Optimal Transport
Table of Contents
Discrete case

So the optimal transport problem becomes

subject to linear constraints

villani09_optim_trans
7. Displacement interpolation
Notation
denotes the action functional
is a certain class of continuous curves
Cost function between an initial point
and final point
:
- Riemannian manifold
- Langrangian
is defined on
Deterministic interpolation via action-minimizing curves
Action is classically given by the time-integral of a *Lagrangian$ along the path:
A continuous curve
is absolutely continuous if there exists a function
s.t. for all intermediate times
in
,
- If
is absolutely continuous, then
is differentiable a.e. and its derivative is integrable.
- If
Examples
Let
be a curve from
to
for some strictly convex
Then, by Jensen's inequality,

which is an equality only when , and thus the action minimizers are lines
, i.e. straight lines:

Also, then we have

since the cost function is defined by the minimizers of .
Interpolation of random variables
is a cost function associated with the Lagrangian action
be two given laws
- Optimal coupling
of
- Random action-minimizing path
joining
to
- => random variable
is an interpolation of
and
; or equivalently
is an interpolation of
and
- This is referred to as displacement interpolation
This should be constrasted with linear interpolation
- A dynamical transference plan
is a probability measure on the space
(i.e. space of curves).
- A dynamical coupling of two probability measures
is a random curve
s.t.
and
.
A dynamical optimal transference plan is a prob. measure on
on
s.t.
- Equivalently,
is the law of a random action-minimizing curve whose endpoints constitute an optimal coupling of
.
- Such a random curve is called a dynamic optimal coupling of
.
- By abuse of language,
is sometimes referred to as the same.
- By abuse of language,
- Equivalently,