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 te 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=-19.532538593517625
Iter:  20, loss=-31.461255298661243
Iter:  30, loss=-36.61195013306527
Iter:  40, loss=-40.11751977083975
Iter:  50, loss=-42.33792029485554
Iter:  60, loss=-43.365989035538256
Iter:  70, loss=-43.82280289707091
Iter:  80, loss=-43.96657611518327
Iter:  90, loss=-44.034595526516014
Iter: 100, loss=-44.06805297544719
Iter: 110, loss=-44.08481945147683
Iter: 120, loss=-44.093910020276276
Iter: 130, loss=-44.099291840246536
Iter: 140, loss=-44.102393258530434
Iter: 150, loss=-44.1042641116981
Iter: 160, loss=-44.10546565667185
Iter: 170, loss=-44.10627559194185
Iter: 180, loss=-44.10683789968571
Iter: 190, loss=-44.10724776836168

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 te 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=-19.61256245960802
Iter:  20, loss=-31.363540342496798
Iter:  30, loss=-36.49361007046405
Iter:  40, loss=-39.979916279390714
Iter:  50, loss=-42.223477598162795
Iter:  60, loss=-43.201975689624035
Iter:  70, loss=-43.64119422752089
Iter:  80, loss=-43.81542027321558
Iter:  90, loss=-43.89587786724255
Iter: 100, loss=-43.93537693832107
Iter: 110, loss=-43.95704629376859
Iter: 120, loss=-43.96910600629429
Iter: 130, loss=-43.97709913774263
Iter: 140, loss=-43.98177341772868
Iter: 150, loss=-43.984410262603674
Iter: 160, loss=-43.98575137601789
Iter: 170, loss=-43.98647621019038
Iter: 180, loss=-43.98687569272949
Iter: 190, loss=-43.9870979148596

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')