# Quick start guide¶

In the following we provide some pointers about which functions and classes to use for different problems related to optimal transport (OT) and machine learning. We refer when we can to concrete examples in the documentation that are also available as notebooks on the POT Github.

Note

For a good introduction to numerical optimal transport we refer the reader to the book by Peyré and Cuturi 15. For more detailed introduction to OT and how it can be used in ML applications we refer the reader to the following OTML tutorial.

## Why Optimal Transport ?¶

### When to use OT¶

Optimal Transport (OT) is a mathematical problem introduced by Gaspard Monge in 1781 that aim at finding the most efficient way to move mass between distributions. The cost of moving a unit of mass between two positions is called the ground cost and the objective is to minimize the overall cost of moving one mass distribution onto another one. The optimization problem can be expressed for two distributions \(\mu_s\) and \(\mu_t\) as

where \(c(\cdot,\cdot)\) is the ground cost and the constraint \(m \# \mu_s = \mu_t\) ensures that \(\mu_s\) is completely transported to \(\mu_t\). This problem is particularly difficult to solve because of this constraint and has been replaced in practice (on discrete distributions) by a linear program easier to solve. It corresponds to the Kantorovitch formulation where the Monge mapping \(m\) is replaced by a joint distribution (OT matrix expressed in the next section) (see Solving optimal transport).

From the optimization problem above we can see that there are two main aspects to the OT solution that can be used in practical applications:

The optimal value (Wasserstein distance): Measures similarity between distributions.

The optimal mapping (Monge mapping, OT matrix): Finds correspondences between distributions.

In the first case, OT can be used to measure similarity between distributions (or datasets), in this case the Wasserstein distance (the optimal value of the problem) is used. In the second case one can be interested in the way the mass is moved between the distributions (the mapping). This mapping can then be used to transfer knowledge between distributions.

#### Wasserstein distance between distributions¶

OT is often used to measure similarity between distributions, especially when they do not share the same support. When the support between the distributions is disjoint OT-based Wasserstein distances compare favorably to popular f-divergences including the popular Kullback-Leibler, Jensen-Shannon divergences, and the Total Variation distance. What is particularly interesting for data science applications is that one can compute meaningful sub-gradients of the Wasserstein distance. For these reasons it became a very efficient tool for machine learning applications that need to measure and optimize similarity between empirical distributions.

Numerous contributions make use of this an approach is the machine learning (ML) literature. For example OT was used for training Generative Adversarial Networks (GANs) in order to overcome the vanishing gradient problem. It has also been used to find discriminant or robust subspaces for a dataset. The Wasserstein distance has also been used to measure similarity between word embeddings of documents or between signals or spectra.

#### OT for mapping estimation¶

A very interesting aspect of OT problem is the OT mapping in itself. When computing optimal transport between discrete distributions one output is the OT matrix that will provide you with correspondences between the samples in each distributions.

This correspondence is estimated with respect to the OT criterion and is found in a non-supervised way, which makes it very interesting on problems of transfer between datasets. It has been used to perform color transfer between images or in the context of domain adaptation. More recent applications include the use of extension of OT (Gromov-Wasserstein) to find correspondences between languages in word embeddings.

### When to use POT¶

The main objective of POT is to provide OT solvers for the rapidly growing area of OT in the context of machine learning. To this end we implement a number of solvers that have been proposed in research papers. Doing so we aim to promote reproducible research and foster novel developments.

One very important aspect of POT is its ability to be easily extended. For
instance we provide a very generic OT solver `ot.optim.cg`

that can solve
OT problems with any smooth/continuous regularization term making it
particularly practical for research purpose. Note that this generic solver has
been used to solve both graph Laplacian regularization OT and Gromov
Wasserstein 30.

Note

POT is originally designed to solve OT problems with Numpy interface and is not yet compatible with Pytorch API. We are currently working on a torch submodule that will provide OT solvers and losses for the most common deep learning configurations.

#### When not to use POT¶

While POT has to the best of our knowledge one of the most efficient exact OT solvers, it has not been designed to handle large scale OT problems. For instance the memory cost for an OT problem is always \(\mathcal{O}(n^2)\) in memory because the cost matrix has to be computed. The exact solver in of time complexity \(\mathcal{O}(n^3\log(n))\) and the Sinkhorn solver has been proven to be nearly \(\mathcal{O}(n^2)\) which is still too complex for very large scale solvers.

If you need to solve OT with large number of samples, we recommend to use entropic regularization and memory efficient implementation of Sinkhorn as proposed in GeomLoss. This implementation is compatible with Pytorch and can handle large number of samples. Another approach to estimate the Wasserstein distance for very large number of sample is to use the trick from Wasserstein GAN that solves the problem in the dual with a neural network estimating the dual variable. Note that in this case you are only solving an approximation of the Wasserstein distance because the 1-Lipschitz constraint on the dual cannot be enforced exactly (approximated through filter thresholding or regularization). Finally note that in order to avoid solving large scale OT problems, a number of recent approached minimized the expected Wasserstein distance on minibtaches that is different from the Wasserstein but has better computational and statistical properties.

## Optimal transport and Wasserstein distance¶

Note

In POT, most functions that solve OT or regularized OT problems have two
versions that return the OT matrix or the value of the optimal solution. For
instance `ot.emd`

returns the OT matrix and `ot.emd2`

returns the
Wassertsein distance. This approach has been implemented in practice for all
solvers that return an OT matrix (even Gromov-Wasserstsein).

### Solving optimal transport¶

The optimal transport problem between discrete distributions is often expressed as

where:

\(M\in\mathbb{R}_+^{m\times n}\) is the metric cost matrix defining the cost to move mass from bin \(a_i\) to bin \(b_j\).

\(a\) and \(b\) are histograms on the simplex (positive, sum to 1) that represent the weights of each samples in the source an target distributions.

Solving the linear program above can be done using the function `ot.emd`

that will return the optimal transport matrix \(\gamma^*\):

```
# a and b are 1D histograms (sum to 1 and positive)
# M is the ground cost matrix
T = ot.emd(a, b, M) # exact linear program
```

The method implemented for solving the OT problem is the network simplex. It is implemented in C from 1. It has a complexity of \(O(n^3)\) but the solver is quite efficient and uses sparsity of the solution.

Hint

Examples of use for `ot.emd`

are available in :

### Computing Wasserstein distance¶

The value of the OT solution is often more interesting than the OT matrix:

It can computed from an already estimated OT matrix with
`np.sum(T*M)`

or directly with the function `ot.emd2`

.

```
# a and b are 1D histograms (sum to 1 and positive)
# M is the ground cost matrix
W = ot.emd2(a, b, M) # Wasserstein distance / EMD value
```

Note that the well known Wasserstein distance between distributions a and b is defined as

\[ \begin{align}\begin{aligned}W_p(a,b)=(\min_{\gamma \in \mathbb{R}_+^{m\times n}} \sum_{i,j}\gamma_{i,j}\|x_i-y_j\|_p)^\frac{1}{p}\\s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0\end{aligned}\end{align} \]

This means that if you want to compute the \(W_2\) you need to compute the
square root of `ot.emd2`

when providing
`M = ot.dist(xs, xt)`

, that uses the squared euclidean distance by default. Computing
the \(W_1\) Wasserstein distance can be done directly with `ot.emd2`

when providing `M = ot.dist(xs, xt, metric='euclidean')`

to use the Euclidean
distance.

### Special cases¶

Note that the OT problem and the corresponding Wasserstein distance can in some special cases be computed very efficiently.

For instance when the samples are in 1D, then the OT problem can be solved in
\(O(n\log(n))\) by using a simple sorting. In this case we provide the
function `ot.emd_1d`

and `ot.emd2_1d`

to return respectively the OT
matrix and value. Note that since the solution is very sparse the `sparse`

parameter of `ot.emd_1d`

allows for solving and returning the solution for
very large problems. Note that in order to compute directly the \(W_p\)
Wasserstein distance in 1D we provide the function `ot.wasserstein_1d`

that
takes `p`

as a parameter.

Another special case for estimating OT and Monge mapping is between Gaussian
distributions. In this case there exists a close form solution given in Remark
2.29 in 15 and the Monge mapping is an affine function and can be
also computed from the covariances and means of the source and target
distributions. In the case when the finite sample dataset is supposed Gaussian,
we provide `ot.da.OT_mapping_linear`

that returns the parameters for the
Monge mapping.

## Regularized Optimal Transport¶

Recent developments have shown the interest of regularized OT both in terms of computational and statistical properties. We address in this section the regularized OT problems that can be expressed as

where :

\(M\in\mathbb{R}_+^{m\times n}\) is the metric cost matrix defining the cost to move mass from bin \(a_i\) to bin \(b_j\).

\(a\) and \(b\) are histograms (positive, sum to 1) that represent the weights of each samples in the source an target distributions.

\(\Omega\) is the regularization term.

We discuss in the following specific algorithms that can be used depending on the regularization term.

### Entropic regularized OT¶

This is the most common regularization used for optimal transport. It has been proposed in the ML community by Marco Cuturi in his seminal paper 2. This regularization has the following expression

The use of the regularization term above in the optimization problem has a very
strong impact. First it makes the problem smooth which leads to new optimization
procedures such as the well known Sinkhorn algorithm 2 or L-BFGS (see
`ot.smooth`

). Next it makes the problem
strictly convex meaning that there will be a unique solution. Finally the
solution of the resulting optimization problem can be expressed as:

where \(u\) and \(v\) are vectors and \(K=\exp(-M/\lambda)\) where the \(\exp\) is taken component-wise. In order to solve the optimization problem, one can use an alternative projection algorithm called Sinkhorn-Knopp that can be very efficient for large values of regularization.

The Sinkhorn-Knopp algorithm is implemented in `ot.sinkhorn`

and
`ot.sinkhorn2`

that return respectively the OT matrix and the value of the
linear term. Note that the regularization parameter \(\lambda\) in the
equation above is given to those functions with the parameter `reg`

.

```
>>> import ot
>>> a = [.5, .5]
>>> b = [.5, .5]
>>> M = [[0., 1.], [1., 0.]]
>>> ot.sinkhorn(a, b, M, 1)
array([[ 0.36552929, 0.13447071],
[ 0.13447071, 0.36552929]])
```

More details about the algorithms used are given in the following note.

Note

The main function to solve entropic regularized OT is `ot.sinkhorn`

.
This function is a wrapper and the parameter `method`

allows you to select
the actual algorithm used to solve the problem:

`method='sinkhorn'`

calls`ot.bregman.sinkhorn_knopp`

the classic algorithm 2.`method='sinkhorn_stabilized'`

calls`ot.bregman.sinkhorn_stabilized`

the log stabilized version of the algorithm 9.`method='sinkhorn_epsilon_scaling'`

calls`ot.bregman.sinkhorn_epsilon_scaling`

the epsilon scaling version of the algorithm 9.`method='greenkhorn'`

calls`ot.bregman.greenkhorn`

the greedy Sinkhorn version of the algorithm 22.`method='screenkhorn'`

calls`ot.bregman.screenkhorn`

the screening sinkhorn version of the algorithm 26.

In addition to all those variants of Sinkhorn, we have another
implementation solving the problem in the smooth dual or semi-dual in
`ot.smooth`

. This solver uses the `scipy.optimize.minimize`

function to solve the smooth problem with `L-BFGS-B`

algorithm. Tu use
this solver, use functions `ot.smooth.smooth_ot_dual`

or
`ot.smooth.smooth_ot_semi_dual`

with parameter `reg_type='kl'`

to
choose entropic/Kullbach Leibler regularization.

**Choosing a Sinkhorn solver**

By default and when using a regularization parameter that is not too small
the default Sinkhorn solver should be enough. If you need to use a small
regularization to get sharper OT matrices, you should use the
`ot.bregman.sinkhorn_stabilized`

solver that will avoid numerical
errors. This last solver can be very slow in practice and might not even
converge to a reasonable OT matrix in a finite time. This is why
`ot.bregman.sinkhorn_epsilon_scaling`

that relie on iterating the value
of the regularization (and using warm start) sometimes leads to better
solutions. Note that the greedy version of the Sinkhorn
`ot.bregman.greenkhorn`

can also lead to a speedup and the screening
version of the Sinkhorn `ot.bregman.screenkhorn`

aim a providing a
fast approximation of the Sinkhorn problem.

Recently Genevay et al. 23 introduced the Sinkhorn divergence that build from entropic
regularization to compute fast and differentiable geometric divergence between
empirical distributions. Note that we provide a function that computes directly
(with no need to precompute the `M`

matrix)
the Sinkhorn divergence for empirical distributions in
`ot.bregman.empirical_sinkhorn_divergence`

. Similarly one can compute the
OT matrix and loss for empirical distributions with respectively
`ot.bregman.empirical_sinkhorn`

and `ot.bregman.empirical_sinkhorn2`

.

Finally note that we also provide in `ot.stochastic`

several implementation
of stochastic solvers for entropic regularized OT 18 19. Those pure Python
implementations are not optimized for speed but provide a robust implementation
of algorithms in 18 19.

Hint

Examples of use for `ot.sinkhorn`

are available in :

### Other regularization¶

While entropic OT is the most common and favored in practice, there exists other
kinds of regularizations. We provide in POT two specific solvers for other
regularization terms, namely quadratic regularization and group Lasso
regularization. But we also provide in `ot.optim`

two generic solvers
that allows solving any smooth regularization in practice.

#### Quadratic regularization¶

The first general regularization term we can solve is the quadratic regularization of the form

This regularization term has an effect similar to entropic regularization by
densifying the OT matrix, yet it keeps some sort of sparsity that is lost with
entropic regularization as soon as \(\lambda>0\) 17. This problem can be
solved with POT using solvers from `ot.smooth`

, more specifically
functions `ot.smooth.smooth_ot_dual`

or
`ot.smooth.smooth_ot_semi_dual`

with parameter `reg_type='l2'`

to
choose the quadratic regularization.

Hint

Examples of quadratic regularization are available in :

#### Group Lasso regularization¶

Another regularization that has been used in recent years 5 is the group Lasso regularization

where \(\mathcal{G}\) contains non-overlapping groups of lines in the OT
matrix. This regularization proposed in 5 promotes sparsity at the group level and for
instance will force target samples to get mass from a small number of groups.
Note that the exact OT solution is already sparse so this regularization does
not make sense if it is not combined with entropic regularization. Depending on
the choice of `p`

and `q`

, the problem can be solved with different
approaches. When `q=1`

and `p<1`

the problem is non-convex but can
be solved using an efficient majoration minimization approach with
`ot.sinkhorn_lpl1_mm`

. When `q=2`

and `p=1`

we recover the
convex group lasso and we provide a solver using generalized conditional
gradient algorithm 7 in function `ot.da.sinkhorn_l1l2_gl`

.

Hint

Examples of group Lasso regularization are available in:

#### Generic solvers¶

Finally we propose in POT generic solvers that can be used to solve any regularization as long as you can provide a function computing the regularization and a function computing its gradient (or sub-gradient).

In order to solve

you can use function `ot.optim.cg`

that will use a conditional gradient as
proposed in 6 . You need to provide the regularization function as parameter
`f`

and its gradient as parameter `df`

. Note that the conditional gradient relies on
iterative solving of a linearization of the problem using the exact
`ot.emd`

so it can be quite slow in practice. However, being an interior point
algorithm, it always returns a transport matrix that does not violates the marginals.

Another generic solver is proposed to solve the problem:

where \(\Omega_e\) is the entropic regularization. In this case we use a
generalized conditional gradient 7 implemented in `ot.optim.gcg`

that
does not linearize the entropic term but
relies on `ot.sinkhorn`

for its iterations.

## Wasserstein Barycenters¶

A Wasserstein barycenter is a distribution that minimizes its Wasserstein distance with respect to other distributions 16. It corresponds to minimizing the following problem by searching a distribution \(\mu\) such that

In practice we model a distribution with a finite number of support position:

where \(a\) is an histogram on the simplex and the \(\{x_i\}\) are the position of the support. We can clearly see here that optimizing \(\mu\) can be done by searching for optimal weights \(a\) or optimal support \(\{x_i\}\) (optimizing both is also an option). We provide in POT solvers to estimate a discrete Wasserstein barycenter in both cases.

### Barycenters with fixed support¶

When optimizing a barycenter with a fixed support, the optimization problem can be expressed as

where \(b_k\) are also weights in the simplex. In the non-regularized case,
the problem above is a classical linear program. In this case we propose a
solver `ot.lp.barycenter()`

that relies on generic LP solvers. By default the
function uses `scipy.optimize.linprog`

, but more efficient LP solvers from
cvxopt can be also used by changing parameter `solver`

. Note that this problem
requires to solve a very large linear program and can be very slow in
practice.

Similarly to the OT problem, OT barycenters can be computed in the regularized
case. When entropic regularization is used, the problem can be solved with a
generalization of the Sinkhorn algorithm based on Bregman projections 3. This
algorithm is provided in function `ot.bregman.barycenter`

also available as
`ot.barycenter`

. In this case, the algorithm scales better to large
distributions and relies only on matrix multiplications that can be performed in
parallel.

In addition to the speedup brought by regularization, one can also greatly
accelerate the estimation of Wasserstein barycenter when the support has a
separable structure 21. In the case of 2D images for instance one can replace
the matrix vector production in the Bregman projections by convolution
operators. We provide an implementation of this algorithm in function
`ot.bregman.convolutional_barycenter2d`

.

Hint

Examples of Wasserstein (`ot.lp.barycenter()`

) and regularized Wasserstein
barycenter (`ot.bregman.barycenter`

) computation are available in :

An example of convolutional barycenter
(`ot.bregman.convolutional_barycenter2d`

) computation
for 2D images is available
in :

### Barycenters with free support¶

Estimating the Wasserstein barycenter with free support but fixed weights corresponds to solving the following optimization problem:

We provide a solver based on 20 in
`ot.lp.free_support_barycenter`

. This function minimize the problem and
return a locally optimal support \(\{x_i\}\) for uniform or given weights
\(a\).

Hint

An example of the free support barycenter estimation is available in :

## Monge mapping and Domain adaptation¶

The original transport problem investigated by Gaspard Monge was seeking for a
mapping function that maps (or transports) between a source and target
distribution but that minimizes the transport loss. The existence and uniqueness of this
optimal mapping is still an open problem in the general case but has been proven
for smooth distributions by Brenier in his eponym theorem. We provide in
`ot.da`

several solvers for smooth Monge mapping estimation and domain
adaptation from discrete distributions.

### Monge Mapping estimation¶

We now discuss several approaches that are implemented in POT to estimate or approximate a Monge mapping from finite distributions.

First note that when the source and target distributions are supposed to be Gaussian
distributions, there exists a close form solution for the mapping and its an
affine function 14 of the form \(T(x)=Ax+b\) . In this case we provide the function
`ot.da.OT_mapping_linear`

that returns the operator \(A\) and vector
\(b\). Note that if the number of samples is too small there is a parameter
`reg`

that provides a regularization for the covariance matrix estimation.

For a more general mapping estimation we also provide the barycentric mapping
proposed in 6. It is implemented in the class `ot.da.EMDTransport`

and
other transport-based classes in `ot.da`

. Those classes are discussed more
in the following but follow an interface similar to scikit-learn classes. Finally a
method proposed in 8 that estimates a continuous mapping approximating the
barycentric mapping is provided in `ot.da.joint_OT_mapping_linear`

for
linear mapping and `ot.da.joint_OT_mapping_kernel`

for non-linear mapping.

Hint

An example of the linear Monge mapping estimation is available in :

### Domain adaptation classes¶

The use of OT for domain adaptation (OTDA) has been first proposed in 5 that also introduced the group Lasso regularization. The main idea of OTDA is to estimate a mapping of the samples between source and target distributions which allows to transport labeled source samples onto the target distribution with no labels.

We provide several classes based on `ot.da.BaseTransport`

that provide
several OT and mapping estimations. The interface of those classes is similar to
classifiers in scikit-learn. At initialization, several parameters such as
regularization parameter value can be set. Then one needs to estimate the
mapping with function `ot.da.BaseTransport.fit`

. Finally one can map the
samples from source to target with `ot.da.BaseTransport.transform`

and
from target to source with `ot.da.BaseTransport.inverse_transform`

.

Here is an example for class `ot.da.EMDTransport`

:

```
ot_emd = ot.da.EMDTransport()
ot_emd.fit(Xs=Xs, Xt=Xt)
Xs_mapped = ot_emd.transform(Xs=Xs)
```

A list of the provided implementation is given in the following note.

Note

Here is a list of the OT mapping classes inheriting from
`ot.da.BaseTransport`

`ot.da.EMDTransport`

: Barycentric mapping with EMD transport`ot.da.SinkhornTransport`

: Barycentric mapping with Sinkhorn transport`ot.da.SinkhornL1l2Transport`

: Barycentric mapping with Sinkhorn + group Lasso regularization 5`ot.da.SinkhornLpl1Transport`

: Barycentric mapping with Sinkhorn + non convex group Lasso regularization 5`ot.da.LinearTransport`

: Linear mapping estimation between Gaussians 14`ot.da.MappingTransport`

: Nonlinear mapping estimation 8

Hint

Examples of the use of OTDA classes are available in:

## Other applications¶

We discuss in the following several OT related problems and tools that has been proposed in the OT and machine learning community.

### Wasserstein Discriminant Analysis¶

Wasserstein Discriminant Analysis 11 is a generalization of Fisher Linear Discriminant Analysis that allows discrimination between classes that are not linearly separable. It consists in finding a linear projector optimizing the following criterion

where \(\#\) is the push-forward operator, \(OT_e\) is the entropic OT
loss and \(\mu_i\) is the
distribution of samples from class \(i\). \(P\) is also constrained to
be in the Stiefel manifold. WDA can be solved in POT using function
`ot.dr.wda`

. It requires to have installed `pymanopt`

and
`autograd`

for manifold optimization and automatic differentiation
respectively. Note that we also provide the Fisher discriminant estimator in
`ot.dr.fda`

for easy comparison.

Warning

Note that due to the hard dependency on `pymanopt`

and
`autograd`

, `ot.dr`

is not imported by default. If you want to
use it you have to specifically import it with `import ot.dr`

.

### Unbalanced optimal transport¶

Unbalanced OT is a relaxation of the entropy regularized OT problem where the violation of the constraint on the marginals is added to the objective of the optimization problem. The unbalanced OT metric between two unbalanced histograms a and b is defined as 25 10:

where KL is the Kullback-Leibler divergence. This formulation allows for
computing approximate mapping between distributions that do not have the same
amount of mass. Interestingly the problem can be solved with a generalization of
the Bregman projections algorithm 10. We provide a solver for unbalanced OT
in `ot.unbalanced`

. Computing the optimal transport
plan or the transport cost is similar to the balanced case. The Sinkhorn-Knopp
algorithm is implemented in `ot.sinkhorn_unbalanced`

and `ot.sinkhorn_unbalanced2`

that return respectively the OT matrix and the value of the
linear term.

Note

The main function to solve entropic regularized UOT is `ot.sinkhorn_unbalanced`

.
This function is a wrapper and the parameter `method`

helps you select
the actual algorithm used to solve the problem:

`method='sinkhorn'`

calls`ot.unbalanced.sinkhorn_knopp_unbalanced`

the generalized Sinkhorn algorithm 25 10.`method='sinkhorn_stabilized'`

calls`ot.unbalanced.sinkhorn_stabilized_unbalanced`

the log stabilized version of the algorithm 10.

Hint

Examples of the use of `ot.sinkhorn_unbalanced`

are available in :

### Unbalanced Barycenters¶

As with balanced distributions, we can define a barycenter of a set of histograms with different masses as a Fréchet Mean:

\[\min_{\mu} \quad \sum_{k} w_kW_u(\mu,\mu_k)\]

where \(W_u\) is the unbalanced Wasserstein metric defined above. This problem
can also be solved using generalized version of Sinkhorn’s algorithm and it is
implemented the main function `ot.barycenter_unbalanced`

.

Note

The main function to compute UOT barycenters is `ot.barycenter_unbalanced`

.
This function is a wrapper and the parameter `method`

helps you select
the actual algorithm used to solve the problem:

`method='sinkhorn'`

calls`ot.unbalanced.barycenter_unbalanced_sinkhorn_unbalanced()`

the generalized Sinkhorn algorithm 10.`method='sinkhorn_stabilized'`

calls`ot.unbalanced.barycenter_unbalanced_stabilized`

the log stabilized version of the algorithm 10.

Hint

Examples of the use of `ot.barycenter_unbalanced`

are available in :

### Partial optimal transport¶

Partial OT is a variant of the optimal transport problem when only a fixed amount of mass m is to be transported. The partial OT metric between two histograms a and b is defined as 28:

Interestingly the problem can be casted into a regular OT problem by adding reservoir points
in which the surplus mass is sent 29. We provide a solver for partial OT
in `ot.partial`

. The exact resolution of the problem is computed in `ot.partial.partial_wasserstein`

and `ot.partial.partial_wasserstein2`

that return respectively the OT matrix and the value of the
linear term. The entropic solution of the problem is computed in `ot.partial.entropic_partial_wasserstein`

(see 3).

The partial Gromov-Wasserstein formulation of the problem

is computed in `ot.partial.partial_gromov_wasserstein`

and in
`ot.partial.entropic_partial_gromov_wasserstein`

when considering the entropic
regularization of the problem.

Hint

Examples of the use of `ot.partial`

are available in:

### Gromov-Wasserstein¶

Gromov Wasserstein (GW) is a generalization of OT to distributions that do not lie in the same space 13. In this case one cannot compute distance between samples from the two distributions. 13 proposed instead to realign the metric spaces by computing a transport between distance matrices. The Gromow Wasserstein alignment between two distributions can be expressed as the one minimizing:

where :\(C1\) is the distance matrix between samples in the source
distribution and \(C2\) the one between samples in the target,
\(L(C1_{i,k},C2_{j,l})\) is a measure of similarity between
\(C1_{i,k}\) and \(C2_{j,l}\) often chosen as
\(L(C1_{i,k},C2_{j,l})=\|C1_{i,k}-C2_{j,l}\|^2\). The optimization problem
above is a non-convex quadratic program but we provide a solver that finds a
local minimum using conditional gradient in `ot.gromov.gromov_wasserstein`

.
There also exists an entropic regularized variant of GW that has been proposed in
12 and we provide an implementation of their algorithm in
`ot.gromov.entropic_gromov_wasserstein`

.

Note that similarly to Wasserstein distance GW allows for the definition of GW barycenters that can be expressed as

where \(Ck\) is the distance matrix between samples in distribution
\(k\). Note that interestingly the barycenter is defined as a symmetric
positive matrix. We provide a block coordinate optimization procedure in
`ot.gromov.gromov_barycenters`

and
`ot.gromov.entropic_gromov_barycenters`

for non-regularized and regularized
barycenters respectively.

Finally note that recently a fusion between Wasserstein and GW, coined Fused
Gromov-Wasserstein (FGW) has been proposed 24.
It allows to compute a similarity between objects that are only partly in
the same space. As such it can be used to measure similarity between labeled
graphs for instance and also provide computable barycenters.
The implementations of FGW and FGW barycenter is provided in functions
`ot.gromov.fused_gromov_wasserstein`

and `ot.gromov.fgw_barycenters`

.

Hint

Examples of computation of GW, regularized G and FGW are available in:

Examples of GW, regularized GW and FGW barycenters are available in:

### GPU acceleration¶

We provide several implementation of our OT solvers in `ot.gpu`

. Those
implementations use the `cupy`

toolbox that obviously need to be installed.

Note

Several implementations of POT functions (mainly those relying on linear
algebra) have been implemented in `ot.gpu`

. Here is a short list on the
main entries:

`ot.gpu.dist()`

: computation of distance matrix`ot.gpu.sinkhorn()`

: computation of sinkhorn`ot.gpu.sinkhorn_lpl1_mm()`

: computation of sinkhorn + group lasso

Note that while the `ot.gpu`

module has been designed to be compatible with
POT, calling its function with `numpy`

arrays will incur a large overhead due to
the memory copy of the array on GPU prior to computation and conversion of the
array after computation. To avoid this overhead, we provide functions
`ot.gpu.to_gpu()`

and `ot.gpu.to_np()`

that perform the conversion
explicitly.

Warning

Note that due to the hard dependency on `cupy`

, `ot.gpu`

is not
imported by default. If you want to
use it you have to specifically import it with `import ot.gpu`

.

## FAQ¶

**How to solve a discrete optimal transport problem ?**The solver for discrete OT is the function

`ot.emd`

that returns the OT transport matrix. If you want to solve a regularized OT you can use`ot.sinkhorn`

.Here is a simple use case:

# a and b are 1D histograms (sum to 1 and positive) # M is the ground cost matrix T = ot.emd(a, b, M) # exact linear program T_reg = ot.sinkhorn(a, b, M, reg) # entropic regularized OT

More detailed examples can be seen on this example: 2D Optimal transport between empirical distributions

**pip install POT fails with error : ImportError: No module named Cython.Build**As discussed shortly in the README file. POT requires to have

`numpy`

and`cython`

installed to build. This corner case is not yet handled by`pip`

and for now you need to install both library prior to installing POT.Note that this problem do not occur when using conda-forge since the packages there are pre-compiled.

See Issue #59 for more details.

**Why is Sinkhorn slower than EMD ?**This might come from the choice of the regularization term. The speed of convergence of Sinkhorn depends directly on this term 22. When the regularization gets very small the problem tries to approximate the exact OT which leads to slow convergence in addition to numerical problems. In other words, for large regularization Sinkhorn will be very fast to converge, for small regularization (when you need an OT matrix close to the true OT), it might be quicker to use the EMD solver.

Also note that the numpy implementation of Sinkhorn can use parallel computation depending on the configuration of your system, yet very important speedup can be obtained by using a GPU implementation since all operations are matrix/vector products.

**Using GPU fails with error: module ‘ot’ has no attribute ‘gpu’**

## References¶

- 1
Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011, December). Displacement nterpolation using Lagrangian mass transport. In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM.

- 2(1,2,3)
Cuturi, M. (2013). Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems (pp. 2292-2300).

- 3(1,2)
Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G. (2015). Iterative Bregman projections for regularized transportation problems. SIAM Journal on Scientific Computing, 37(2), A1111-A1138.

- 5(1,2,3,4,5)
N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, Optimal Transport for Domain Adaptation, in IEEE Transactions on Pattern Analysis and Machine Intelligence , vol.PP, no.99, pp.1-1

- 6(1,2)
Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3), 1853-1882.

- 7(1,2)
Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized conditional gradient: analysis of convergence and applications. arXiv preprint arXiv:1510.06567.

- 8(1,2)
M. Perrot, N. Courty, R. Flamary, A. Habrard (2016), Mapping estimation for discrete optimal transport, Neural Information Processing Systems (NIPS).

- 9(1,2)
Schmitzer, B. (2016). Stabilized Sparse Scaling Algorithms for Entropy Regularized Transport Problems. arXiv preprint arXiv:1610.06519.

- 10(1,2,3,4,5,6)
Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016). Scaling algorithms for unbalanced transport problems. arXiv preprint arXiv:1607.05816.

- 11
Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016). Wasserstein Discriminant Analysis. arXiv preprint arXiv:1608.08063.

- 12
Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016), Gromov-Wasserstein averaging of kernel and distance matrices International Conference on Machine Learning (ICML).

- 13(1,2)
Mémoli, Facundo (2011). Gromov–Wasserstein distances and the metric approach to object matching. Foundations of computational mathematics 11.4 : 417-487.

- 14(1,2)
Knott, M. and Smith, C. S. (1984). On the optimal mapping of distributions, Journal of Optimization Theory and Applications Vol 43.

- 15(1,2)
Peyré, G., & Cuturi, M. (2018). Computational Optimal Transport .

- 16
Agueh, M., & Carlier, G. (2011). Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis, 43(2), 904-924.

- 17
Blondel, M., Seguy, V., & Rolet, A. (2018). Smooth and Sparse Optimal Transport. Proceedings of the Twenty-First International Conference on Artificial Intelligence and Statistics (AISTATS).

- 18(1,2)
Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) Stochastic Optimization for Large-scale Optimal Transport. Advances in Neural Information Processing Systems (2016).

- 19(1,2)
Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet, A.& Blondel, M. Large-scale Optimal Transport and Mapping Estimation. International Conference on Learning Representation (2018)

- 20
Cuturi, M. and Doucet, A. (2014) Fast Computation of Wasserstein Barycenters. International Conference in Machine Learning

- 21
Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A. & Guibas, L. (2015). Convolutional wasserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphics (TOG), 34(4), 66.

- 22(1,2)
J. Altschuler, J.Weed, P. Rigollet, (2017) Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration, Advances in Neural Information Processing Systems (NIPS) 31

- 23
Genevay, A., Peyré, G., Cuturi, M., Learning Generative Models with Sinkhorn Divergences, Proceedings of the Twenty-First International Conference on Artficial Intelligence and Statistics, (AISTATS) 21, 2018

- 24
Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N. (2019). Optimal Transport for structured data with application on graphs Proceedings of the 36th International Conference on Machine Learning (ICML).

- 25(1,2)
Frogner C., Zhang C., Mobahi H., Araya-Polo M., Poggio T. : Learning with a Wasserstein Loss, Advances in Neural Information Processing Systems (NIPS) 2015

- 26
Alaya M. Z., Bérar M., Gasso G., Rakotomamonjy A. (2019). Screening Sinkhorn Algorithm for Regularized Optimal Transport <https://papers.nips.cc/paper/9386-screening-sinkhorn-algorithm-for-regularized-optimal-transport>, Advances in Neural Information Processing Systems 33 (NeurIPS).

- 28
Caffarelli, L. A., McCann, R. J. (2020). Free boundaries in optimal transport and Monge-Ampere obstacle problems <http://www.math.toronto.edu/~mccann/papers/annals2010.pdf>, Annals of mathematics, 673-730.

- 29
Chapel, L., Alaya, M., Gasso, G. (2019). Partial Gromov-Wasserstein with Applications on Positive-Unlabeled Learning <https://arxiv.org/abs/2002.08276>, arXiv preprint arXiv:2002.08276.

- 30
Flamary, Rémi, et al. “Optimal transport with Laplacian regularization: Applications to domain adaptation and shape matching.” NIPS Workshop on Optimal Transport and Machine Learning OTML. 2014.