Optimal transport with Laplacian regularization

Rémi Flamary
Laboratoire Lagrange UMR CNRS 7293
Observatoire de la Côte d’Azur
Université de Nice, France
remi.flamary@unice.fr

Nicolas Courty
IRISA UMR CNRS 6074
Université de Bretagne Sud
56000 Vannes, France
ncourty@irisa.fr

Alain Rakotomamonjy
LITIS EA 4108
Université de Rouen
76000, Saint Etienne du Rouvray, France
alain.rakoto@insa-rouen.fr

Devis Tuia
Department of Geography
University of Zurich
8057 Zurich, Switzerland
devis.tuia@geo.uzh.ch

Abstract
We propose a method based on optimal transport for empirical distributions with
Laplacian regularization (LOT). Laplacian regularization is a graph-based regularization that can encode neighborhood similarity between samples either on the
final position of the transported samples or on their displacement as in the work of
Ferradans. In both cases, LOT is expressed as a quadratic programming problem
and can be solved with a Frank-Wolfe algorithm with optimal step size. Resultson domain adaptation and a shape matching problems show the interest of using
this regularization in optimal transport.

1

Introduction

Recently, there has been a renewed interest for the study and the application of optimal transport(OT) [1]. OT aims at matching two probability distributions while minimizing a displacement cost.
To attain this objective, it uses a similarity measure on probability distributions denoted as the optimal transport distance or the Wasserstein distance. When considering discrete distributions, OT
is the solution of a convex linear programming problem, but several recent works in the machinelearning literature proposed to use additional regularization terms [2, 3, 4].
In the light of these recent efforts, optimal transport has also impacted several domains, in which
it has been applied successfully. For instance, in computer vision, the Earth Mover Distance has
been shown to work very well to compare color and texture distributions [5]. In machine learning,
two recent applications have shown the potential of optimal transport for computing barycenters of
distributions [6] and for domain adaptation [7]. In the latter, authors introduced a solution for OT,
which is regularized by the class memberships in the source domain.
In this work, we investigate a new regularization scheme for optimal transport of discrete distributions. Similarly to the work of Ferradans et al. [2], in which authors proposed to smooth the
optimal transport plan by controlling the displacement of pairs of points, our idea is to achieve a
structure-preserving optimal transport. For this purpose, we propose two regularization schemes
based on Laplacian regularization. These regularizers favor OT solutions, that preserve the spatial
relationships between the elements after interpolation. The first, named LOTpos regularizes samples
by their final position after transport, while the second, named LOTdisp , acts on the transportation
plan itself.
1

The rest of the paper is as follows: in Section 2 we introduce the framework of optimal transport for
empirical distributions. In Section 3 we present the proposed regularizers for LOT, whose effects
are illustrated in Fig. 2. Section 4 provides details about the Frank-Wolfe optimization proposed tosolve the LOT problem. Finally, Section 5 presents results on a synthetic domain adaptation problem
and an application of OT in 3D human meshes registration.

2

Optimal transport for discrete distributions

In this section, we shortly introduce OT and how it can be used for interpolating discrete collectionsof samples expressed as probability distributions. OT aims at computing a minimal cost transportation between a source probability distribution µs , and a target distribution µt generally corresponding to continuous distributions. In this work we consider the case when those distributions are only
available through a finite number of samples, and can be written as
nt
ns
X
X
(1)
pti δxtiµs =psi δxsi , µt =
i=1

i=1

t
where δxi is the Dirac at location xi ∈ Rd . psi and pP
are probability
associated to thei n
Pnt massesssti-th sample, belonging to the probability simplex, i.e.
i=1 pi =i=1 pi = 1. The source andtarget samples are in matrices Xs = [xs1 , . . . , xsns ]> ∈ Rns ×d and Xt = [xt1 , . . . , xtns ]> ∈ Rns ×d ,respectively. The set of probabilistic couplings between these two distributions is then the set of
doubly stochastic matrices P defined asP =γ ∈ (R+ )ns ×nt | γ1nt = µs , γ T 1ns = µt(2)where 1d is a d-dimensional vector of ones. The Kantorovitch formulation of the optimal transport [8] is:γ 0 = argmin hγ, CiF(3)γ∈P

where h., .iF is the Frobenius dot product and C ≥ 0 is the cost function matrix of term C(i, j)
related to the energy needed to move a probability mass from xsi to xtj . In our setting, this cost is
chosen as the Euclidian distance between the two locations, i.e. C(i, j) = ||xsi − xtj ||2 , but othertypes of metric could be considered, such as Riemannian distances over a manifold [1].
When the optimal transportation plan γ has been estimated, one can transport [2] the source distribution by expressing the new transported distribution X̂s as linear combination of elements from
Xt . The corresponding convex combination relates to the corresponding transportation plan (i.e. theprojection over the unit simplex of the corresponding line of γ) through the following relation:X̂s = Tγ 0 (Xs ) = diag(γ 0 1nt )−1 γ 0 Xt

and

>
−1 >
X̂t = T−1
γ 0 Xt . (4)γ 0 (Xt ) = diag(γ 0 1ns )Note that in this work we focus on uniform distributions with psi = n1s and pti = n1t . In this case, asimpler transportation formulation can be derived:>X̂s = Tγ 0 (Xs ) = ns γ 0 Xt and X̂t = T−1(5)
γ 0 (Xt ) = nt γ 0 Xs .
Using this last transportation formula avoids inverting a diagonal matrix that depends on γ and thiswill greatly simplify the optimization presented in Section 4. In the more general case where theweights are non-uniform, we refer the reader to [2, sec. 3.3].

Using this simple ingredients, recent works have solved domain adaptation problems, either using
entropy-based regularization (below referred to as OTsinkhorn [3]) or class-label group sparse regularization [7] and achieved state of the art results with respect to current domain adaptation literature.
In what follows, we assume that data live on a manifold and propose two Laplacian regularization
schemes as an alternative to entropy regularization.

3

Laplacian regularization for optimal transport

In the following, we discuss two Laplacian regularization schemes for OT. The first one is an application of graph regularization on the transported samples. This regularization term will promote
samples that are neighbors in the graph to stay neighbors after transportation. The second one isa direct transposition of the graph regularization proposed in [2], where the graph regularization isapplied on the displacement of the samples during transportation.
2

3.1

Regularizing the transported samples positions

Regularizing the transported sample positions leads to the following optimization problem:
λs X sλt X tγ 0 = argmin hγ, CiF + 2
Si,j kx̂si − x̂sj k2 + 2
S kx̂t − x̂tj k2
Ns i,jNt i,j i,j i
γ∈P

(6)

The first term is the classical optimal transport linear loss that encodes the transportation cost between the source and the target samples. The two other terms are Laplacian regularization terms thatwill promote the respect of the proximities observed in the original distribution after the transport.
Note that the graph regularization is defined by symmetric similarity matrices Ss and St computed
between the source and target samples in their original configurations.
The optimization problem (6) does not explicitly illustrate the dependency of the regularization
term on the variable γ. This dependency exists through the use of the transported x̂ samples. The
regularization terms can be easily expressed for the source regularization as
X
s
>
Si,j
kx̂si − x̂sj k2 = Ns2 Tr(X>(7)
t γ Ls γXt )i,jwhere Ls = diag(Ss 1) − Ss is the Laplacian of the graph Ss ; and for the target regularization as
X
t
>
Si,j
kx̂ti − x̂tj k2 = Nt2 Tr(X>(8)
s γLt γ Xs )i,jwhere Lt = diag(St 1) − St . Finally, optimization problem (6) car be reformulated as>>>
γ 0 = argmin hγ, CiF + λs Tr(X>t γ Ls γXt ) + λt Tr(Xs γLt γ Xs ),

(9)

γ∈P

which is a quadratic program w.r.t. γ. This raises problems of high computational complexity,
particularly due to the fact that the quadratic loss matrix is full. This transportation of the samples
will be referred to as LOTpos .
3.2

Regularizing the transported samples displacement

As discussed previously, [2] proposed to adapt image histograms through the transportation of the
pixels from one image to another. One of their contributions was to propose a graph regularization of
the displacement of the samples during interpolation (4). This regularization leads to the following
optimization problem
λs X sλt X tγ 0 = argmin hγ, CiF + 2
S̃ k(x̂ti −xti )−(x̂tj −xtj )k2
Si,j k(x̂si −xsi )−(x̂sj −xsj )k2 + 2
Ns i,jNt i,j i,jγ∈P(10)
As for the LOTpos regularization above, one can reformulate the first regularization term as
X
s
>
>
>
Si,j
k(x̂si − xsi ) − (x̂sj − xsj )k2 = Ns2 Tr(X>t γ Ls γXt ) + γ, −Ns (Ls + Ls )Xs Xt ) F + cs
i,jwith cs = Tr(X>s Ls Xs ) a constant w.r.t. γ ; and the second regularization term as
X
t
>
>
>
Si,j
k(x̂ti − xti ) − (x̂tj − xtj )k2 = Nt2 Tr(X>s γLt γ Xs ) + γ, −Nt Xs Xt (Lt + Lt ) F + ct
i,jwith ct = Tr(X>t Lt Xt ). Finally, the optimization problem (10) can be expressed as>>>
γ 0 = argmin hγ, C + λs Cs + λt Ct iF + λs Tr(X>t γ Ls γXt ) + λt Tr(Xs γLt γ Xs )

(11)

γ∈P

1
>
>
>
with Cs = − N1s (Ls + L>s )Xs Xt and Ct = − Nt Xs Xt (Lt + Lt ) linear costs associated tothe source and target samples respectively. Optimization problem (11) is expressed in [2, Eq. 3.3]
and is extremely similar to (9), their only difference being a linear term taking into account theregularization terms Cs and Ct . This transportation of the samples will be referred to as LOTdisp .

3

4

Optimization

As discussed in [2], the Frank-Wolfe algorithm can be used to solve the optimization problems (9)and (11). This approach is particularly interesting for these problems, since it works by solving at
each iteration a linearization of the loss function under the linear constraints (a LP problem).
The Frank-Wolfe (FW) algorithm is an interior point method that has been proposed in [9] and has
drawn some attention recently thanks to its ability to handle large scale optimization problems [10].
It addresses the minimization of f (γ) subject to γ ∈ P, where f (·) is a convex and differentiable
function and P are convex constraints. The algorithm with optimal step size is defined as follows:
0. Initialize k = 0 and γ 0 ∈ P an initial value. (either γ 0 = µs µ>t or solution of the linear
problem with no quadratic regularization)
1. Compute the solution of the following problem
γ ∗ = argminγ∈Pγ, ∇γ f (γ k ) F

2. Find the optimal step 0 ≤ αk ≤ 1 with descent direction ∆γ = γ ∗ − γ k such thatαk = argmin f (γ k + α∆γ)
0≤α≤1

3. γ

k+1

k

k

← γ + α ∆γ, set k ← k + 1 and go to step 1.

We want to empathize the importance of an interior point method such as FW. In practice, the exact
solution of the optimization problem is often not necessary and an early stopping strategy is used. In
our case, an interior point method ensures that even if the iterations are stopped before convergence,
γ k belongs to P and leads to a proper transportation in the convex hull of the target data. Note thatthe gradient of objective values term in (9) and (11) can be easily derived from derivative of trace[11, Eq. (116)].
Finally, we discuss how the optimal step is found when optimizing problem (9). One want to minimize the following function w.r.t. α:g(α) = f (γ k + α∆γ) = aα2 + bα + cwhere a, b, and c are constants at each iteration k. Since g(·) is a second order polynomial, theoptimal step can be computed with g 0 (α) = 2aα + b = 0. When applied to the optimization
problem (9), this gives
>
>
k
>
>
k>
Xs )
1 h∆γ, CiF + λs Tr(X>t ∆γ (Ls + Ls )γ Xt ) + λt Tr(Xs ∆γ(Lt + Lt )γ
>
>
>
>
2
λs Tr(Xt ∆γ Ls ∆γXt ) + λt Tr(Xs ∆γLt ∆γ Xs )(12)
The optimal step for problem (11) is given in [2] and can be computed by replacing C by C +λs Cs + λt Ct in equation (12).

αk = −

5

Numerical experiments

In this section, we first illustrate the effect of the two regularizers proposed. Then, we show theresults obtained on a synthetic domain adaptation problem and on a shape matching dataset.
5.1

Illustration of the regularized optimal transport

In Fig. 1, we illustrate on a simple dataset the behavior of the different regularization terms in term ofsample transportation. In our example dataset, both the source and target distributions consist in two
clusters (see Fig. 1.a). The similarity graph Ss is then chosen to be 1 for samples belonging to thesame cluster and 0 otherwise (Fig. 1.b). To ease interpretation, we use the Laplacian regularization
only in the source domain (λt = 0).
The results of adaptation are illustrated in Figure 2 for OTSinkhorn , LOTpos and LOTdisp with asmall and large regularization term λs . In the figure we can appreciate the shrinkage effect to thecenter of mass of the data of the Sinkhorn regularization (top row). Similarly, LOTpos leads to4Source samples xsi
Target samples xsi

Sim graph with Si,js >0(b) Similarity graph Ss(a) Initial distributions

Figure 1: Simulated distribution with two clusters in each domain.

a shrinkage but takes into account the graph structure with a per-cluster shrinkage (middle row).
This is interesting, as it can be seen as a de-noising of the source samples during the transport.
The displacement regularization LOTdisp leads to a smaller shrinkage, since it aims at preserving
the shape of each cluster while transporting onto the target data (the obtained displacements are allparallel and of same length per cluster). Even if one could argue that the best regularization depends
on the dataset, one can suppose that LOTpos might be a better fit when a non-rigid transformation of
the data is required.

5.2

Domain adaptation on the two-moons dataset

In this experiment, we consider a rotating two-moons toy example that was used for domain adaptation in [12]. The source domain consists in the standard two entangled moons data, where eachmoon is associated to a specific class. The target domain is built by applying a rotation to the twomoons, which allows to consider an adaptation problem with an increasing difficulty as a function of
the rotation angle. Examples of the datasets at 50◦ , 70◦ and 90◦ are reported in the top row of Fig. 3.
This example is notably interesting because the corresponding transformation is clearly non-linear,and because the input dimensionality is small. We follow the same experimental setup as in [12]: thisallows for a direct comparison with the state-of-the-art results presented in [12]. Both domains arebuilt with two moons made of 150 samples each. After adaptation between the two distributions, thegeneralization power is tested over a set of 1000 samples from the target domain. The experiments
are conducted 10 times and we consider the mean classification error as a comparison criterion. The
classification is conducted by an SVM classifier with a Gaussian kernel, whose parameters were set
by 5-fold cross-validation. We compare the adaptation results with two state-of-the-art methods:
the DA-SVM approach [13] and the more recent PBDA [12], which has proved to give competitive
results on this dataset. The graph regularization Ss and St are computed using a Gaussian kernel
with γ = 0.1 and the regularization term are set to λs = λt = λ selected empirically. Results
are reported in Table 1 and the resulting classifiers for classical OT LP, OTSinkhorn and OTpos areplotted on Figure 3 for the 50◦ , 70◦ and 90◦ cases.
Our first observation is that all the methods based on optimal transport behave better than the competing methods. This is particularly striking when the amount of rotation is important (> 50◦ ). This
clearly shows the capacity of our method to handle strong distortions between domains. The entropy
regularization (OTsinkhorn ) leads to a dramatic performance gain w.r.t. the classical OT LP. OTpos
is the best performing method thanks to its Laplacian regularization. In this dataset, it clearly outperforms OTdisp . Our interpretation is that, in this case, non-rigid transport is necessary. Note that
we are aware that this dataset, despite its wide use in the domain adaptation community, is clearlybiased toward the Laplacian regularization of the position of the samples.
5Transported samples x̂si

Transported samples x̂si(a) OTSinkhorn with small regularization(b) OTSinkhorn with large regularization

Transported samples x̂si

Transported samples x̂si(c) LOTpos with small regularization(d) LOTpos with large regularization

Transported samples x̂si

Transported samples x̂si(e) LOTdisp with small regularization(f) LOTdisp with large regularization

Figure 2: Illustration of the different transport regularizations on the example of Figure 1. The transported samples and their displacement (in red) are illustrated for OTsinkhorn , LOTpos and LOTdisp
for a small (left column) and large (right column) regularization term.

5.3

Non-rigid shape matching

The last example considers non-rigid 3D matching of humanoid shapes. This problem has attracted
over the past years a lot of interest in both the domains of computer vision and graphics, where 3D
shapes need to be registered, either for reconstruction or tracking purposes (see [14] for a survey).
Among those approaches, the optimal transport distances were envisaged [15] to measure the distortion between objects. Yet, most of the approaches based on such distances encountered difficulties
in the practical computation of the matching.
6

SVM (no adpat)
DASVM [13]PBDA [12]OT LP
OTsinkhorn
LOTpos
LOTdisp

10◦
0.000
0.000
0.000
0.000
0.000
0.000
0.000

20◦
0.104
0.000
0.094
0.000
0.000
0.000
0.000

30◦
0.24
0.259
0.103
0.031
0.000
0.000
0.000

40◦
0.312
0.284
0.225
0.102
0.000
0.000
0.000

50◦
0.4
0.334
0.412
0.166
0.013
0.000
0.000

70◦
0.764
0.747
0.626
0.292
0.202
0.022
0.067

90◦
0.828
0.82
0.687
0.441
0.386
0.152
0.384

Table 1: Mean error rate over 10 realizations of the two-moons classification problem. The target
domain is given by a rotation of the source domain of angle given in the first row.

1
2
3
4
5
6

methodOT LP)
LOTpos
LOTdisp
OT LP
LOTpos
LOTdisp
OT LP
LOTpos
LOTdisp
OT LP
LOTpos
LOTdisp
OT LP
LOTpos
LOTdisp
OT LP
LOTpos
LOTdisp1–––
7.7 (57.0)
7.0 (39.2)
6.6 (41.5)
51.3 (113.7)
49.2 (108.7)
51.4 (109.7)
3.2 (48.7)
3.4 (47.8)
3.3 (45.0)
46.8 (113.5)
45.0 (109.0)
47.2 (110.9 )43.9 (112.6)
41.3 (107.5)
44.0 (108.7)23
8.0 (54.5) 46.3 (141.2)
7.5 (37.2) 44.0 (133.2)
7.1 (38.9) 44.3 (132.0)–
40.2 – 133.2)–
38.0 (90.3 )–38.5 (89.6)
41.3 (85.2)–
39.1 (78.7)–
40.8 (80.0)–
12.8 (71.6) 43.8 (143.7)
11.8 (54.5 ) 41.3 (132.4)
10.9 (66.0) 41.9 (131.0)
36.6 (79.3)
5.7 (53.1)
34.9 (79.3)
5.6 (49.8)
36.6 (79.8 ) 5.2 (44.6 )32.1 (84.2) 12.3 (54.2)
30.0 (77.7) 12.1 (49.0 )32.1 (79.6) 11.6 (45.4)4
3.4 (57.6)
3.7 (41.1)
3.5 (42.6)
12.0 (58.7)
11.1 (48.7)
10.5 (47.5)
48.5 (113.2)
46.4 (107.7)
48.7 (108.8)–––
49.8 (112.9)
48.1 (108.6)
50.2 (110.7)
46.3 -112.1)
43.8 (107.0)
46.4 (108.0)

550.1 (160.9)
46.5 (139.1 )47.9 (142.5)
44.1 (138.7)
40.7 (100.9)
42.4 (100.0)
6.1 (50.3)
6.0 (47.5)
5.7 (48.3)
54.0 (178.5)
50.3 (109.5)
51.4 (149.8)–––
7.5 (50.7)
7.5 (44.3)
7.1 (45.6)

634.0 (125.3)
31.1 (102.5)
31.9 (108.0)
26.8 (100.8)
24.3 (60.4)
25.5 (61.5)
11.4 (49.4)
11.0 (46.9)
10.8 (46.0)
37.4 (144.1)
34.8 (109.5)
35.3 (114.5)
6.6 (41.2)
6.0 (36.1)
5.9 (37.4)–––

Table 2: Shape registration experiments. For each pairs of possible matching, we report in centimeters the mean average error and, between parenthesis, the maximum error for the three considered
methods.

We use the FAUST dataset [16], which provides ground truth correspondences between shapes, since
every scan comes with a registered template watertight mesh. This allows to compute numericalerrors over the possible registrations. Our primary goal here is not to compare LOT with state-ofthe-art methods in shape registration, but to study the benefits of our the Laplacian regularizationof the transport for this registration purpose. Laplacian methods seem to be particularly adapted tosolve this problem, since its objective is to align surfaces, while preserving their graph structures.
Among the different shapes offered in the dataset, we selected 6 shapes that are related to different
poses of the same subject (see Figure 4). The corresponding registration problem is very complex,
particularly because the magnitude of deformations differs from one pair of shapes to the other. We
directly use the corresponding template meshes to assess for the quality of the matching. Figure 5
illustrates the mapping over two pairs of shapes, respectively (#1,#2) and (#3,#6) by the LOTpos
regularization. We note here that the original transport leads to a one-to-one assignment between
vertices, whereas the Laplacian regularization yields a one-to-many mapping. Consequently, weevaluate the correctness of the mapping by looking at the distance between vertices of the transported
mesh (equation (5)) and their expected location on the target mesh. Table 2 reports both the average
distance error (in centimeters) over all the vertices of the mesh and the corresponding maximum
error (as a measure of the most irrelevant mapping). From those results, one can see that both
LOTpos and LOTdisp regularization strategies lead to smaller average and maximal matching errors
compared to OT LP, with a slight advantage for the regularization over the positions of the vertices.
7

(a) rotation=50◦(a) rotation=70◦(a) rotation=90◦

Figure 3: Illustration of the classification decision boundary over the two moons example. The target
domain is depicted as points in grey (best viewed with colors).

6

Conclusion

In this work we proposed to use two Laplacian regularization schemes in the optimal transport
definition, acting either on the position or on the displacement of the transported samples. The
resolution of the resulting optimization problem is discussed and the different regularization terms
are tested on an illustrative example, the complex two-moons domain adaptation dataset and a 3D
shape matching problem. In all examples, the proposed LOT showed promising performances andachieved at least state-of-art results.
8

(a) #1(b) #2(c) #3(d) #4(e) #5(f) #6

Figure 4: The 6 different human shapes from the FAUST dataset considered in our non-rigid registration experiment.

Figure 5: Illustrations of the shape matching results for shapes pairs (#1,#2) and (#3,#6) whenusing LOTpos regularization. The optimal coupling matrix is also represented in the bottom rightcorner of each panel. As the points are ordered in the same ways in both meshes, a diagonal dominance of the matrix indicates the success of the matching process.
We plan in future works to perform more in-depth numerical experiments with new real life datasetsand to address the problem of large scale optimization. The size of the data in terms of number ofsamples is indeed limited by the use of a LP solver at each iteration of the Frank-Wolfe algorithm
and new optimization strategies will be necessary for exploiting this type of methodologies withlarge scale datasets.

References[1] C. Villani. Optimal transport: old and new. Grundlehren der mathematischen Wissenschaften.
Springer, 2009.
[2] S. Ferradans, N. Papadakis, G. Peyré, and J-F. Aujol. Regularized discrete optimal transport.
SIAM Journal on Imaging Sciences, 7(3), 2014.
[3] M. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transportation. In NIPS,
pages 2292–2300. 2013.
[4] N. Papadakis, G. Peyré, and E. Oudet. Optimal Transport with Proximal Splitting. SIAM
Journal on Imaging Sciences, 7(1):212–238, 2014.
[5] Y. Rubner, C. Tomasi, and L.J. Guibas. A metric for distributions with applications to image
databases. In ICCV, pages 59–66, Jan 1998.
9

[6] M. Cuturi and A. Doucet. Fast computation of wasserstein barycenters. In Proceedings of the
31st International Conference on Machine Learning (ICML-14), jun 2014.
[7] N. Courty, R. Flamary, and D. Tuia. Domain adaptation with regularized optimal transport.
In European Conference on Machine Learning and Principles and Practice of Knowledge
Discovery in Databases (ECML PKDD), 2014.
[8] L. Kantorovich. On the translocation of masses. C.R. (Doklady) Acad. Sci. URSS (N.S.),
37:199–201, 1942.
[9] Marguerite Frank and Philip Wolfe. An algorithm for quadratic programming. Naval researchlogistics quarterly, 3(1-2):95–110, 1956.
[10] Martin Jaggi. Revisiting {Frank-Wolfe}: Projection-free sparse convex optimization. In Proceedings of the 30th International Conference on Machine Learning (ICML-13), pages 427–
435, 2013.
[11] K.B. Petersen and M.S. Pedersen. The matrix cookbook. Technical University of Denmark,2006.
[12] P. Germain, A. Habrard, F. Laviolette, and E. Morvant. A PAC-Bayesian Approach for Domain
Adaptation with Specialization to Linear Classifiers. In ICML, pages 738–746, Atlanta, USA,2013.
[13] L. Bruzzone and M. Marconcini. Domain adaptation problems: A dasvm classification technique and a circular validation strategy. Pattern Analysis and Machine Intelligence, IEEE
Transactions on, 32(5):770–787, May 2010.
[14] O. van Kaick, H. Zhang, G. Hamarneh, and D. Cohen-Or. A survey on shape correspondence.
Computer Graphics Forum, 30(6):1681–1707, 2011.
[15] F. Mémoli. Gromov-wasserstein distances and the metric approach to object matching. Foundations of Computational Mathematics, pages 1–71, 2011.
[16] Federica Bogo, Javier Romero, Matthew Loper, and Michael J. Black. FAUST: Dataset andevaluation for 3D mesh registration. In Proceedings IEEE Conf. on Computer Vision and
Pattern Recognition (CVPR), Piscataway, NJ, USA, June 2014. IEEE.

10