Signal Processing - July 2017 - 52

O (N 2 ) [14] or O (N log (N )) using the convolutional
Wasserstein distance presented in [47] (compared to O (N 3) of
the linear programming methods), where N is the number of
delta masses in each of the measures. The disadvantage is that
it is difficult to obtain high-accuracy approximations of the
optimal transport plan. The entropy-regularized p-Wasserstein
distance, also known as the Sinkhorn distance, between PDFs
I0 and I1 defined on the metric space (X, d) is defined as
p

W p, m (I 0, I 1) = infc ! MP
+m #

X#X

#X # X d p (x, y) c (x, y) dxdy
c (x, y) ln (c (x, y)) dxdy,

(10)

where the regularizer is the negative entropy of the plan. We
p
note that this is not a true metric since W p, m (I 0, I 1) 2 0. Since
the entropy term is strictly concave, the overall optimization
in (10) becomes strictly convex. It is shown in [14] that the
entropy-regularized p-Wasserstein distance in (10) can be
reformulated as
p

W p, m (I 0, I 1) = m * inf
KL (c | K m),
! MP
c

where K m (x, y) = exp (- d p (x, y) /m) and KL (c | K m) is the
Kullback-Leibler (KL) divergence between c and K m . In
short, the regularizer enforces the plan to be within 1/m radius
in the KL-divergence sense from the transport plan
*
c 3 (x, y) = I 0 (x) I 1 (y) .
Cuturi shows that the optimal transport plan c in (10) is of
the form D v K m D w, where Dv and Dw are diagonal matrices
with diagonal entries v, w ! R N [14]; therefore, the number of
unknowns in the regularized formulation is reduced from N2
to 2N. The new problem can then be solved through computationally efficient algorithms such as the iterative proportional
fitting procedure, also known as the iterative proportional fitting procedure algorithm, or, alternatively, through the Sinkhorn-Knopp algorithm.

k=0

k = 20

k = 40

k = 60

k = 80

φk

∇φk

Flow minimization (AHT)
Angenent, Haker, and Tannenbaum [2], proposed a flow minimization scheme to obtain the optimal transport map from
the Monge problem. The method was used in several imageregistration applications [22], pattern recognition [27], [50],
and computer vision [26]. A brief review of the method is
provided here.
Let I 0: X " R + and I 1: Y " R + be continuous probability
densities defined on convex domains X, Y 3 R d . To find the
optimal transport map, f *, AHT starts with an initial transport map, f0: X " Y calculated from the Knothe-Rosenblatt
coupling [49]. Then it updates f0 to minimize the transport cost
while constraining it to remain a transport map from I0 to I1.
The updated equation for finding the optimal transport map in
AHT is calculated to be
fk + 1 (x) = fk (x) + e 1 Dfk ( fk - d (D -1 div ( fk))),
I0
where e is the step size, Df k is the Jacobian matrix, and D -1
is the Poisson solver with Neumann boundary conditions.
AHT show that for infinitesimal step size, e, fk (x) converges
to the optimal transport map. For a detailed derivation of the
preceding equation, see [2] and [24].
The AHT method is, in essence, a gradient descent method
on the Monge formulation of the optimal transport problem.
Chartrand, Wohlberg, Vixie, and Bollt (CWVB) [11] proposed
an alternative gradient-descent method based on Kantorovich's
dual formulation of the transport problem that updates the
optimal potential transport field, h (x), where f (x) = dh (x) .
Figure 6 presents the iterations of the CWVB method for two
face images taken from the YaleB face database.

Monge-Ampère equation
The Monge-Ampère PDE is defined as
det (Hz) = h (x, z, Dz)
for some functional h and where Hz is the Hessian matrix of
z. The Monge-Ampère PDE is closely related to the Monge
problem for the quadratic cost function. According to Bernier's
theorem (discussed in the "Basic Properties" section), when I0
and I1 are absolutely continuous PDFs defined on sets
X, Y 1 R n, the optimal transport map that minimizes the
2-Wasserstein metric is uniquely characterized as the gradient of a convex function z: X " Y. Moreover, we showed that
the mass-preserving constraint of the Monge problem can be
written as det (Df ) I 1 ( f ) = I 0 . Combining these results, one
can have

Ik

det (D (dz (x))) =
Ik = det(D∇ηk)I1(∇ηk), φk(x) = 1 x 2 - ηk (x)
2

FIGURE 6. A visualization of the iterative update of the transport potential
and correspondingly the transport displacement map through CWVB iterations. (Face portraits courtesy of the public Extended Yale Face Database B.)

52

I 0 (x)
,
I 1 (dz)

(11)

where Ddz = Hz, and, therefore, the equation shown above
is in the form of the Monge-Ampère PDE. Now, if z is a convex function on X satisfying dz (X) = Y and solving (11),
then f * = dz is the optimal transportation map from I0 to I1.

IEEE SIGNAL PROCESSING MAGAZINE

|

July 2017

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Table of Contents for the Digital Edition of Signal Processing - July 2017

Signal Processing - July 2017 - Cover1
Signal Processing - July 2017 - Cover2
Signal Processing - July 2017 - 1
Signal Processing - July 2017 - 2
Signal Processing - July 2017 - 3
Signal Processing - July 2017 - 4
Signal Processing - July 2017 - 5
Signal Processing - July 2017 - 6
Signal Processing - July 2017 - 7
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Signal Processing - July 2017 - 196
Signal Processing - July 2017 - Cover3
Signal Processing - July 2017 - Cover4
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