References

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Bibliography

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.. bibliography::

    :cited:

Glossary

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.. 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).