ott.geometry package#

OTT ground geometries: Classes and cost functions to instantiate them.

This package implements several classes to define a geometry, arguably the most influential ingredient of optimal transport problem. In its full generality, a Geometry defines source points (input measure), target points (target measure) and a ground cost function (resp. a positive kernel function) that quantifies how expensive (resp. easy) it is to displace a unit of mass from any of the input points to the target points.

The geometry package proposes a few simple geometries. The simplest of all would be that for which input and target points coincide, and the geometry between them simplifies to a symmetric cost or kernel matrix. In the very particular case where these points happen to lie on grid (a cartesian product in full generality, e.g. 2 or 3D grids), the Grid geometry will prove useful.

For more general settings where input/target points do not coincide, one can alternatively instantiate a Geometry through a rectangular cost matrix.

However, it is often preferable in applications to define ground costs “symbolically”, by listing instead points in the input/target point clouds, to specify directly a cost function between them. Such functions should follow the CostFn class description. We provide a few standard cost functions that are meaningful in an OT context, notably the (unbalanced, regularized) Bures distances between Gaussians 1. That cost can be used for instance to compute a distance between Gaussian mixtures, as proposed in 2 and revisited in 3.

To be useful with Sinkhorn solvers, Geometries typically need to provide an epsilon regularization parameter. We propose either to set that value once for all, or implement an annealing scheduler.

Geometries#

geometry.Geometry([cost_matrix, ...])

Base class to define ground costs/kernels used in optimal transport.

pointcloud.PointCloud(x[, y, cost_fn, ...])

Defines geometry for 2 pointclouds (possibly 1 vs itself) using CostFn.

grid.Grid([x, grid_size, cost_fns, num_a, ...])

Class describing the geometry of points taken in a cartestian product.

low_rank.LRCGeometry(cost_1, cost_2[, bias, ...])

Low-rank Cost Geometry defined by two factors.

epsilon_scheduler.Epsilon([target, ...])

Scheduler class for the regularization parameter epsilon.

Cost Functions#

costs.CostFn()

A generic cost function, taking two vectors as input.

costs.Euclidean()

Squared Euclidean distance CostFn.

costs.Cosine([ridge])

Cosine distance CostFn.

costs.Bures(dimension, **kwargs)

Bures distance between a pair of (mean, cov matrix) raveled as vectors.

costs.UnbalancedBures(dimension[, gamma, sigma])

Regularized, unbalanced Bures distance between triplets.

References#

1
  1. Janati et al., Entropic Optimal Transport between Unbalanced Gaussian Measures has a Closed Form , NeurIPS 2020.

2
  1. Chen et al., Optimal Transport for Gaussian Mixture Models , IEEE Access (7)

3
  1. Delon and A. Desolneux, A Wasserstein-Type Distance in the Space of Gaussian Mixture Models , SIIMS (13)-2, 936–970