Low-Rank GW#

We use the low-rank (LR) Gromov-Wasserstein solver, proposed by [Scetbon et al., 2022], as a follow up to the LR Sinkhorn solver in [Scetbon et al., 2021], see Low-rank Sinkhorn for more information.

import sys

if "google.colab" in sys.modules:
    !pip install -q git+https://github.com/ott-jax/ott@main
import jax
import jax.numpy as jnp

import matplotlib.pyplot as plt

from ott.geometry import pointcloud
from ott.problems.quadratic import quadratic_problem
from ott.solvers.quadratic import gromov_wasserstein
def create_points(
    rng: jax.random.PRNGKeyArray, n: int, m: int, d1: int, d2: int
    rngs = jax.random.split(rng, 5)
    x = jax.random.uniform(rngs[0], (n, d1))
    y = jax.random.uniform(rngs[1], (m, d2))
    a = jax.random.uniform(rngs[2], (n,))
    b = jax.random.uniform(rngs[3], (m,))
    a = a / jnp.sum(a)
    b = b / jnp.sum(b)
    z = jax.random.uniform(rngs[4], (m, d1))
    return x, y, a, b, z

rng = jax.random.PRNGKey(0)
n, m, d1, d2 = 24, 17, 2, 3
x, y, a, b, z = create_points(rng, n, m, d1, d2)

Create two point clouds of heterogeneous size, and add a third geometry to formulate a fused problem [Vayer et al., 2020].

geom_xx = pointcloud.PointCloud(x)
geom_yy = pointcloud.PointCloud(y)
geom_xy = pointcloud.PointCloud(x, z)
prob = quadratic_problem.QuadraticProblem(

Solve the problem using the LRSinkhorn solver class.

solver = gromov_wasserstein.GromovWasserstein(rank=6)
ot_gwlr = solver(prob)

Run it with the widespread entropic GromovWasserstein solver for the sake of comparison.

solver = gromov_wasserstein.GromovWasserstein(epsilon=0.05)
ot_gw = solver(prob)

One can notice that their outputs are quantitatively similar.

def plot_ot(ot, leg):
    plt.imshow(ot.matrix, cmap="Purples")
    plt.title(leg + " cost: " + str(ot.costs[ot.costs > 0][-1]))

plot_ot(ot_gwlr, "Low rank")
plot_ot(ot_gw, "Entropic")
../../_images/631cbd22dbbabdd161bf20933bde93b78f25ca0a76b9038e5d3568b34e89e636.png ../../_images/ba6554ed1f591337eb9b828f86f4a3dce5f775e1aa83992461077048f6a855cd.png