.. -- mode: rst --

smooth-OT

Python implementation of smooth optimal transport.

What is it?

Entropic regularization of optimal transport, as popularized by [1], is quickly emerging as a new standard. However, other strongly convex regularizations are possible. In particular, squared L2 regularization is interesting, since it typically results in sparse transportations plans. As shown in [2], OT with general strongly convex regularization can be solved using alternate Bregman projection algorithms. This package follows instead the dual and semi-dual approaches introduced in [3], which converge much faster in practice, especially when the regularization is low.

Supported features

• L-BFGS solver for the smoothed dual
• L-BFGS solver for the smoothed semi-dual
• Regularizations: negative entropy, squared L2 norm

Example

.. code-block:: python

import numpy as np

from smoothot.dual_solvers import solve_dual, solve_semi_dual
from smoothot.dual_solvers import NegEntropy, SquaredL2
from smoothot.dual_solvers import get_plan_from_dual, get_plan_from_semi_dual

# Generate artificial data.
rng = np.random.RandomState(0)
a = rng.rand(5)
b = rng.rand(8)
a /= np.sum(a)
b /= np.sum(b)
C = rng.rand(5, 8)

# Set regularization term.
# Can also use NegEntropy(gamma=1.0).
regul = SquaredL2(gamma=1.0)

# Solve smoothed dual.
alpha, beta = solve_dual(a, b, C, regul, max_iter=1000)
T = get_plan_from_dual(alpha, beta, C, regul)
print("Dual")
print("Value:", np.sum(T * C) + regul.Omega(T))
print("Sparsity:", np.sum(T != 0) / T.size)
print()

# Solve smoothed semi-dual.
alpha = solve_semi_dual(a, b, C, regul, max_iter=1000)
T = get_plan_from_semi_dual(alpha, b, C, regul)
print("Semi-dual")
print("Value:", np.sum(T * C) + regul.Omega(T))
print("Sparsity:", np.sum(T != 0) / T.size)
print()

Installation

This project can be installed from its git repository.

1. Obtain the sources by::

   git clone https://github.com/mblondel/smooth-ot.git

or, if git is unavailable, download as a ZIP from GitHub <https://github.com/mblondel/smooth-ot/archive/master.zip>_.

2. Install the dependencies::

   via pip

   pip install numpy scipy scikit-learn nose

   via conda

   conda install numpy scipy scikit-learn nose

3. Install smooth-ot::

   cd smooth-ot sudo python setup.py install

References

.. [1] Marco Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In: Proc. of NIPS 2013.

.. [2] Arnaud Dessein, Nicolas Papadakis, Jean-Luc Rouas. Regularized optimal transport and the rot mover’s distance. arXiv preprint arXiv:1610.06447, 2016.

.. [3] Mathieu Blondel, Vivien Seguy, Antoine Rolet. Smooth and Sparse Optimal Transport. In: Proc. of AISTATS 2018. [PDF <https://arxiv.org/abs/1710.06276>_]

Author

• Mathieu Blondel, 2018