Dual OT solvers for entropic and quadratic regularized OT with Pytorch

# Author: Remi Flamary <remi.flamary@polytechnique.edu>
#
# License: MIT License

# sphinx_gallery_thumbnail_number = 3

import numpy as np
import matplotlib.pyplot as pl
import torch
import ot
import ot.plot

Data generation

torch.manual_seed(1)

n_source_samples = 100
n_target_samples = 100
theta = 2 * np.pi / 20
noise_level = 0.1

Xs, ys = ot.datasets.make_data_classif("gaussrot", n_source_samples, nz=noise_level)
Xt, yt = ot.datasets.make_data_classif(
    "gaussrot", n_target_samples, theta=theta, nz=noise_level
)

# one of the target mode changes its variance (no linear mapping)
Xt[yt == 2] *= 3
Xt = Xt + 4

Plot data

pl.figure(1, (10, 5))
pl.clf()
pl.scatter(Xs[:, 0], Xs[:, 1], marker="+", label="Source samples")
pl.scatter(Xt[:, 0], Xt[:, 1], marker="o", label="Target samples")
pl.legend(loc=0)
pl.title("Source and target distributions")

Text(0.5, 1.0, 'Source and target distributions')

Convert data to torch tensors

xs = torch.tensor(Xs)
xt = torch.tensor(Xt)

Estimating dual variables for entropic OT

u = torch.randn(n_source_samples, requires_grad=True)
v = torch.randn(n_source_samples, requires_grad=True)

reg = 0.5

optimizer = torch.optim.Adam([u, v], lr=1)

# number of iteration
n_iter = 200


losses = []

for i in range(n_iter):
    # generate noise samples

    # minus because we maximize the dual loss
    loss = -ot.stochastic.loss_dual_entropic(u, v, xs, xt, reg=reg)
    losses.append(float(loss.detach()))

    if i % 10 == 0:
        print("Iter: {:3d}, loss={}".format(i, losses[-1]))

    loss.backward()
    optimizer.step()
    optimizer.zero_grad()


pl.figure(2)
pl.plot(losses)
pl.grid()
pl.title("Dual objective (negative)")
pl.xlabel("Iterations")

Ge = ot.stochastic.plan_dual_entropic(u, v, xs, xt, reg=reg)

Iter:   0, loss=0.202049490022473
Iter:  10, loss=-19.327822113197268
Iter:  20, loss=-31.865002074175298
Iter:  30, loss=-36.986166992556264
Iter:  40, loss=-39.73519694176019
Iter:  50, loss=-41.56008336494196
Iter:  60, loss=-42.21166745709296
Iter:  70, loss=-42.53350497382068
Iter:  80, loss=-42.63558227022633
Iter:  90, loss=-42.67411132826723
Iter: 100, loss=-42.693469703267404
Iter: 110, loss=-42.70486184956903
Iter: 120, loss=-42.711021495144855
Iter: 130, loss=-42.714899950378275
Iter: 140, loss=-42.71745076358151
Iter: 150, loss=-42.71925119281369
Iter: 160, loss=-42.72054452517881
Iter: 170, loss=-42.72148135988978
Iter: 180, loss=-42.72217217424471
Iter: 190, loss=-42.72268860726768

Plot the estimated entropic OT plan

pl.figure(3, (10, 5))
pl.clf()
ot.plot.plot2D_samples_mat(Xs, Xt, Ge.detach().numpy(), alpha=0.1)
pl.scatter(Xs[:, 0], Xs[:, 1], marker="+", label="Source samples", zorder=2)
pl.scatter(Xt[:, 0], Xt[:, 1], marker="o", label="Target samples", zorder=2)
pl.legend(loc=0)
pl.title("Source and target distributions")

Text(0.5, 1.0, 'Source and target distributions')

Estimating dual variables for quadratic OT

u = torch.randn(n_source_samples, requires_grad=True)
v = torch.randn(n_source_samples, requires_grad=True)

reg = 0.01

optimizer = torch.optim.Adam([u, v], lr=1)

# number of iteration
n_iter = 200


losses = []


for i in range(n_iter):
    # generate noise samples

    # minus because we maximize the dual loss
    loss = -ot.stochastic.loss_dual_quadratic(u, v, xs, xt, reg=reg)
    losses.append(float(loss.detach()))

    if i % 10 == 0:
        print("Iter: {:3d}, loss={}".format(i, losses[-1]))

    loss.backward()
    optimizer.step()
    optimizer.zero_grad()


pl.figure(4)
pl.plot(losses)
pl.grid()
pl.title("Dual objective (negative)")
pl.xlabel("Iterations")

Gq = ot.stochastic.plan_dual_quadratic(u, v, xs, xt, reg=reg)

Iter:   0, loss=-0.0018442196020623663
Iter:  10, loss=-19.532271752852665
Iter:  20, loss=-31.583880242291595
Iter:  30, loss=-36.765155220728005
Iter:  40, loss=-39.339254737797425
Iter:  50, loss=-41.35171572521136
Iter:  60, loss=-41.965794464951514
Iter:  70, loss=-42.36475439472597
Iter:  80, loss=-42.49856891995049
Iter:  90, loss=-42.53732345351729
Iter: 100, loss=-42.56919906223859
Iter: 110, loss=-42.58344215103438
Iter: 120, loss=-42.59102466559467
Iter: 130, loss=-42.595819066291625
Iter: 140, loss=-42.59912375010589
Iter: 150, loss=-42.60147400812687
Iter: 160, loss=-42.60299922670597
Iter: 170, loss=-42.6039595571292
Iter: 180, loss=-42.604538990522954
Iter: 190, loss=-42.60489520794249

Plot the estimated quadratic OT plan

pl.figure(5, (10, 5))
pl.clf()
ot.plot.plot2D_samples_mat(Xs, Xt, Gq.detach().numpy(), alpha=0.1)
pl.scatter(Xs[:, 0], Xs[:, 1], marker="+", label="Source samples", zorder=2)
pl.scatter(Xt[:, 0], Xt[:, 1], marker="o", label="Target samples", zorder=2)
pl.legend(loc=0)
pl.title("OT plan with quadratic regularization")

Text(0.5, 1.0, 'OT plan with quadratic regularization')