Python Optimal Transport (POT)

POT provides a wide range of Optimal Transport (OT) solvers and machine learning applications:

Generic OT Solvers

• Linear/Exact OT: Network Simplex solver for Earth Mover's Distance.
• Regularized OT: Conditional gradient, Generalized conditional gradient, and various Sinkhorn Knopp algorithms (including stabilized, greedy, and screening versions).
• Barycenters: Wasserstein, Gromov-Wasserstein (GW), convolutional, and debiased Sinkhorn barycenters.
• Gromov-Wasserstein: Exact and regularized GW distances, Fused-GW (FGW) distances, and GW barycenters.
• Unbalanced & Partial OT: Unbalanced OT (UOT) with KL relaxation, partial Wasserstein, and partial Gromov-Wasserstein solvers.
• Specialized OT: Sliced Wasserstein, Wasserstein distance on the circle, Spherical Sliced Wasserstein, and Gaussian Mixture Model (GMM) OT.
• Large-scale OT: Stochastic solvers and differentiable losses for large-scale problems.

Machine Learning Solvers

• Domain Adaptation: OT for domain adaptation with group lasso, Laplacian, or semi-supervised regularization, and the JCPOT algorithm.
• Mapping Estimation: Linear OT mapping and Joint OT mapping estimation.
• Graph Neural Networks: TFGW and TW (OT-GNN) layers.
• Other: Wasserstein Discriminant Analysis (requires autograd and pymanopt).

POT (Python Optimal Transport) is designed for solving Optimal Transport (OT) problems in the context of machine learning. You should use POT when you need to:

1. Measure similarity between distributions: Use the Wasserstein distance (the optimal value of the OT problem) to compare datasets, word embeddings, signals, or spectra. It is particularly useful when distributions have disjoint supports.
2. Find correspondences (Mapping Estimation): Use the OT matrix (the optimal mapping) to find non-supervised correspondences between samples in discrete distributions. This is useful for tasks like color transfer between images or domain adaptation.
3. Perform research with custom regularization: Use generic solvers like ot.optim.cg to solve OT problems with any smooth/continuous regularization term (e.g., graph Laplacian regularization or Gromov-Wasserstein).

Note on Scale: POT is optimized for efficiency but is not designed for extremely large-scale problems due to the $\mathcal{O}(n^2)$ memory requirement for the cost matrix. For very large sample sizes, consider using 'lazy' Sinkhorn implementations or neural network-based dual approximations.

POT provides a wide range of examples covering various Optimal Transport (OT) tasks. You can find specialized implementation examples in the following categories:

• OT and regularized OT: Standard OT problems and entropic regularization.
• Differentiable OT with PyTorch: Using OT within deep learning workflows and backends.
• Gromov-Wasserstein (GW) and Fused GW: Comparing distributions in different metric spaces.
• Unbalanced and Partial OT: Handling distributions with different total masses.
• OT in 1D and Sliced Wasserstein: Efficient OT computations for high-dimensional data using projections.
• OT on Gaussian and Gaussian Mixture Models: Analytical or specialized OT for parametric models.
• Factored and Low-Rank OT: Scalable OT for large datasets.
• Wasserstein and (F)GW barycenters: Computing the mean/centroid of multiple distributions.
• Domain adaptation with OT: Using OT for aligning different data distributions.

POT provides several Domain Adaptation (OTDA) classes that inherit from ot.da.BaseTransport. These classes follow a scikit-learn-like interface:

1. Initialize: Set parameters like regularization.
2. Fit: Call .fit(Xs=Xs, Xt=Xt) to estimate the mapping between source samples Xs and target samples Xt.
3. Transform: Use .transform(Xs=Xs) to map source samples to the target distribution, or .inverse_transform(Xt=Xt) to map target samples back to the source.

Available implementations:

• ot.da.EMDTransport: Barycentric mapping with EMD transport.
• ot.da.SinkhornTransport: Barycentric mapping with Sinkhorn transport.
• ot.da.SinkhornL1l2Transport: Barycentric mapping with Sinkhorn + group Lasso regularization.
• ot.da.SinkhornLpl1Transport: Barycentric mapping with Sinkhorn + non-convex group Lasso regularization.
• ot.da.LinearTransport: Linear mapping estimation between Gaussians.
• ot.da.MappingTransport: Nonlinear mapping estimation.

ot_emd = ot.da.EMDTransport()
ot_emd.fit(Xs=Xs, Xt=Xt)
Xs_mapped = ot_emd.transform(Xs=Xs)