This toolbox is a basic implementation of Sinkhorn-like algorithms to solve for OT-related problems.

An overview of the toolbox is provided as a Jupyter notebook.

The computational methods and relevant bibliography can be found in:

L. Chizat, G. Peyré, B. Schmitzer, F-X. Vialard. Scaling Algorithms for Unbalanced Transport Problems. Preprint Arxiv:1607.05816, 2016.

Comparison of balanced vs. unbalanced interpolation computed using barycenters with varying weights:

The unbalanced interpolation is able to correctly interpolate between the modes of distribution by introducing mass creation/destruction during the interpolation.

Main features

This toolbox implements log-domain computations, so it is always stable, even for small values of epsilon. It implements both balanced (exact marginal constraints) and unbalanced transport (relaxed marginal constraints). It uses a heavy-ball-like extrapolation to speed-up convergence.

Computational functions

• sinkhorn_log.m: computation of OT couplings between two distributions.
• barycenter_log.m: computation of OT barycenters between two or more distributions.

Parameters

The main parameters are:

• epsilon is the entropic regularization strength. Increasing speeds-up the convergence but leads to blurrier results.
• options.rho controls balanced (tau=Inf) vs unbalanced (0<tau<Inf) trade-off.
• options.tau introduces extrapolation acceleration (tau=0 means no extrapolation, -1<tau<0 means extrapolation).

Scripts

• test_sinkhorn.m
• test_barycenters.m: test the computation of barycenters between two distributions, and compares in particular the balanced to the unbalanced cases.

Copyright

Copyright (c) 2017 Gabriel Peyré