve
Wasserstein distances. Furthermore, on a Riemannian manifold one can find
measure-preserving mappings whose conjugation make the Wasserstein distance
arbitrarily small.

####  2.1.2. Topological considerations

Given that they are defined for different classes of probability measures, it
might be expected that the $p$-Wasserstein distance and KL divergence induce
different topologies, and this is indeed the case. Due to the asymmetry of the
KL divergence, there are actually two separate topologies depending on whether
one considers open balls with respect to the first or second argument (see,
e.g., [Bel15]). However, convergence in either topology implies convergence in
total variation, so both are fairly strong notions of convergence which are
invariant under measure-preserving mappings.

On the other hand, the Wasserstein distance metrizes the weak topology of
$\mathbb{P}_{p}$, in that convergence in Wasserstein distance implies weak
convergence of measures and convergence of their first $p$-th moments (Theorem
6.9 [Vil09]).

As a result of this, we generally (but not always) expect that the topology
induced by the Wasserstein distance will be coarser than that induced by the
Kullback-Liebler divergence. As such, we can often find bounds on the
Wasserstein distance in terms of the KL divergence, but essentially never the
other way round (except for finite spaces). However, when the measures are
further controlled in a stronger sense, it is sometimes possible to prove
interpolation results between the Wasserstein distance and KL divergence. Most
famously, this idea shows up in the “HWI-inequalities” of Otto and Villani
[OV00].

###  2.2. Entropy-regularized transport

One link between optimal transport and information geometry comes in the form
of entropy-regularized optimal transport. To explain this, it will be
convenient to specialize to the case of discrete probability measures (instead
of probability density functions) which are supported on finite sets.666See
Section 2.4 for related work in the continuous setting. Assuming the source is
supported on $X=\\{1,\cdots,n\\}$ and the target on $Y=\\{1,\cdots,m\\}$, with
a cost $C_{ij}$ incurred from transferring from $i\in X$ to $j\in Y$. The
coupling can be written as

$\Pi(p,q)=\left\\{P_{ij}:P_{ij}\geq
0,\,\sum_{ij}P_{ij}=1,\,\sum_{j}P_{ij}=p_{i},\,\sum_{i}P_{ij}=q_{j}\right\\}.$

Here,

$\sum_{i}p_{i}=\sum_{j}q_{j}=1,$

and the objective function for transport is:

$W(p,q)=\inf_{P\in\Pi(p,q)}\langle
P,C\rangle=\inf_{P\in\Pi(p,q)}\sum_{i,j}P_{ij}C_{ij}.$

As a linear-programming problem, there are standard method for solving optimal
transport over discrete probabilities, but this may be computationally
inefficient.

Cuturi [Cut13] proposed a relaxation to the above problem by introducing a
regularization term using the Shannon entropy $H(P)$:

(11) $\displaystyle W_{\epsilon}(p,q)$ $\displaystyle=$
$\displaystyle\inf_{P\in\Pi(p,q)}\langle P,C\rangle-\epsilon H(P)$
$\displaystyle=$
$\displaystyle\inf_{P\in\Pi(p,q)}\sum_{i,j}P_{ij}C_{ij}+\epsilon\sum_{i,j}P_{ij}\log
P_{ij}.$

Applying the Lagrange multiplier method to this entropy-regularized (also
called entropy-relaxed) problem, Cuturi showed that the solution is of the
form

(12)
$P^{opt}_{ij}=\exp\left[\frac{1}{\epsilon}\left(\alpha_{i}+\beta_{j}-C_{ij}\right)\right]=u_{i}K_{ij}v_{j}$

where $\alpha$ and $\beta$ are vectors of Lagrange multipliers, which are
specified by the following equations:

$\sum_{j}P^{opt}_{ij}=p_{i},\,\,\,\,\sum_{i}P^{opt}_{ij}=q_{j},\quad\forall
i,j$

and the matrix $K$ and vectors $u$ and $v$ denote

$K_{ij}=\exp\left(-\frac{1}{\epsilon}C_{ij}\right),\,\,\,\,u_{i}=\exp\left(\frac{\alpha_{i}}{\epsilon}\right),\,\,\,\,v_{j}=\exp\left(\frac{\beta_{j}}{\epsilon}\right).$

Note that using entropy $H(P)$ as the regularization term is equivalent to
using KL divergence $K[P||p\otimes q]$ as a regularizer. Though
$W_{\epsilon}(p,q)$ may no longer be a distance, it approaches the Wasserstein
distance $W(p,q)$ in the limiting case of $\epsilon$ tending zero.

Amari, Karakida, Oizumi [AKO18] observed that the solutions $P^{opt}$ from
Equation (12) form an exponential family, where $p,q$ are the expectation
parameters, and $\alpha,\beta$ are the natural parameters. Viewed in this way,
the Sinkhorn algorithm used to iteratively solve (12) has the information
geometric interpretation of successive applications of the $e$-projection to a
pair of $m$-autoparallel submanifolds. One can also define a Bregman-like
divergence function on these autoparallel submanifolds (the so-called Cuturi
function). A similar geometric interpretation, in terms iterative Bregman
projections, was given by Benamou et al. [BCC+15].

Muzellec et al. [MNPN17] extended Cuturi’s entropy-regularized framework to
the optimal transport problem by using Tsallis entropy. Of course, more
general regularizers were investigated under this “regularized transport”
framework. The work by Kurose, Yoshizawa, Amari [KYA21] in this issue extended
the regularizer from using Tsallis entropy to using the general deformed-
entropy, a standard move adopted by researchers of deformation theory since
Naudts [Nau04]. The work by Tsuisui [Tsu21] in this issue investigated a
generic strictly concave function as the regularizer. Explicit expressions
were obtained in these works, allowing solutions to be obtained iteratively
with computational efficiency.

####  2.2.1. Computational optimal transport

One main advantage to considering the entropy-regularized problem is that it
is possible to solve it much more quickly than the unregularized problem. For
a more complete reference on this subject, we refer the reader to the recent
text of Peyrè and Cuturi [PC19].

As mentioned previously, a major insight of Kantorovich was that the dual
problem is a linear program, which can be solved using the simplex algorithm
(or a similar such algorithm for linear programming). In practice, the
convergence to the solution of an optimal transport may happen very slowly. In
the worst case scenario, algorithms for solving linear programs may have
exponential complexity (although their average complexity is generally
polynomial).

To avoid slow convergence, one often does not try to solve such a problem
exactly, but instead tries to find an approximate solution which is close to
being optimal. Using an auction algorithm, one can find an arbitrarily good
approximation in polynomial time. More precisely, for $X$ and $Y$ finite sets
(say, both of size $N$), one can find an algorithm which terminates in
$O(N^{3}\log\|C\|_{\infty})$ steps, where $\|C\|_{\infty}$ is the $L^{\infty}$
norm of the cost function ([Ber98], page 264).

Using the entropy-regularized optimal transport, it is possible to do even
better. In particular, Cuturi also showed that solution (12) to the entropy-
regulated problem $W_{\epsilon}(p,q)$ can be obtained by the Sinkhorn
algorithm. In this algorithm, the $\alpha$ and $\beta$ vectors need not be
evaluated; only the vectors $u$ and $v$, which are iteratively calculated.
Sinkhorn’s algorithm (with entropy of size $\epsilon$) can be solved with time
complexity $O\left(N^{2}\log^{4}(N)\epsilon^{-3}\right)$, which saves a factor
of (almost) $N$ compared to the auction algorithm [PC19, ANWR17].

To explain why regularizing speeds up the convergence, it is helpful to
understand the behavior of a non-regularized optimal transport problem. For
instance, in many settings one will use the discretized optimal transport as
an approximation for a continuous optimal transport problem. If the associated
continuous problem satisfies the hypotheses of Theorem 2, the solution to the
continuous problem will be a Monge solution, which the Kantorovich solution
will approximate in a weak sense. As a result, the solution will concentrate
on a sparse region of the product space $X\times Y$ (in this context, a set of
size $O(N)$). This phenomena makes solving such problems difficult, since most
points in $X\times Y$ will not be in the support of the optimal coupling.

On the other hand, the solution to the entropy-regularized problem will be
more diffuse, which makes it more amenable to numeric computation. Along with
faster convergence for the computational algorithms, this phenomena also leads
to faster convergence in a statistical sense (page 59 [PC19]).

###  2.3. Extending Legendre duality to $\lambda$-duality

Legendre duality plays a crucial role in information geometry as it provides
the canonical divergence in a dually flat space ((for exponential or mixture
families of probability density functions, for instance)). In optimal
transport, a more generalized notion of $c$-duality appears, based on the
notion of $c$-conjugation (2), which generalizes the Legendre conjugation, in
which $c(x,y)$ is simply to be taken as $-\langle x,y\rangle$.

In a series of papers, Pal and Wong [PW16, PW18, Won19, PW20] applied optimal
transport to mathematical finance by invoking a special cost function
$c(x,y)=\log(1-\langle x,y\rangle)$. A new divergence function, called a
logarithmic divergence [PW16], was proposed for a suitably generalized convex
function $\psi$ that deforms the Bregman divergence. Wong [Won18] then
extended this to a one-parameter family of cost functions
$\tilde{c}(x,y)=\kappa_{\lambda}(\langle x,y\rangle)$, where $\lambda$ is a
real number777Originally [Won18] used $\alpha$ and treated $\alpha>0$ and
$\alpha<0$ cases separately. This is unified by [WZ21] based on the notion of
$\lambda$-exponential convexity. indexing the single-variable function
$\kappa_{\lambda}(\cdot)$:

$\kappa_{\lambda}(t)=\frac{1}{\lambda}\log(1+\lambda t),$

and $\log a$ is defined to be $-\infty$ if $a\leq 0$. The work of [Won18]
showed that this generates a class of deformed exponential families for which
the underlying space has _constant curvature_. A one-parameter family of
divergence, called $\lambda$-divergence was found which includes the
logarithmic divergence ($\lambda=-1$) and Bregman divergence ($\lambda=0$) as
special cases [Won18, WZ21]:

$\displaystyle D_{\lambda,\psi}(x,x^{\prime})$ $\displaystyle=$
$\displaystyle\psi(x)-\psi(x^{\prime})-\kappa_{\lambda}\left(\langle\mathsf{D}\psi(x^{\prime}),x-x^{\prime}\rangle\right)$
$\displaystyle=$
$\displaystyle\psi(x)-\psi(x^{\prime})-\frac{1}{\lambda}\log\left(1+\lambda\langle\mathsf{D}\psi(x^{\prime}),x-x^{\prime}\rangle\right)$
(14) $\displaystyle=$
$\displaystyle\psi(x)-\psi(x^{\prime})-\kappa_{\lambda}\left(\langle
x,u^{\prime}\rangle\right)+\kappa_{\lambda}\left(\langle
x^{\prime},u^{\prime}\rangle\right)$

where $\mathsf{D}\psi$ is the usual gradient operation on $\psi$,
$\mathsf{D}^{\lambda}\psi$ is the deformed gradient (appropriate for the
$\lambda$-deformed duality):

$\mathsf{D}^{\lambda}\psi(x)=\frac{\mathsf{D}\psi(x)}{1-\lambda\langle
x,\mathsf{D}\psi(x)\rangle},$

and

$u^{\prime}=\mathsf{D}^{\lambda}\psi(x^{\prime}),u=\mathsf{D}^{\lambda}\psi(x)\,\,\Longleftrightarrow\,\,x^{\prime}=\mathsf{D}^{\lambda}\psi^{\lambda}(u^{\prime}),x=\mathsf{D}^{\lambda}\psi^{\lambda}(u).$

Here and below, $\psi^{\lambda}$ is the $\lambda$-conjugate function of $\psi$
defined, as a special case of (2), by

$\psi^{\lambda}(u)=\sup_{x}\left(\frac{1}{\lambda}\log(1+\lambda\langle
x,u\rangle)-\psi(x)\right).$

When the $\lambda$-divergence is expressed in the conjugate variables
$u,u^{\prime}$, a generalized version of reference–representation biduality
results:

$D_{\lambda,\psi^{\lambda}}(u^{\prime},u)=D_{\lambda,\psi}(x,x^{\prime}).$

This result generalizes the reference–representation biduality characteristic
of Bregman divergence and the canonical divergence for dually flat spaces (see
[Zha05, Zha13]).

In the above theory, the $\kappa_{\lambda}$ function associated to
$c$-duality, which originates as a cost function, is viewed as a deformation
to the standard Legendre duality. This $\lambda$-deformation theory, fully
developed in Zhang and Wong [ZW22], enables $\lambda$-deformed Legendre
duality and the standard Legendre duality to be mutually transformable to each
other based on a reparameterization of one of the $\lambda$-conjugate
variables $x,u$. This fact underlies the finding of [WZ21], where a
$\lambda$-deformed exponential family has two apparent expressions, i.e., one
with subtractive normalization and the other with divisive normalization. The
former corresponds to the $q$-deformed exponential family associated to the
Tsallis entropy and divergence, whereas the latter corresponds to the deformed
exponential families studied by [Won18] associated to the Réyni entropy and
divergence. The $\lambda$-deformation framework, inspired by the $c$-duality
in optimal transport specializing the function form of $\kappa_{\lambda}$,
appears to provide a canonical extension of the dually flat geometry to an
underlying manifold with constant curvature [Won18, WZ21, ZW21, ZW22].

###  2.4. Relating Otto calculus to information geometry

The $2$-Wasserstein geometry is special in that it admits a formal Riemannian
structure (in addition to being a distance space), so one can apply Otto
calculus (to the upstairs space) in connection with statistical structure
(modeling the downstairs manifold both in the finite and infinite dimensional
settings). For the terminology ”downstairs manifold” and “upstairs space,” see
Section 1.7.

####  2.4.1. Upstairs and downstairs probability measures

Figure 2. Upstairs and downstairs probability measures

One natural setting for exploring the relationship between information
geometry and Otto calculus is to consider the case when the downstairs
manifold $X$ is itself a statistical manifold instead of simply a Riemannian
manifold (as was considered in Otto’s work). A statistical manifold is a
Riemannian manifold (with Fisher-Rao metric) along with a pair of conjugate
connections; the statistical structure can be induced by a proper divergence
function, such as KL divergence.

For instance, if we consider discrete probability measures on a finite set
$S=\\{1,\ldots,n\\}$, the space of all such probability measures is the
probability simplex $\triangle_{n-1}$. Here, each point on $\triangle_{n-1}$
is a discrete measure (a mixture of point measures). Taking the entirety of
discrete measures supported on $S$ to be the space $X$, the Fisher-Rao metric
(on the probability simplex) gives the Riemannian metric of the space of
discrete measure (probability mixture). The statistical manifold of
$\triangle_{n-1}$ is isometric to the positive orthant of an $(n-1)$-sphere,
and is a well-known space in information geometry.

Next, we take $\triangle_{n-1}$ to be the downstairs manifold, and consider
probability measures $\rho$ supported on $\triangle_{n-1}$ (in Bayesian
language, this is the hyper-prior on discrete probability measures). As such,
we denote the space of all such probability measures by
$\mathbb{P}_{2}(\triangle_{n-1})$; this forms the upstairs space. In this
example, the Fisher-Rao metric (of the downstairs manifold) has bounded
radius, so all probability measures (on the upstairs space) will have finite
second-moment. Figure 2 depicts both the upstairs space
$\mathbb{P}_{2}(\triangle_{n-1})$ and the downstairs manifold
$\triangle_{n-1}$.

In linking Otto calculus with information geometry, one must be mindful that
an element in the (infinite-dimensional) upstairs space
$\mathbb{P}_{2}(\triangle_{n-1})$ should not be confused with an element in
the (finite-dimensional) downstairs space $\triangle_{n-1}$, even though both
represent probability measures. However, one can embed $\triangle_{n-1}$
isometrically in $\mathbb{P}_{2}(\triangle_{n-1})$ via the the map
$x\mapsto\delta_{x}$.

For a more general statistical manifold $X$, it is necessary to restrict one’s
attention to $\mathbb{P}_{2}(X)$, so that the Wasserstein metric (7) is well-
defined, but the essential idea is the same. We can then study optimal
transport on $\mathbb{P}_{2}(X)$ where the underlying cost is the distance-
squared induced by the Fisher-Rao metric. It is worth noting that in the
original work of Pal and Wong [PW18], a very similar framework was adopted,
except that the cost for the transport was taken to be the underlying
divergence, rather than the squared-distance.

In a novel approach to statistical structure of a manifold, Khesin et al.
[KLMP13] studied the space of densities on a Riemannian manifold
$\mathcal{M}=X$ in the context of its diffeomorphism group $Diff(X)$. More
precisely, they treat the space of densities on $X$ as the right cosets
arising from the quotient $Diff(X)/Diff_{0}(X)$, where $Diff_{0}(X)$ is the
subgroup of diffeomorphisms of $\mathcal{M}$ that preserve its Riemannian
volume. Notably, this work derived the Fisher-Rao metric and the Amari-
Chentsov tensor on $\mathcal{M}$ in this setting. As such, this framework
demonstrates a remarkably deep link to information geometry through intrinsic
geometric objects such as infinite-dimensional Lie groups. It should be noted
that this density manifold approach also links to optimal transport (through
studying hydrodynamics of incompressible fluids, i.e., Euler equation as
geodesic equation on $Diff_{0}(X)$), and provides metrics which interpolate
between the Fisher-Rao, Sobolev $H^{1}$, and Wasserstein metrics by parametric
variation). See also the work of Bauer, Bruveris, and Michor [BBM16] who
further demonstrated the uniqueness of the Fisher-Rao metric and the Amari-
Chentsov tensor on this density manifold $\mathcal{M}$.

There have been other work studying metrics which interpolate between the
Wasserstein metric and Fisher-Rao geometry in the continuous setting [CPSV18,
KMV16, LMS18]. This line of work has applications in functional analysis and
partial differential equations, and is an active area of research. This work
is closely related to the entropy-regularized optimal transport in the
continuous setting, since the Fisher-Rao metric is induced by the second
variation of the relative entropy, whereas the Wasserstein metric is induced
by optimal transport.

####  2.4.2. Transport information geometry

In contrast to the previous way of relating 2-Wasserstein geometry and
information geometry, where the 2-Wasserstein geometry of the upstairs space
relates to a statistical structure downstairs (following Otto calculus),
another approach is to consider the pullback of the formal Riemannian metric
to the downstairs statistical manifold $X$. This idea first appeared in
McCann’s work on displacement interpolation [McC97], where he observed that
the statistical manifold of normal distributions is geodesically convex in
$\mathbb{P}_{2}(\mathbb{R}^{n})$. This geometry was studied in detail by
Takatsu [Tak11]. In related work, she also studied $q$-Gaussian measures
[Tak12] and $\phi$-exponentials [Tak13] using this approach.

In the past five years, a related line of work was advanced by Li and his
colleagues, [LM18, LM20, CL20, Li21a, Li21b, LLLM21], under the name
_transport information geometry_. Specifically, this work compares the Fisher-
Rao metric with its dual connections and the 2-Wasserstein metric “side-by-
side.” In the case where the underlying space is a finite set with a distance
$d$, the statistical manifold $X=\triangle_{n-1}$ is the probability simplex,
and the distance $d$ can be used to define the 2-Wasserstein metric on $X$.
This is a case of particular interest because it is analytically trackable, so
we mention it now.

For instance, Li and Montúfar [LM18] used transport information geometry to
construct a natural gradient flow, which they called the Wasserstein gradient
flow, using the Otto’s prescription of the Riemannian metric. In this issue,
Li [Li21a] studies the Bregman divergence and KL divergence for transport
information geometry in the case where the underlying space is finite. The
readers may wish to compare and contrast the above approach with the entropy-
regularized transport framework (see Section 2.2).

When this approach is applied to the space of measures on a finite graph $G$
with a distance function $d$, interesting results were obtained when comparing
the associated Wasserstein distance (induced by the graph metric) with the
Fisher-Rao metric, see [Li21b] of this issue.

Due to the differing topologies induced by the Wasserstein geometry and
Fisher-Rao metric (see Section 2.1.2), comparing the Wasserstein and Fisher-
Rao geometries in the infinite-dimensional setting presents several
challenges. To circumvent these issues, Chen and Li [CL20] consider
statistical manifolds $\mathcal{P}_{\theta}$ as finite-dimensional
submanifolds of $\mathbb{P}_{2}(X)$. By pulling back Otto’s Riemannian
structure to $\mathcal{P}_{\theta}$, this induces a Riemannian metric
associated to optimal transport. It is worth noting that except for the case
of normal distributions and a few other special cases, the manifold
$\mathcal{P}_{\theta}$ will not be totally geodesic in $\mathbb{P}_{2}(X)$, so
the minimizing paths in $\mathcal{P}_{\theta}$ will not be the same as those
in $\mathbb{P}_{2}(X)$.

###  2.5. Other papers included in this Special Issue

Previously, Malago, Montrucchio and Pistone [MMP18] studied $2$-Wasserstein
geometry of multivariate normal distributions (i.e., Gaussian measures) and
carried out explicit calculations of Riemannian geodesics and Levi-Civita
covariant derivatives over the space of symmetric matrices, which parametrizes
the Gaussian measure as an exponential model. In this issue, Montrucchio and
Pistone [MP21] investigated the $1$-Wasserstein distance for finite metric
space and representation of $l_{1}$-embeddable metrics. They performed in-
depth analysis of weighted trees and tree-like spaces. This work contains
novel results about optimal transport on trees, e.g., deriving the closed-form
expression for the $1$-Wasserstein transport between measures on a tree in
term of cut norms, and showing that the cut norm is equivalent to the Arens-
Eells norm in this setting.

Mallasto, Gerolin, and Minh [MGM21] in this issue investigated entropy-
regularized 2-Wasserstein distance between Gaussian measures. Through involved
matrix calculations, they provided a closed-form expression of this distance.
It is well-known that the $2$-Wasserstein metric between Gaussians is the
Bures metric comparing quantum states in quantum information geometry. By
incorporating a regularization parameter, the above authors are able to
compare it with the Sinkorn divergence as its limiting case. Their work is
among several other concurrent groups that independently provided calculations
about this important case of Gaussian measure transport (e.g., [Tak11]).

Finally, Sei [Sei21] in this Special Issue studied multi-dimensional
probability distributions and showed that under certain regularity conditions,
there exists a unique coordinate-wise transformation such that the transformed
distribution satisfies a Stein-type identity. These transformations naturally
arise in copula theory modeling joint distributions as having uniform
marginals. The main results of that paper include a variational
characterization of the possible target measures, a proof of uniqueness of
such maps, and a sufficient condition (namely, copositivity) for their
existence. The proofs used some arguments from optimal transport theory; it
was based on an energy minimization problem over a totally geodesic subset of
the Wasserstein space. Sei’s result is considered as an alternative to Sklar’s
theorem regarding copulas, and is also interpreted as a generalization of a
diagonal scaling theorem.

##  3\. Kähler versus para-Kähler Geometries for Optimal Transport

A fruitful interaction between information geometry and optimal transport
stems from attempts to prescribe a “canonical” geometry to optimal transport
problems. There are multiple ways to geometrize optimal transport, and we will
mention two such theories which have close links with information geometry.

###  3.1. The para-Kähler geometry of optimal transport

While studying the regularity theory of the Monge problem, Kim and McCann
[KM10] found a natural geometric formulation of optimal transport in terms of
para-Kähler geometry. More precisely, given two smooth $n$-dimensional
manifolds $X$ and $Y$ with Borel measures $\mu$ and $\nu$ and a cost function
$c:X\times Y\to\mathbb{R}$ which satisfies several regularity and non-
degeneracy conditions, they considered the space $X\times Y$ as a pseudo-
Riemannian manifold with the pseudo-metric

(15) $h:=\frac{1}{2}\left(\begin{array}[]{cc}0&-c_{i,j}\\\
-c_{i,j}&0\end{array}\right).$

Here $c_{i,j}=\partial_{x^{i}}\partial_{y^{j}}c(x,y)$. This induces the
product space with a pseudo-metric of signature $(n,n)$ (i.e., $n$ positive
eigenvalues and $n$ negative ones). It turns out [CMMS04, Vil08] that this is
also an example of a para-Kähler manifold, which is a pseudo-Riemannian metric
with an associated para-complex structure $K$ satisfying

$h(K\eta,K\xi)=-h(\eta,\xi).$

Here the para-complex structure $K$ induces the map $K(\eta,\xi)=(\eta,-\xi)$
for $\eta\in T_{x}X$ and $\xi\in T_{y}Y$, and $K^{2}$ equals the identity map.
Para-Kähler metrics of the above form were discussed by the second named
author in the context of information geometry [ZF16].

To study an optimal transport problem on this space, Kim, McCann and Warren
[KMW10] consider the conformal deformation

(16)
$h_{c}^{\rho,\bar{\rho}}=\left(\frac{\rho(x)\bar{\rho}(\bar{x}))}{|\operatorname{det}(c_{i,j})|}\right)^{\frac{1}{n}}\left(\begin{array}[]{cc}0&-c_{i,j}\\\
-c_{i,j}&0\end{array}\right)$

which is another pseudo-metric on $X\times Y$. In this new geometry, the
solution to the Monge problem is an $n$-dimensional stable maximal surface.
Furthermore, they developed a theory of calibrations for pseudo-Riemannian
metrics and showed that the solution to the optimal transport problem is
calibrated by the function

(17) $\Phi(x,\bar{x})=\frac{\rho(x)dx+\bar{\rho}(\bar{x})d\bar{x}}{2}.$

Wong and Yang [WY21] considered the framework of a $c$-divergence (based on
[PW18] and [Won18]) and considered the graph of the optimal transport as a
(possibly disconnected) statistical manifold equipped with the c-divergence
which is induced by the cost function and the pair of Kantorovich potentials.
In this work, they show that the dual connections $\nabla$ and
$\nabla^{\star}$ are the projections of the Levi-Civita connection of the
pseudo-Riemannian metric (15) on the product space. This gives an embedding of
the dualistic information geometry into pseudo-Riemmanian geometry, and
demonstrates a deep connection between optimal transport and information
geometry.

Another striking feature of the para-Kähler framework of optimal transport is
that the curvature of the product space encodes the MTW condition. In the Kim-
McCann geometry, the MTW tensor is the curvature of certain light-like planes.
This interpretation gives intrinsic differential geometric structure to the
regularity problem and immediately explains many of the properties of
$\mathfrak{S}$, such as why it transforms tensorially under change of
coordinates. We refer the reader to ([KM10], Section 5) for some interesting
remarks on how various aspects of optimal transport can be understood by
studying the geometry of this space.

###  3.2. Kähler geometry of optimal transport

For cost functions which are induced by a convex potential $\Psi$ (such as
$c(x,y)=\Psi(x-y)$ or a $\mathcal{D}_{\Psi}^{(\alpha)}$-divergence [Zha04]),
there is a separate formulation of the regularity theory in terms of Kählerian
information geometry [KZ20a]. In particular, given such a cost function, one
can use the convex potential to define a Kähler metric whose curvature encodes
the MTW tensor.

For this, we consider a convex potential $\Psi:\Omega\to\mathbb{R}$, where
$\Omega$ is assumed to contain the relevant Minkowski linear combination of
the sets $X$ and $Y$ so that the cost function is well defined.

When one treats the domain $\Omega$ as a Hessian manifold (with a global
chart), its tangent bundle $T\Omega$ admits a canonical Kähler structure which
has a natural $\mathbb{R}^{n}$ symmetry induced by translations in the fiber
directions.888It is a general fact that the tangent bundle of any Hessian
manifold admits a Kähler metric (see, e.g. [Dom62, Sat07]).

In order to relate the MTW tensor to the curvature of the tangent bundle
$T\Omega$, it is necessary to lift tangent vectors from $\Omega$ to
corresponding vectors in $T\Omega$. For this, we use the _polarized lift_.

######  Definition 3.

Given a tangent vector $v\in T_{x_{0}}\Omega$ defined as

(18) $v=\sum_{i}v^{i}\frac{\partial}{\partial{x^{i}}},$

we define its polarized lift to be the $(1,0)$ vector

(19) $v^{pol.}:=\sum_{i}v^{i}\frac{\partial}{\partial{z^{i}}}\in
T_{(x_{0},y)}^{(1,0)}T\Omega.$

Here, $\\{z^{i}\\}_{i=1}^{n}$ are the holomorphic coordinates
$x^{i}+\sqrt{-1}y^{i}$. Since the metric is translation symmetric in the
fibers, we are free to pick the fiber coordinate $y$ at will. These vectors
are polarized in the sense that their components are real (in the
$z$-coordinates). By considering polarized $(1,0)$ vectors, we can make sense
of the notion of positive or negative anti-bisectional curvature.

######  Definition 4 (Anti-bisectional curvature).

Given two $(1,0)$ vectors $V$ and $W$, the anti-bisectional curvature is
defined to be

(20) $\mathfrak{A}(V,W)=R(V,\overline{W},V,\overline{W}).$

At first, it does not seem meaningful to discuss positive and negative anti-
bisectional curvature; this is because
$\mathfrak{A}(\sqrt{-1}V,W)=-\mathfrak{A}(V,W)$. However, by restricting our
attention to polarized vectors, it is meaningful to talk about this quantity
as having a sign, and it turns out to be proportional to the MTW tensor.

######  Theorem 5 ([KZ20a]).

For a cost which is induced by a convex potential $\Psi:\Omega\to\mathbb{R}$,
the MTW tensor $\mathfrak{S}(\xi,\eta)$ is proportional999The proportionality
constant depends on how the convex potential is used to induce the cost
function. to the anti-bisectional curvature of an associated Kähler-Sasaki
metric on the tangent bundle $T\Omega$.

(21) $\mathfrak{S}(\xi,\eta)\propto
R\left(\xi^{pol.},\overline{(\eta^{\sharp})^{pol.}},\xi^{pol.},\overline{(\eta^{\sharp})^{pol.}}\right).$

Furthermore, the conjugate connection (of the associated Hessian manifold
$\Omega$) encodes the notion of relative $c$-convexity.

######  Proposition 6.

For a cost of the form $c(x,y)=\Psi(x-y)$, a set $Y$ is $c$-convex relative to
$X$ if and only if, for all $x\in X$, the set $x-Y$ is geodesically convex
with respect to the dual connection $\nabla^{*}.$

At present, this geometry only applies to cost functions which are induced by
a convex potential $\Psi$. However, this includes some important examples. In
particular, using this geometry the present authors [KZ20a] were able to
establish a regularity result for relative pseudo-arbitrages, which solved a
question posed by Pal and Wong in their work [PW20] relating information
geometry, optimal transport, and mathematical finance.

####  3.2.1. Some open problems related to this geometry

One open problem is to extend this work to a non-Kähler theory for costs
defined on a Lie group $G$. More precisely, on any Lie group $G$ we can
construct a curvature-free connection $\nabla_{G}$ whose torsion is induced by
the associated Lie algebra. The connection can then be used to define an
almost complex structure on the tangent bundle $TG$, which is non-integrable
whenever $G$ is non-Abelian. Then, given a potential function
$\Psi:G\to\mathbb{R}$, it may be possible to define a structure analogous to a
metric on $G$ (although care must be taken since $\nabla_{G}^{2}\Psi$ will
generally not be symmetric, so does not define a Riemannian metric). If this
can be done, the hope is that it can then be used to define a Sasaki-type
structure on the tangent bundle $TG$ whose curvature encodes the MTW tensor of
the cost $c(x,y)=\Psi(xy^{-1})$. From a high level perspective, the Kähler
formulation of optimal transport is precisely this construction for the
_Abelian_ Lie group $\mathbb{R}^{n}$ (and where $\Psi$ may be defined on an
open subset of $\mathbb{R}^{n}$ rather than the entire space).

######  Question 2.

Is there a non-Kähler version of Theorem 5 for costs of the form

(22) $c(x,y)=\Psi(xy^{-1})$

where $x$ and $y$ are elements of a Lie group?

As a particular application of the previous question, one might hope to find
examples of cost functions which satisfy the MTW(0) condition. In particular,
the squared distance function for a left-invariant metric on $SO(3)$ is
precisely a function of the form (22), which raises the following question.

######  Question 3.

Is it possible to classify left-invariant metrics on $SO(3)$ whose squared
distance costs have non-negative MTW tensor?

Another natural question about the complex geometry of optimal transport is
how to interpret the geometric meaning of the solution of an optimal transport
problem. It seems that this is intrinsically linked with displacement
interpolation, but it is not clear how to make this precise.

Using displacement interpolation, it is possible to understand the solution to
optimal transport in terms of a section $s$ of $TX$. To do so, we consider a
flow which follows the $c$-exponential map and whose initial condition is $s$.
In general, this flow will satisfy a particular semi-spray equation induced by
the cost function. In the setting of 2-Wasserstein geometry, the flow lines
will be geodesics in the usual sense. For costs induced by a convex potential,
the flow lines will be geodesics with respect to the _dual_ connection. Along
this flow, we can solve the continuity equation with the initial measure
$\mu=\mu(0)$. The section $s$ encodes a viable transport plan if the induced
measure at time $t=1$ is our desired target measure $\nu$. Furthermore, this
transport plan will be optimal if the induced cost from this flow is minimal.

This gives a way to encode the solutions to optimal transport in terms of
sections to the tangent bundle, but it is not clear what geometric properties
such a section will need to satisfy in order to induce the optimal transport
between $\mu$ and $\nu$. Furthermore, it is not clear what role the curvature
of $T\Omega$ plays (which may be a different space from $TX$).

######  Question 4.

When does a section of $TX$ induce an optimal flow (in the sense of
displacement interpolation)? What is the relationship between these sections
and the curvature of the tangent bundle $T\Omega$?

Recall that in the para-complex geometry, the MTW is the curvature of certain
light-like planes, whereas in the complex setting, the MTW tensor is the
orthogonal anti-bisectional curvature. The link between the Kähler geometry
approach and the para-Kähler geometry approach is not well understood.

######  Question 5.

How can one relate the pseudo-Riemannian geometry to the complex geometry of
optimal transport?

Before moving on, it is worth noting that the Kähler manifolds which appear in
this theory also have applications outside optimal transport. In particular,
they exhibit a form of mirror symmetry induced by the information-geometric
duality of the Hessian metric [ZK20]. As shown in [KZ21], this duality appears
to relate to the theory of automorphic forms encountered in analytic number
theory.

###  3.3. Kähler-Ricci flow and other smoothing flows

Although the complex geometric perspective is less general than the pseudo-
Riemannian framework, there are several advantages associated to using a
Hermitian, rather than an indefinite, pseudo-metric (para-Hermitian metric).
In particular, it is possible to use techniques from complex geometry, such as
Kähler-Ricci flow, to deform the cost functions and study optimal transport.

Ricci flow [Ham82] is a geometric flow which deforms a Riemannian metric by
its Ricci curvature:

(23) $\frac{\partial g}{\partial t}=-2\,\textrm{Ric}(g).$

When the underlying manifold is Kähler, this flow preserves the complex
structure and the Kählerian nature of the metric [Cao85], so is known as the
Kähler-Ricci flow. Recently, the first named author and Zheng investigated the
behavior of the MTW tensor under Kähler-Ricci flow [KZ20b].

######  Theorem 7 ([KZ20b]).

Suppose that $\Omega\subset\mathbb{R}^{n}$ is a convex domain and
$\Psi:\Omega\to\mathbb{R}$ is a strongly convex function so that the
associated Kähler manifold $(T\Omega,\omega_{0})$ is complete and has bounded
curvature.

  1. (1)

For $n=2$, _non-negative orthogonal anti-bisectional curvature_ (i.e., the
MTW(0) condition) is preserved by Kähler-Ricci flow.

  2. (2)

For all $n$, _non-positive anti-bisectional curvature_ is preserved under
Kähler-Ricci flow.

This result allows us to prove regularity for the two-dimensional Monge
problem for cost functions which are induced by a convex potential and are
much rougher than $C^{4}$, which is the regularity needed so that Equation (6)
is meaningful (see also, [GK15, LT21]). For instance, it is possible to prove
a Hölder estimate on the transport for costs which are $W^{2,p}$ for $p>2$ and
which satisfy the MTW condition in the sense that the positive-time potentials
induce Kähler metrics with non-negative anti-bisectional curvature.

####  3.3.1. Smoothing flows in optimal transport

This line of work suggests that smoothing flows can play a role in optimal
transport, especially in cases with low regularity. Parabolic flows for the
transport potential $\psi$ mentioned in Section 1.4 had previously been
considered [Kit12] but in this context Kähler-Ricci flow is used to deform the
cost function.

Currently, Theorem 7 is limited to costs which are induced by a convex
potential because it is not possible to define a Ricci flow naively for
general cost functions. The reason for this is that the Ricci flow is not
weakly parabolic in the pseudo-Riemannian setting. More concretely, the
natural analogue of the Kähler-Ricci flow equation in the Kim-McCann geometry
is the following:

(24) $\frac{\partial}{\partial t}c(x,y)=\log\det[c_{i,j}(x,y)].$

Solutions to this equation are not unique. For instance, given one solution
$c(x,y,t)$, there are infinitely many other solutions of the form

$c(x,y,t)+f(x,t)+g(y,t)$

where $f(x,t)$ and $g(y,t)$ are arbitrary functions of $(x,t)$ and $(y,t)$,
respectively. In other words, this flow cannot be smoothing, and it is not
even clear when solutions exist at all. It would be of interest to find a
smoothing flow for more general costs, which we leave as an open problem.

######  Question 6.

For more general cost functions, is it possible to define a smoothing flow?

When the cost function is the squared distance on a Riemannian manifold, one
natural idea for a smoothing flow is to evolve the underlying metric by Ricci
flow (see, e.g., [MT10, Top09]). This is distinct from (24), and will be
smoothing away from the cut-locus. This raises the following question.

######  Question 7.

For the squared-distance cost on a Riemannian manifold, does the Ricci flow
preserve non-negativity of the MTW tensor?

We suspect that the answer to this question is positive, at least in
dimensions two and three. The reason to believe this conjecture is that for
the squared distance cost, the MTW tensor restricted to the diagonal (i.e.,
$x=y$) is two-thirds the sectional curvature. As such, the MTW(0) condition is
a non-local strengthening of non-negative sectional curvature [Loe09]. In two
and three dimensions, the Ricci flow preserves positivity of the sectional
curvature and Hamilton showed that if a metric starts with positive sectional
curvature, it converges to a metric of constant curvature with the scalar
curvature satisfying a differential Harnack estimate. As such, two- and three-
dimensional manifolds with non-negative MTW tensor converge quickly to metrics
of constant curvature. Furthermore, Figalli, Rifford and Villani showed that
in all dimensions, small deformations (in the $C^{4}$-sense) of the round
sphere satisfy the MTW condition [FRV12].

Together, these results show that if the initial metric satisfies the MTW
condition, there will be a neighborhood of the diagonal for which the MTW
tensor is positive for all positive time of the flow. Furthermore, after a
definite amount of time, the MTW condition will hold everywhere. In higher
dimensions, Ricci flow need not preserve positivity of the sectional
curvature, so it might be necessary to make the additional assumption that the
curvature operator is positive (and to formulate the correct version of
positivity for the MTW operator). In spirit, these results can be established
using Hamilton’s tensor maximum principle (at least away from the cut locus),
but the computations needed are extremely involved.

##  4\. Conclusion

In this paper, we have surveyed several lines of fruitful interaction between
optimal transport and information geometry: entropy-regularized (also called
entropy-relaxed) transport, $c$-duality as generalizing Legendre duality,
relating Wasserstein metric and Otto calculus to the statistical structure,
para-Kähler versus Kähler geometry, etc. Our survey is admittedly incomplete:
we have focused largely on the work published in the Springer journal
_Information Geometry_ , including the current Special Issue, with an intended
audience of information geometers. For this reason, we have omitted a large
amount of work on optimal transport/Wasserstein geometry that does not have an
apparent information geometry connection.

We conclude that the budding intersection between optimal transport and
information geometry is a highly promising area of research. Given that both
Wasserstein geometry and information geometry have, in their own rights, seen
wide applications in statistics, machine learning, computer vision, etc., it
is all the more remarkable that there are some deep, internal links of these
two mathematical sub-disciplines. Future research will continue to illuminate
the relation between these two geometric frameworks about probability measures
(and probability density functions).

Acknowledgement The authors would like to thank Leonard Wong for his helpful
comments on the manuscript. G.K. is partially supported by a Simons
Collaboration Grant 849022 (“Kähler-Ricci flow and optimal transport”), while
J.Z. is partially supported by United States Air Force Office for Scientific
Research, grant number AFOSR-FA9550-19-1-0213.

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