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Quick start
===========
In the following we provide some pointers about which functions and classes
to use for different problems related to optimal transport (OT).
Optimal transport and Wasserstein distance
------------------------------------------
Solving optimal transport
^^^^^^^^^^^^^^^^^^^^^^^^^
The optimal transport problem between discrete distributions is often expressed
as
.. math::
\gamma^* = arg\min_\gamma \sum_{i,j}\gamma_{i,j}M_{i,j}
s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0
where :
- :math:`M\in\mathbb{R}_+^{m\times n}` is the metric cost matrix defining the cost to move mass from bin :math:`a_i` to bin :math:`b_j`.
- :math:`a` and :math:`b` are histograms (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 :any:`ot.emd`
that will return the optimal transport matrix :math:`\gamma^*`:
.. code:: python
# a,b are 1D histograms (sum to 1 and positive)
# M is the ground cost matrix
T=ot.emd(a,b,M) # exact linear program
.. hint::
Examples of use for :any:`ot.emd` are available in the following examples:
- :any:`auto_examples/plot_OT_2D_samples`
- :any:`auto_examples/plot_OT_1D`
- :any:`auto_examples/plot_OT_L1_vs_L2`
Computing Wasserstein distance
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The value of the OT solution is often more of interest that the OT matrix :
.. math::
W(a,b)=\min_\gamma \sum_{i,j}\gamma_{i,j}M_{i,j}
s.t. \gamma 1 = a; \gamma^T 1= b; \gamma\geq 0
where :math:`W(a,b)` is the `Wasserstein distance
<https://en.wikipedia.org/wiki/Wasserstein_metric>`_ between distributions a and b
It is a metrix that has nice statistical
properties. It can computed from an already estimated OT matrix with
:code:`np.sum(T*M)` or directly with the function :any:`ot.emd2`.
.. code:: python
# a,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
.. hint::
Examples of use for :any:`ot.emd2` are available in the following examples:
- :any:`auto_examples/plot_compute_emd`
.. 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. Fir
instance :any:`ot.emd` return the OT matrix and :any:`ot.emd2` return the
Wassertsein distance.
Regularized Optimal Transport
-----------------------------
Wasserstein Barycenters
-----------------------
Monge mapping and Domain adaptation with Optimal transport
----------------------------------------
Other applications
------------------
GPU acceleration
----------------
How to?
-------
1. **How to solve a discrete optimal transport problem ?**
The solver for discrete is the function :py:mod:`ot.emd` that returns
the OT transport matrix. If you want to solve a regularized OT you can
use :py:mod:`ot.sinkhorn`.
Here is a simple use case:
.. code:: python
# a,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
:doc:`auto_examples/plot_OT_2D_samples`
2. **Compute a Wasserstein distance**
|
