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‎docs/source/index.rst‎

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--------
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.. toctree::
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:maxdepth: 3
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:maxdepth: 2
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self
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quickstart

‎docs/source/quickstart.rst‎

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Quick start
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===========
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Quick start guide
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=================
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In the following we provide some pointers about which functions and classes
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to use for different problems related to optimal transport (OT).
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Optimal transport and Wasserstein distance
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------------------------------------------
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.. note::
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In POT, most functions that solve OT or regularized OT problems have two
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versions that return the OT matrix or the value of the optimal solution. For
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instance :any:`ot.emd` return the OT matrix and :any:`ot.emd2` return the
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Wassertsein distance.
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Solving optimal transport
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^^^^^^^^^^^^^^^^^^^^^^^^^
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# M is the ground cost matrix
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T=ot.emd(a,b,M) # exact linear program
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The method used for solving the OT problem is the network simplex, it is
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implemented in C from [1]_. It has a complexity of :math:`O(n^3)` but the
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solver is quite efficient and uses sparsity of the solution.
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.. hint::
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Examples of use for :any:`ot.emd` are available in the following examples:
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- :any:`auto_examples/plot_compute_emd`
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.. note::
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In POT, most functions that solve OT or regularized OT problems have two
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versions that return the OT matrix or the value of the optimal solution. Fir
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instance :any:`ot.emd` return the OT matrix and :any:`ot.emd2` return the
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Wassertsein distance.
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Regularized Optimal Transport
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-----------------------------
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Entropic regularized OT
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^^^^^^^^^^^^^^^^^^^^^^^
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Other regularization
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^^^^^^^^^^^^^^^^^^^^
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Stochastic gradient decsent
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^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Wasserstein Barycenters
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-----------------------
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How to?
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-------
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FAQ
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---
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2. **Compute a Wasserstein distance**
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References
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----------
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.. [1] Bonneel, N., Van De Panne, M., Paris, S., & Heidrich, W. (2011,
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December). `Displacement nterpolation using Lagrangian mass transport
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<https://people.csail.mit.edu/sparis/publi/2011/sigasia/Bonneel_11_Displacement_Interpolation.pdf>`__.
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In ACM Transactions on Graphics (TOG) (Vol. 30, No. 6, p. 158). ACM.
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.. [2] Cuturi, M. (2013). `Sinkhorn distances: Lightspeed computation of
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optimal transport <https://arxiv.org/pdf/1306.0895.pdf>`__. In Advances
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in Neural Information Processing Systems (pp. 2292-2300).
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.. [3] Benamou, J. D., Carlier, G., Cuturi, M., Nenna, L., & Peyré, G.
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(2015). `Iterative Bregman projections for regularized transportation
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problems <https://arxiv.org/pdf/1412.5154.pdf>`__. SIAM Journal on
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Scientific Computing, 37(2), A1111-A1138.
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.. [4] S. Nakhostin, N. Courty, R. Flamary, D. Tuia, T. Corpetti,
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`Supervised planetary unmixing with optimal
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transport <https://hal.archives-ouvertes.fr/hal-01377236/document>`__,
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Whorkshop on Hyperspectral Image and Signal Processing : Evolution in
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Remote Sensing (WHISPERS), 2016.
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.. [5] N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, `Optimal Transport
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for Domain Adaptation <https://arxiv.org/pdf/1507.00504.pdf>`__, in IEEE
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Transactions on Pattern Analysis and Machine Intelligence , vol.PP,
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no.99, pp.1-1
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.. [6] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014).
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`Regularized discrete optimal
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transport <https://arxiv.org/pdf/1307.5551.pdf>`__. SIAM Journal on
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Imaging Sciences, 7(3), 1853-1882.
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.. [7] Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). `Generalized
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conditional gradient: analysis of convergence and
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applications <https://arxiv.org/pdf/1510.06567.pdf>`__. arXiv preprint
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arXiv:1510.06567.
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.. [8] M. Perrot, N. Courty, R. Flamary, A. Habrard (2016), `Mapping
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estimation for discrete optimal
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transport <http://remi.flamary.com/biblio/perrot2016mapping.pdf>`__,
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Neural Information Processing Systems (NIPS).
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.. [9] Schmitzer, B. (2016). `Stabilized Sparse Scaling Algorithms for
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Entropy Regularized Transport
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Problems <https://arxiv.org/pdf/1610.06519.pdf>`__. arXiv preprint
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arXiv:1610.06519.
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.. [10] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2016).
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`Scaling algorithms for unbalanced transport
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problems <https://arxiv.org/pdf/1607.05816.pdf>`__. arXiv preprint
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arXiv:1607.05816.
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.. [11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016).
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`Wasserstein Discriminant
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Analysis <https://arxiv.org/pdf/1608.08063.pdf>`__. arXiv preprint
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arXiv:1608.08063.
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.. [12] Gabriel Peyré, Marco Cuturi, and Justin Solomon (2016),
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`Gromov-Wasserstein averaging of kernel and distance
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matrices <http://proceedings.mlr.press/v48/peyre16.html>`__
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International Conference on Machine Learning (ICML).
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.. [13] Mémoli, Facundo (2011). `Gromov–Wasserstein distances and the
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metric approach to object
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matching <https://media.adelaide.edu.au/acvt/Publications/2011/2011-Gromov%E2%80%93Wasserstein%20Distances%20and%20the%20Metric%20Approach%20to%20Object%20Matching.pdf>`__.
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Foundations of computational mathematics 11.4 : 417-487.
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.. [14] Knott, M. and Smith, C. S. (1984).`On the optimal mapping of
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distributions <https://link.springer.com/article/10.1007/BF00934745>`__,
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Journal of Optimization Theory and Applications Vol 43.
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.. [15] Peyré, G., & Cuturi, M. (2018). `Computational Optimal
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Transport <https://arxiv.org/pdf/1803.00567.pdf>`__ .
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.. [16] Agueh, M., & Carlier, G. (2011). `Barycenters in the Wasserstein
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space <https://hal.archives-ouvertes.fr/hal-00637399/document>`__. SIAM
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Journal on Mathematical Analysis, 43(2), 904-924.
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.. [17] Blondel, M., Seguy, V., & Rolet, A. (2018). `Smooth and Sparse
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Optimal Transport <https://arxiv.org/abs/1710.06276>`__. Proceedings of
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the Twenty-First International Conference on Artificial Intelligence and
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Statistics (AISTATS).
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.. [18] Genevay, A., Cuturi, M., Peyré, G. & Bach, F. (2016) `Stochastic
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Optimization for Large-scale Optimal
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Transport <https://arxiv.org/abs/1605.08527>`__. Advances in Neural
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Information Processing Systems (2016).
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.. [19] Seguy, V., Bhushan Damodaran, B., Flamary, R., Courty, N., Rolet,
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A.& Blondel, M. `Large-scale Optimal Transport and Mapping
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Estimation <https://arxiv.org/pdf/1711.02283.pdf>`__. International
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Conference on Learning Representation (2018)
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.. [20] Cuturi, M. and Doucet, A. (2014) `Fast Computation of Wasserstein
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Barycenters <http://proceedings.mlr.press/v32/cuturi14.html>`__.
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International Conference in Machine Learning
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.. [21] Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A.,
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Nguyen, A. & Guibas, L. (2015). `Convolutional wasserstein distances:
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Efficient optimal transportation on geometric
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domains <https://dl.acm.org/citation.cfm?id=2766963>`__. ACM
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Transactions on Graphics (TOG), 34(4), 66.
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.. [22] J. Altschuler, J.Weed, P. Rigollet, (2017) `Near-linear time
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approximation algorithms for optimal transport via Sinkhorn
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iteration <https://papers.nips.cc/paper/6792-near-linear-time-approximation-algorithms-for-optimal-transport-via-sinkhorn-iteration.pdf>`__,
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Advances in Neural Information Processing Systems (NIPS) 31
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.. [23] Aude, G., Peyré, G., Cuturi, M., `Learning Generative Models with
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Sinkhorn Divergences <https://arxiv.org/abs/1706.00292>`__, Proceedings
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of the Twenty-First International Conference on Artficial Intelligence
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and Statistics, (AISTATS) 21, 2018
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.. [24] Vayer, T., Chapel, L., Flamary, R., Tavenard, R. and Courty, N.
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(2019). `Optimal Transport for structured data with application on
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graphs <http://proceedings.mlr.press/v97/titouan19a.html>`__ Proceedings
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of the 36th International Conference on Machine Learning (ICML).

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