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references.rst
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References
==========
Bibliography
------------
.. bibliography::
:cited:
Glossary
--------
.. glossary::
OT
An `optimal transport <https://en.wikipedia.org/wiki/Transportation_theory_(mathematics)>`_ problem is defined
as a matching task between distributions, e.g. sets of cells.
transport matrix
The output of a discrete :term:`OT` problem indicating how much mass from data point :math:`x_i` in row
:math:`i` is transported to data point :math:`y_j` in column :math:`j`.
entropic regularization
Entropy regularization of :term:`OT` problems :cite:`cuturi:2013` reduces the time complexity and allows for
more desirable statistical properties. The higher the entropy regularization, the more diffused the OT solution.
marginals
An :term:`OT` problem matches distributions, e.g. set of cells. The distribution is defined by the location
of a cell, e.g. in gene expression space, and the weight assigned to one cell.
balanced OT problem
:term:`OT` problem where the :term:`marginals` are fixed. Each data point (cell) of the source distribution
emits a certain amount of mass given by the source :term:`marginals`, and each data point (cell) of the target
distribution receives a certain amount of mass given by the target :term:`marginals`.
unbalanced OT problem
:term:`OT` problem where the :term:`marginals` are not fixed. If beneficial, a data point might emit or
receive more or less mass than prescribed by the :term:`marginals`. The larger the unbalancedness parameters
``tau_a`` and ``tau_b``, the more the mass emitted, and received, respectively, can deviate from the
:term:`marginals` :cite:`chizat:18`.
linear problem
:term:`OT` problem only containing a :term:`linear term` and no :term:`quadratic term`.
linear term
Term of the cost function on the shared space, e.g. gene expression space.
quadratic problem
:term:`OT` problem containing a :term:`quadratic term` and possibly a :term:`linear term`.
quadratic term
Term of the cost function comparing two different spaces.
Gromov-Wasserstein
:term:`OT` problem between two distributions where a data point, e.g. a cell. in the source distribution
does not live in the same space as a data point in the target distribution. Such problem is a
:term:`quadratic problem`.
fused Gromov-Wasserstein
:term:`OT` problem between two distributions where a data point, e.g. a cell, of the source distribution
has both features in the same space as the target distribution (:term:`linear term`) and features in a
different space than a data point in the target distribution (:term:`quadratic term`). Such problem is a
:term:`quadratic problem`.
dual potentials
Potentials obtained by the :term:`Sinkhorn` algorithm which define the solution of a :term:`linear problem`
:cite:`cuturi:2013`. These weights are referred to as `marginals`.
Sinkhorn
The Sinkhorn algorithm :cite:`cuturi:2013` is used for solving a :term:`linear problem`, and is also used
in inner iterations for solving a :term:`quadratic problem`.
low-rank OT
`low-rank <https://en.wikipedia.org/wiki/Low-rank_approximation>`_ OT approximates full-rank :term:`OT`,
which allows for faster computations and lower memory complexity
:cite:`scetbon:21a,scetbon:21b,scetbon:22b,scetbon:23`. The :term:`transport matrix`
will be :term:`low-rank`.
low-rank
If the OT problem is solved with a `low-rank <https://en.wikipedia.org/wiki/Low-rank_approximation>`_ solver,
the :term:`transport matrix` is the product of several low-rank matrices (i.e. lower than the number of data
points in the source distribution and the target distribution).