Optimal transport for machine learning
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1 Optimal transport for machine learning Practical sessions Rémi Flamary, Nicolas Courty, Marco Cuturi Data SCience Summer School (DS3) 2018, Paris, France 1
2 Course organization A day in Optimal Transport 09:00-10:30 Introduction to OT 11:00-12:30 Practical session 1 14:00-15:30 Advanced OT 1. Intro to OT with POT 2. Regularized OT with Sinkhorn 3. Word Mover s Distance on text 16:00-17:30 Practical session 2 Domain Adaptation on digits Color grading Wasserstein GAN 2 / 7
3 Files and solution Get the files Solution for all practical sessions Github repository: All files: /archive/master.zip Where [NUMBER] is replaced by the integer part of the value of the Wasserstein distance obtained in Practical Session 1 Part 1 using the Manhattan/Cityblock ground metric. 3 / 7
4 Required Python libraries Install POT (Python Optimal Transport) On Anaconda (in terminal) : conda install -c conda-forge pot With pip (requires C compiler) : pip install pot Test install by executing: import ot Install Keras (optional, only for WGAN session) On anaconda (in terminal) : conda install -c conda-forge keras With pip (requires C compiler) : pip install keras 4 / 7
5 Practical Session 1 Part 1: Intro to OT with POT File: 0 Intro OT.ipynb Problem of Cafés and Bakeries (in Manhattan). Visualize the problem (on the map and in matrix form). Solve OT with different ground metrics. Interpret the OT matrix. 5 / 7
![Practical Session 1 Part 2: Implement Sinkhorn [Cuturi, 2013] File: 0](/docs-images/93/112321724/images/6-0.jpg "Intro OT.ipynb Implement the Sinkhorn-Knopp loop. Interpret the OT matrix.")
6 Practical Session 1 Part 2: Implement Sinkhorn [Cuturi, 2013] File: 0 Intro OT.ipynb Implement the Sinkhorn-Knopp loop. Interpret the OT matrix. 5 / 7
7 Practical Session 1 Part 3: Word Mover s Distance [Kusner et al., 2015] File: 4 WMD.ipynb Reproduce figure above. Interpret the OT matrix on words. Perform regression for sentence similarity. 5 / 7
8 Course organization A day in Optimal Transport 09:00-10:30 Introduction to OT 11:00-12:30 Practical session 1 14:00-15:30 Advanced OT 1. Intro to OT with POT 2. Regularized OT with Sinkhorn 3. Word Mover s Distance on text 16:00-17:30 Practical session 2 Domain Adaptation on digits Color grading Wasserstein GAN 6 / 7
9 Practical Session 2 Domain adaptation with OT [Courty et al., 2016] File: 1 DomainAdaptation.ipynb Adapting between MNIST and USPS digits datasets. Solve OT and apply approximate Monge Mapping. Look at displaced samples and train classifier. 7 / 7
10 Practical Session 2 Dataset Optimal transport Classification on transported samples Class 1 Class 2 Samples Samples Classifier on Samples Samples Samples Samples Classifier on Domain adaptation with OT [Courty et al., 2016] File: 1 DomainAdaptation.ipynb Adapting between MNIST and USPS digits datasets. Solve OT and apply approximate Monge Mapping. Look at displaced samples and train classifier. 7 / 7
11 Practical Session 2 Color Grading [Ferradans et al., 2014] File: 2 ColorGrading.ipynb Adapt between paintings from Gustav Klimt and Egon Schiele. Represent image as distribution of pixels in 3D. Compute OT and apply approximate Monge mapping. Reconstruct image. 7 / 7
![Practical Session 2 Color Grading [Ferradans et al., 2014] File: 2 ColorGrading.](/docs-images/93/112321724/images/12-0.jpg "ipynb Adapt between paintings from Gustav Klimt and Egon Schiele. Represent image as distribution of pixels in 3D.")
12 Practical Session 2 Color Grading [Ferradans et al., 2014] File: 2 ColorGrading.ipynb Adapt between paintings from Gustav Klimt and Egon Schiele. Represent image as distribution of pixels in 3D. Compute OT and apply approximate Monge mapping. Reconstruct image. 7 / 7
13 Practical Session 2 Wasserstein GAN [Arjovsky et al., 2017] File: 3 WGAN.ipynb Requires Keras and knowledge of neural networks. Design and learn Wasserstein GAN for 2D samples. Implement both original WGAN and and improved WGAN (gradient penalty [Gulrajani et al., 2017]). Convergence in few minutes on laptops. 7 / 7
14 References i Arjovsky, M., Chintala, S., and Bottou, L. (2017). Wasserstein gan. arxiv preprint arxiv: Courty, N., Flamary, R., Tuia, D., and Rakotomamonjy, A. (2016). Optimal transport for domain adaptation. Pattern Analysis and Machine Intelligence, IEEE Transactions on. Cuturi, M. (2013). Sinkhorn distances: Lightspeed computation of optimal transportation. In Neural Information Processing Systems (NIPS), pages / 7
15 References ii Ferradans, S., Papadakis, N., Peyré, G., and Aujol, J.-F. (2014). Regularized discrete optimal transport. SIAM Journal on Imaging Sciences, 7(3). Gulrajani, I., Ahmed, F., Arjovsky, M., Dumoulin, V., and Courville, A. C. (2017). Improved training of wasserstein gans. In Advances in Neural Information Processing Systems, pages Kusner, M., Sun, Y., Kolkin, N., and Weinberger, K. (2015). From word embeddings to document distances. In International Conference on Machine Learning, pages / 7
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