1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
|
# -*- coding: utf-8 -*-
"""
Dimension reduction with optimal transport
"""
import autograd.numpy as np
from pymanopt.manifolds import Stiefel
from pymanopt import Problem
from pymanopt.solvers import SteepestDescent, TrustRegions
def dist(x1,x2):
""" Compute squared euclidean distance between samples
"""
x1p2=np.sum(np.square(x1),1)
x2p2=np.sum(np.square(x2),1)
return x1p2.reshape((-1,1))+x2p2.reshape((1,-1))-2*np.dot(x1,x2.T)
def sinkhorn(w1,w2,M,reg,k):
"""
Simple solver for Sinkhorn algorithm with fixed number of iteration
"""
K=np.exp(-M/reg)
ui=np.ones((M.shape[0],))
vi=np.ones((M.shape[1],))
for i in range(k):
vi=w2/(np.dot(K.T,ui))
ui=w1/(np.dot(K,vi))
G=ui.reshape((M.shape[0],1))*K*vi.reshape((1,M.shape[1]))
return G
def split_classes(X,y):
"""
split samples in X by classes in y
"""
lstsclass=np.unique(y)
return [X[y==i,:].astype(np.float32) for i in lstsclass]
def wda(X,y,p=2,reg=1,k=10,solver = None,maxiter=100,verbose=0):
"""
Wasserstein Discriminant Analysis [11]_
The function solves the following optimization problem:
.. math::
P = \\text{arg}\min_P \\frac{\\sum_i W(PX^i,PX^i)}{\\sum_{i,j\\neq i} W(PX^i,PX^j)}
where :
- :math:`P` is a linear projection operator in the Stiefel(p,d) manifold
- :math:`W` is entropic regularized Wasserstein distances
- :math:`X^i` are samples in the dataset corresponding to class i
Parameters
----------
X : numpy.ndarray (n,d)
Training samples
y : np.ndarray (n,)
labels for training samples
p : int, optional
size of dimensionnality reduction
reg : float, optional
Regularization term >0 (entropic regularization)
solver : str, optional
None for steepest decsent or 'TrustRegions' for trust regions algorithm
else shoudl be a pymanopt.sovers
verbose : int, optional
Print information along iterations
Returns
-------
P : (d x p) ndarray
Optimal transportation matrix for the given parameters
proj : fun
projectiuon function including mean centering
References
----------
.. [11] Flamary, R., Cuturi, M., Courty, N., & Rakotomamonjy, A. (2016). Wasserstein Discriminant Analysis. arXiv preprint arXiv:1608.08063.
"""
mx=np.mean(X)
X-=mx.reshape((1,-1))
# data split between classes
d=X.shape[1]
xc=split_classes(X,y)
# compute uniform weighs
wc=[np.ones((x.shape[0]),dtype=np.float32)/x.shape[0] for x in xc]
def cost(P):
# wda loss
loss_b=0
loss_w=0
for i,xi in enumerate(xc):
xi=np.dot(xi,P)
for j,xj in enumerate(xc[i:]):
xj=np.dot(xj,P)
M=dist(xi,xj)
G=sinkhorn(wc[i],wc[j+i],M,reg,k)
if j==0:
loss_w+=np.sum(G*M)
else:
loss_b+=np.sum(G*M)
# loss inversed because minimization
return loss_w/loss_b
# declare manifold and problem
manifold = Stiefel(d, p)
problem = Problem(manifold=manifold, cost=cost)
# declare solver and solve
if solver is None:
solver= SteepestDescent(maxiter=maxiter,logverbosity=verbose)
elif solver in ['tr','TrustRegions']:
solver= TrustRegions(maxiter=maxiter,logverbosity=verbose)
Popt = solver.solve(problem)
def proj(X):
return (X-mx.reshape((1,-1))).dot(Popt)
return Popt, proj
|