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diff --git a/docs/source/quickstart.rst b/docs/source/quickstart.rst index 978eaff..d56f812 100644 --- a/docs/source/quickstart.rst +++ b/docs/source/quickstart.rst @@ -645,6 +645,53 @@ implemented the main function :any:`ot.barycenter_unbalanced`. - :any:`auto_examples/plot_UOT_barycenter_1D` +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]_: + +.. math:: + \gamma = \arg\min_\gamma <\gamma,M>_F + + s.t. + \gamma\geq 0 \\ + \gamma 1 \leq a\\ + \gamma^T 1 \leq b\\ + 1^T \gamma^T 1 = m \leq \min\{\|a\|_1, \|b\|_1\} + + +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 :any:`ot.partial`. The exact resolution of the problem is computed in :any:`ot.partial.partial_wasserstein` +and :any:`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 :any:`ot.partial.entropic_partial_wasserstein` +(see [3]_). + +The partial Gromov-Wasserstein formulation of the problem + +.. math:: + GW = \min_\gamma \sum_{i,j,k,l} L(C1_{i,k},C2_{j,l})*\gamma_{i,j}*\gamma_{k,l} + + s.t. + \gamma\geq 0 \\ + \gamma 1 \leq a\\ + \gamma^T 1 \leq b\\ + 1^T \gamma^T 1 = m \leq \min\{\|a\|_1, \|b\|_1\} + +is computed in :any:`ot.partial.partial_gromov_wasserstein` and in +:any:`ot.partial.entropic_partial_gromov_wasserstein` when considering the entropic +regularization of the problem. + + +.. hint:: + + Examples of the use of :any:`ot.partial` are available in : + + - :any:`auto_examples/plot_partial` + + + Gromov-Wasserstein ^^^^^^^^^^^^^^^^^^ @@ -921,3 +968,20 @@ References .. [25] 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). + +.. [27] Redko I., Courty N., Flamary R., Tuia D. (2019). Optimal Transport for Multi-source + Domain Adaptation under Target Shift <http://proceedings.mlr.press/v89/redko19a.html>, + Proceedings of the Twenty-Second International Conference on Artificial Intelligence + and Statistics (AISTATS) 22, 2019. + +.. [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. |