--- a +++ b/docs/references.rst @@ -0,0 +1,81 @@ +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).