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# -*- coding: utf-8 -*-
"""
==================================
Regularized OT with generic solver
==================================
Illustrates the use of the generic solver for regularized OT with
user-designed regularization term. It uses Conditional gradient as in [6] and
generalized Conditional Gradient as proposed in [5,7].
[5] N. Courty; R. Flamary; D. Tuia; A. Rakotomamonjy, Optimal Transport for
Domain Adaptation, in IEEE Transactions on Pattern Analysis and Machine
Intelligence , vol.PP, no.99, pp.1-1.
[6] Ferradans, S., Papadakis, N., Peyré, G., & Aujol, J. F. (2014).
Regularized discrete optimal transport. SIAM Journal on Imaging
Sciences, 7(3), 1853-1882.
[7] Rakotomamonjy, A., Flamary, R., & Courty, N. (2015). Generalized
conditional gradient: analysis of convergence and applications.
arXiv preprint arXiv:1510.06567.
"""
# sphinx_gallery_thumbnail_number = 5
import numpy as np
import matplotlib.pylab as pl
import ot
import ot.plot
##############################################################################
# Generate data
# -------------
# %% parameters
n = 100 # nb bins
# bin positions
x = np.arange(n, dtype=np.float64)
# Gaussian distributions
a = ot.datasets.make_1D_gauss(n, m=20, s=5) # m= mean, s= std
b = ot.datasets.make_1D_gauss(n, m=60, s=10)
# loss matrix
M = ot.dist(x.reshape((n, 1)), x.reshape((n, 1)))
M /= M.max()
##############################################################################
# Solve EMD
# ---------
# %% EMD
G0 = ot.emd(a, b, M)
pl.figure(1, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, G0, "OT matrix G0")
##############################################################################
# Solve EMD with Frobenius norm regularization
# --------------------------------------------
# %% Example with Frobenius norm regularization
def f(G):
return 0.5 * np.sum(G**2)
def df(G):
return G
reg = 1e-1
Gl2 = ot.optim.cg(a, b, M, reg, f, df, verbose=True)
pl.figure(2)
ot.plot.plot1D_mat(a, b, Gl2, "OT matrix Frob. reg")
##############################################################################
# Solve EMD with entropic regularization
# --------------------------------------
# %% Example with entropic regularization
def f(G):
return np.sum(G * np.log(G))
def df(G):
return np.log(G) + 1.0
reg = 1e-3
Ge = ot.optim.cg(a, b, M, reg, f, df, verbose=True)
pl.figure(3, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Ge, "OT matrix Entrop. reg")
##############################################################################
# Solve EMD with Frobenius norm + entropic regularization
# -------------------------------------------------------
# %% Example with Frobenius norm + entropic regularization with gcg
def f(G):
return 0.5 * np.sum(G**2)
def df(G):
return G
reg1 = 1e-3
reg2 = 1e-1
Gel2 = ot.optim.gcg(a, b, M, reg1, reg2, f, df, verbose=True)
pl.figure(4, figsize=(5, 5))
ot.plot.plot1D_mat(a, b, Gel2, "OT entropic + matrix Frob. reg")
pl.show()
# %%
# Comparison of the OT matrices
nvisu = 40
pl.figure(5, figsize=(10, 4))
pl.subplot(2, 2, 1)
pl.imshow(G0[:nvisu, :])
pl.axis("off")
pl.title("Exact OT")
pl.subplot(2, 2, 2)
pl.imshow(Gl2[:nvisu, :])
pl.axis("off")
pl.title("Frobenius reg.")
pl.subplot(2, 2, 3)
pl.imshow(Ge[:nvisu, :])
pl.axis("off")
pl.title("Entropic reg.")
pl.subplot(2, 2, 4)
pl.imshow(Gel2[:nvisu, :])
pl.axis("off")
pl.title("Entropic + Frobenius reg.")
|
