C-Transform

92 Semi-discrete Optimal Transport

0 0.5 1

-0.2

0

0.2

0.4

0.6

0.5

1

1.5

2

p = 1/2 p = 1 p = 3/2 p = 2

Figure 5.1: Top: examples of semi-discrete ¯

c-transforms g

¯

c in 1-D, for ground

d the useful indicator function notation (4.42).

alternate minimization on either f or g leads to the im-

n of c-transform:

’ y œ Y, fc(y) def.

= inf

xœX

c(x, y) ≠ f(x), (5.1)

’ x œ X, g¯

c(x) def.

= inf

yœY

c(x, y) ≠ g(y), (5.2)

oted ¯

c(y, x) def.

= c(x, y). Indeed, one can check that

œ argmax

g

E(f, g) and g¯

c œ argmax

f

E(f, g). (5.3)

ese partial minimizations deﬁne maximizers on the sup-

tively – and —, while the deﬁnitions (5.1) actually deﬁne

he whole spaces X and Y. This is thus a way to extend in

ay solutions of (2.22) on the whole spaces. When X = Rd

Îx ≠ yÎp, then the c-transform (5.1) fc is the so-called

n between ≠f and Î·Îp. The deﬁnition of fc is also often

a “Hopf-Lax formula”.

(f, g) œ C(X) ◊ C(Y) ‘æ (g¯

c, fc) œ C(X) ◊ C(Y) replaces

s by “better” ones (improving the dual objective E). Func-

be written in the form fc and g¯

c are called c-concave and

ctions. In the special case c(x, y) = Èx, yÍ in X = Y = Rd,

n coincides with the usual notion of concave functions.

turally Proposition 3.1 to a continuous case, one has the

operations are replaced by a “soft-min”.

Using (5.3), one can reformulate (2.22) as an unconstrained

program over a single potential

Lc

(–, —) = max

fœC

(

X

)

⁄

X

f(x)d–(x) +

⁄

Y

fc(y)d—(y),

= max

gœC

(

Y

)

⁄

X

g¯

c(x)d–(x) +

⁄

Y

g(y)d—(y).

Since one can iterate the map (f, g) ‘æ (g¯

c, fc), it is possible to

these optimization problems the constraint that f is ¯

c-concave

is c-concave, which is important to ensure enough regularity on

potentials and show for instance existence of solutions to (2.22)

5.2 Semi-discrete Formulation

A case of particular interest is when — =

q

j bj

”

yj

is discrete (of

the same construction applies if – is discrete by exchanging the

(–, —)). One can adapt the deﬁnition of the ¯

c transform (5.1)

setting by restricting the minimization to the support (y

j

)

j

of —

’ g œ Rm, ’ x œ X, g

¯

c(x) def.

= min

jœJmK

c(x, y

j

) ≠ gj

.

This transform maps a vector g to a continuous function g

¯

c œ

Note that this deﬁnition coincides with (5.1) when imposing th

space X is equal to the support of —. Figure 5.1 shows some ex

of such discrete ¯

c-transforms in 1-D and 2-D.

Using this discrete ¯

c-transform, in this semi-discrete case, (

equivalent to the following ﬁnite dimensional optimization