Wasserstein Discriminant Analysis

This example illustrate the use of WDA as proposed in [11].

[11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016). Wasserstein Discriminant Analysis.

# Author: Remi Flamary <remi.flamary@unice.fr>
#
# License: MIT License

# sphinx_gallery_thumbnail_number = 2

import numpy as np
import matplotlib.pylab as pl

from ot.dr import wda, fda

Generate data

n = 1000  # nb samples in source and target datasets
nz = 0.2

np.random.seed(1)

# generate circle dataset
t = np.random.rand(n) * 2 * np.pi
ys = np.floor((np.arange(n) * 1.0 / n * 3)) + 1
xs = np.concatenate(
    (np.cos(t).reshape((-1, 1)), np.sin(t).reshape((-1, 1))), 1)
xs = xs * ys.reshape(-1, 1) + nz * np.random.randn(n, 2)

t = np.random.rand(n) * 2 * np.pi
yt = np.floor((np.arange(n) * 1.0 / n * 3)) + 1
xt = np.concatenate(
    (np.cos(t).reshape((-1, 1)), np.sin(t).reshape((-1, 1))), 1)
xt = xt * yt.reshape(-1, 1) + nz * np.random.randn(n, 2)

nbnoise = 8

xs = np.hstack((xs, np.random.randn(n, nbnoise)))
xt = np.hstack((xt, np.random.randn(n, nbnoise)))

Plot data

pl.figure(1, figsize=(6.4, 3.5))

pl.subplot(1, 2, 1)
pl.scatter(xt[:, 0], xt[:, 1], c=ys, marker='+', label='Source samples')
pl.legend(loc=0)
pl.title('Discriminant dimensions')

pl.subplot(1, 2, 2)
pl.scatter(xt[:, 2], xt[:, 3], c=ys, marker='+', label='Source samples')
pl.legend(loc=0)
pl.title('Other dimensions')
pl.tight_layout()

Compute Fisher Discriminant Analysis

p = 2

Pfda, projfda = fda(xs, ys, p)

Compute Wasserstein Discriminant Analysis

p = 2
reg = 1e0
k = 10
maxiter = 100

P0 = np.random.randn(xs.shape[1], p)

P0 /= np.sqrt(np.sum(P0**2, 0, keepdims=True))

Pwda, projwda = wda(xs, ys, p, reg, k, maxiter=maxiter, P0=P0)

Optimizing...
Iteration    Cost                       Gradient norm
---------    -----------------------    --------------
  1          +8.3042776946697494e-01    5.65147154e-01
  2          +4.4401037686381040e-01    2.16760501e-01
  3          +4.2234351238819928e-01    1.30555049e-01
  4          +4.2169879996364462e-01    1.39115407e-01
  5          +4.1924746118060602e-01    1.25387848e-01
  6          +4.1177409528990749e-01    6.70993539e-02
  7          +4.0862213476139048e-01    3.52716830e-02
  8          +4.0747229322240269e-01    3.34923131e-02
  9          +4.0678766065261684e-01    2.74029183e-02
 10          +4.0621337155459647e-01    2.03651803e-02
 11          +4.0577080390746939e-01    2.59605592e-02
 12          +4.0543140912472148e-01    3.28883715e-02
 13          +4.0470236926310577e-01    1.47528039e-02
 14          +4.0445628467498224e-01    5.03183254e-02
 15          +4.0364189455866245e-01    3.31006504e-02
 16          +4.0303977563823823e-01    1.39885352e-02
 17          +4.0301476238242911e-01    2.17467624e-02
 18          +4.0292344306414324e-01    1.79959907e-02
 19          +4.0271888325518124e-01    6.94408237e-03
 20          +4.0183214741002155e-01    1.98322994e-02
 21          +3.9762636217090053e-01    1.03196875e-01
 22          +3.8225627240876070e-01    1.36012863e-01
 23          +3.0855506616050116e-01    1.92702943e-01
 24          +2.8001027160864295e-01    2.01920255e-01
 25          +2.3687486090807947e-01    9.01780640e-02
 26          +2.3431203993360381e-01    7.23716793e-02
 27          +2.3118645266923005e-01    2.90753137e-02
 28          +2.3067593392325469e-01    1.02767925e-02
 29          +2.3064856262240019e-01    8.07925279e-03
 30          +2.3060699763593800e-01    1.95215754e-03
 31          +2.3060442760754873e-01    2.77368118e-05
 32          +2.3060442709529139e-01    5.34108449e-06
 33          +2.3060442708435561e-01    3.52599061e-06
 34          +2.3060442707674844e-01    1.07742368e-06
 35          +2.3060442707600512e-01    2.36125504e-07
Terminated - min grad norm reached after 35 iterations, 8.68 seconds.

Plot 2D projections

xsp = projfda(xs)
xtp = projfda(xt)

xspw = projwda(xs)
xtpw = projwda(xt)

pl.figure(2)

pl.subplot(2, 2, 1)
pl.scatter(xsp[:, 0], xsp[:, 1], c=ys, marker='+', label='Projected samples')
pl.legend(loc=0)
pl.title('Projected training samples FDA')

pl.subplot(2, 2, 2)
pl.scatter(xtp[:, 0], xtp[:, 1], c=ys, marker='+', label='Projected samples')
pl.legend(loc=0)
pl.title('Projected test samples FDA')

pl.subplot(2, 2, 3)
pl.scatter(xspw[:, 0], xspw[:, 1], c=ys, marker='+', label='Projected samples')
pl.legend(loc=0)
pl.title('Projected training samples WDA')

pl.subplot(2, 2, 4)
pl.scatter(xtpw[:, 0], xtpw[:, 1], c=ys, marker='+', label='Projected samples')
pl.legend(loc=0)
pl.title('Projected test samples WDA')
pl.tight_layout()

pl.show()