ott-jax 0.2.6

OTT: Optimal Transport Tools in Jax.

Optimal Transport Tools (OTT).

See full documentation.

What is OTT-JAX?

Optimal transport theory can be loosely described as the branch of mathematics and optimization that studies matching problems: given two sets of points, how to find (given some prior information, typically a cost function) a good way to associate bijectively every point in the first set with another in the second. A typical matching problem arises, for instance, when sorting numbers (when sorting, one associates to numbers [3.1, -4.2, -18, 5.4] the ranks [3, 2, 1, 4] that reorder them in increasing fashion) or when matching submitted papers with reviewers at ML conferences!

These problems are easy to describe yet quite hard to solve. For instance, matching optimally two sets of n points using a pairwise cost can be solved with the Hungarian algorithm, but this requires an effort that scales as $n^3$. Sometimes, the two sets can have different sizes; sometimes a relevant matching cost function is not given (it remains an open problem to set a score to qualify how well a reviewer is likely to be to check a paper!). Fortunately, optimal transport theory has made decisive progresses since the Hungarian algorithm was proposed, and can count on many efficient algorithms and extensions that can handle such situations.

OTT-JAX is a toolbox providing sturdy, scalable and efficient solvers for those problems. OTT builds upon the JAX framework. Some of JAX features include JIT, auto-vectorization and implicit differentiation.

The typical ingredient in OT problems consists of two discrete measures (an efficient way to encode weighted sets of points), and a cost function comparing points. By default, OTT assumes that the measures are supported on vectors, and that these vectors are compared with a squared Euclidean distance, as given below:

Example

import jax
import jax.numpy as jnp
from ott.tools import transport
# Samples two point clouds and their weights.
rngs = jax.random.split(jax.random.PRNGKey(0),4)
n, m, d = 12, 14, 2
x = jax.random.normal(rngs[0], (n,d)) + 1
y = jax.random.uniform(rngs[1], (m,d))
a = jax.random.uniform(rngs[2], (n,))
b = jax.random.uniform(rngs[3], (m,))
a, b = a / jnp.sum(a), b / jnp.sum(b)
# Computes the couplings via Sinkhorn algorithm.
ot = transport.solve(x, y, a=a, b=b)
P = ot.matrix

The call to sinkhorn above works out the optimal transport solution by storing its output. That transoprt matrix links each point from the first point cloud to one or more points from the second, as illustrated below.

Overall description of source code

Currently implements the following classes and functions:

• The geometry folder describes tools that to encapsulate the essential ingredients of OT problems: measures and cost functions.

  • The CostFn class in costs.py and its descendants define cost functions between points. A few simple costs are considered, Euclidean between vectors, and Bures, between a pair of mean vector and covariance (p.d.) matrix.

  • The Geometry class in geometry.py describes a cost structure between two measures. That cost structure is accessed through various member functions, either used when running the Sinkhorn algorithm (typically kernel multiplications, or log-sum-exp row/column-wise application) or after (to apply the OT matrix to a vector).

    • In its generic Geometry implementation, as in geometry.py, an object can be initialized with either a cost_matrix along with an epsilon regularization parameter (or scheduler), or with a kernel_matrix.

    • If one wishes to compute OT between two weighted point clouds $x=\left(x_1, \ldots, x_n\right)$ and $y=\left(y_1, \ldots, y_m\right)$ endowed with a given cost function (e.g. Euclidean) $c$, the PointCloud class in pointcloud.py can be used to define the corresponding kernel $K_{i j}=\exp\left(-c\left(x_{i}, y_{j}\right) / \epsilon\right)$. When the number of these points grows very large, this geometry can be instantiated with an online=True parameter, to avoid storing the kernel matrix and choose instead to recompute the matrix on the fly at each application.

    • Simlarly, if all measures to be considered are supported on a separable grid (e.g. $\{1, \ldots, n\}^{d}$), and the cost is separable along all axis, i.e. the cost between two points on that grid is equal to the sum of (possibly $d$ different) cost functions evaluated on each of the $d$ pairs of coordinates, then the application of the kernel is much simplified, both in log space or on the histograms themselves. This particular case is exploited in the Grid geometry in grid.py which can be instantiated as a hypercube using a grid_size parameter, or directly through grid locations in x.

    • LRCGeometry, low-rank cost geometries, of which a PointCloud endowed with a squared-Euclidean distance is a particular example, can efficiently carry apply their cost to another matrix. This is leveraged in particular in the low-rank Sinkhorn (and Gromov-Wasserstein) solvers.

• In the core folder,

  • The sinkhorn function in sinkhorn.py is a wrapper around the Sinkhorn solver class, running the Sinkhorn algorithm, with the aim of solving approximately one or various optimal transport problems in parallel. An OT problem is defined by a Geometry object, and a pair $\left(a, b\right)$ (or batch thereof) of histograms. The function's outputs are stored in a SinkhornOutput named t-uple, containing potentials, regularized OT cost, sequence of errors and a convergence flag. Such outputs (with the exception of errors and convergence flag) can be differentiated w.r.t. any of the three inputs (Geometry, a, b) either through backprop or implicit differentiation of the optimality conditions of the optimal potentials f and g.

  • A later addition in sinkhorn_lr.py is focused on the LRSinkhorn solver class, which is able to solve OT problems at larger scales using an explicit factorization of couplings as being low-rank.

  • In discrete_barycenter.py: implementation of discrete Wasserstein barycenters : given $N$ histograms all supported on the same Geometry, compute a barycenter of theses measures, using an algorithm by Janati et al. (2020).

  • In continuous_barycenter.py: implementation of continuous Wasserstein barycenters : given probability measures described as points which can be compared with an arbitrary cost function, compute a barycenter of theses measures, supported at most $k$ points on using an algorithm by Cuturi and Doucet (2014).

  • In gromov_wasserstein.py: implementation of two Gromov-Wasserstein solvers (both entropy-regularized and low-rank) to compare two measured-metric spaces, here encoded as a pair of Geometry objects, geom_xx, geom_xy along with weights a and b. Additional options include using a fused term by specifying geom_xy.

• In the tools folder,

  • In soft_sort.py: implementation of soft-sorting operators, notably soft-quantile transforms

  • The sinkhorn_divergence function in sinkhorn_divergence.py, implements the unbalanced formulation of the Sinkhorn divergence, a variant of the Wasserstein distance that uses regularization and is computed by centering the output of sinkhorn when comparing two measures.

  • The Transport class in sinkhorn_divergence.py, provides a simple wrapper to the sinkhorn function defined above when the user is primarily interested in computing and storing an OT matrix.

  • The gaussian_mixture folder provides novel tools to compare and estimate GMMs with an OT perspective.

Citation

If you have found this work useful, please consider citing us:

@article{cuturi2022optimal,
  title={Optimal Transport Tools (OTT): A JAX Toolbox for all things Wasserstein},
  author={Cuturi, Marco and Meng-Papaxanthos, Laetitia and Tian, Yingtao and Bunne, Charlotte and Davis, Geoff and Teboul, Olivier},
  journal={arXiv preprint arXiv:2201.12324},
  year={2022}
}